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prelim.v
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prelim.v
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(** * Preliminaries to lBsystems. To be moved to upstream files later.
by Vladimir Voevodsky, file created on Jan. 6, 2015 *)
Unset Automatic Introduction.
Require Export Foundations.hlevel2.finitesets.
(* To upsetream files *)
Notation isaproptotal2 := ( isofhleveltotal2 1 ) .
Lemma gth0_to_geh1 { n : nat } ( gt0 : n > 0 ) : n >= 1 .
Proof.
intros.
induction n as [ | n IHn ] .
assert ( absd : empty ) .
apply ( negnatgthnn _ gt0 ) .
destruct absd .
apply ( natgehn0 n ) .
Defined.
Lemma geh1_to_gth0 { n : nat } ( geh1 : n >= 1 ) : n > 0 .
Proof.
intros .
apply ( natgehgthtrans _ _ _ geh1 ( natgthsnn 0 ) ) .
Defined.
Lemma natminusind ( m n : nat ) : m - S n = ( m - 1 ) - n .
Proof . intros .
induction m as [ | m IHm ] . apply idpath . simpl . rewrite ( natminuseqn m ) .
apply idpath .
Defined.
Lemma natgthnnmius1 { n : nat } ( gt : n > 0 ) : n > n - 1 .
Proof.
intros . induction n as [ | n ] .
induction ( negnatgthnn _ gt ) .
change ( S n > n - 0 ) .
rewrite natminuseqn .
exact ( natgthsnn _ ) .
Defined.
Lemma natminusnn ( n : nat ) : n - n = 0 .
Proof.
intro . induction n as [ | n IHn ] .
exact ( idpath _ ) .
exact IHn .
Defined.
Lemma natminusmequalsn { m n : nat } ( ge : n >= m ) ( eq0 : n - m = 0 ) : n = m .
Proof .
intro m . induction m as [ | m IHm ] .
intros .
rewrite natminuseqn in eq0 .
exact eq0 .
intros .
induction n as [ | n ] .
assert ( absd : empty ) .
exact ( ge ( natgthsn0 _ ) ) .
destruct absd .
apply ( maponpaths S ) . exact ( IHm n ge eq0 ) .
Defined.
Lemma natmiusmius1mminus1 { n m : nat } ( gt1 : n > 0 ) ( gt2 : m > 0 ) :
( n - 1 ) - ( m - 1 ) = n - m .
Proof .
intro n . induction n as [ | n IHn ] .
intros .
destruct ( negnatgthnn _ gt1 ) .
intros .
simpl . rewrite natminuseqn . induction m as [ | m ] .
destruct ( negnatgthnn _ gt2 ) .
simpl . rewrite natminuseqn . apply idpath .
Defined.
Lemma natgthtominus1geh { m n : nat } : m > n -> m - 1 >= n .
Proof.
intro m . induction m as [ | m IHm ] .
intros n gt . induction ( natgehn0 _ gt ) .
intros n gt .
change ( m - 0 >= n ) . rewrite ( natminuseqn m ) . apply natgthsntogeh . exact gt .
Defined .
Lemma natgthminus1togeh { n m : nat } : m > n - 1 -> m >= n .
Proof.
intro n . induction n as [ | n IHn ] .
intros m gt . exact ( natgehn0 _ ) .
intros m gt .
change ( S n - 1 ) with ( n - 0 ) in gt . rewrite ( natminuseqn n ) in gt .
apply natgthtogehsn . exact gt .
Defined .
Lemma nat1plusminusgt { n m : nat } ( gt : 1 + m > n ) : ( 1 + m ) - n = 1 + ( m - n ) .
Proof.
intros .
apply ( natplusrcan _ _ n ) . rewrite ( natplusassoc _ _ n ) .
rewrite ( minusplusnmm _ n ( natgthtogeh _ _ gt ) ) .
rewrite ( minusplusnmm _ n ) .
apply idpath .
apply ( natgthsntogeh _ _ gt ) .
Defined.
Lemma lB_2014_12_07_l1 { m n : nat } ( gt : m > n ) : m - n = 1 + (( m - 1 ) - n ) .
Proof.
intros. induction m as [ | m IHm ] . induction ( natgehn0 _ gt ) .
clear IHm. change ( S m - n = S ( m - 0 - n ) ) . rewrite ( natminuseqn m ) .
exact ( nat1plusminusgt gt ) .
Defined.
Lemma natminusmmk { m k : nat } ( ge : m >= k ) : m - ( m - k ) = k .
Proof.
intros .
apply ( natplusrcan _ _ ( m - k ) ) .
rewrite minusplusnmm .
rewrite natpluscomm .
rewrite minusplusnmm .
apply idpath .
apply ge .
apply natminuslehn.
Defined.
Lemma natleftplustorightminus ( n m k : nat ) : n + m = k -> n = k - m .
Proof.
intros n m k e .
assert ( ge : k >= m ) . rewrite <- e . exact ( natgehplusnmm _ _ ) .
apply ( natplusrcan _ _ m ) . rewrite ( minusplusnmm _ _ ge ) .
exact e .
Defined.
Definition natassocpmeq ( n m k : nat ) ( ge : m >= k ) : ( n + m ) - k = n + ( m - k ) .
Proof.
intros. apply ( natplusrcan _ _ k ) . rewrite ( natplusassoc n _ k ) .
rewrite ( minusplusnmm _ k ge ) .
set ( ge' := istransnatgeh _ _ _ ( natgehplusnmm n m ) ge ) .
rewrite ( minusplusnmm _ k ge' ) . apply idpath.
Defined.
Definition natassocmpeq ( n m k : nat ) ( isnm : n >= m ) ( ismk : m >= k ) :
( n - m ) + k = n - ( m - k ) .
Proof. intros. apply ( natplusrcan _ _ ( m - k ) ) .
assert ( is' : natleh ( m - k ) n ) .
apply ( istransnatleh _ _ _ (natminuslehn _ _ ) isnm ) .
rewrite ( minusplusnmm _ _ is' ) . rewrite (natplusassoc _ k _ ) .
rewrite ( natpluscomm k _ ) . rewrite ( minusplusnmm _ _ ismk ) .
rewrite ( minusplusnmm _ _ isnm ) . apply idpath.
Defined.
Definition natminusassoc ( n m k : nat ) : ( n - m ) - k = n - ( m + k ) .
Proof. intros n m . revert n .
induction m as [ | m IHm ] . intros . simpl . rewrite ( natminuseqn n ) . apply idpath .
intros . rewrite ( natminusind n m ) . rewrite ( IHm ( n - 1 ) k ) .
rewrite ( ! ( natminusind _ _ ) ) . apply idpath .
Defined.
Definition natminuscomm ( n m k : nat ) : ( n - m ) - k = ( n - k ) - m .
Proof .
intros .
rewrite natminusassoc .
rewrite natminusassoc .
rewrite natpluscomm .
apply idpath .
Defined.
Definition natminusinter { n m k : nat } ( ge1 : n >= m ) ( ge2 : m >= k ) :
n - k = ( n - m ) + ( m - k ) .
Proof.
intros .
assert ( int1 : n - m + (m - k) = n - ( m - ( m - k ))). refine ( natassocmpeq _ _ _ _ _ ) .
exact ge1 .
exact ( natminuslehn _ _ ) .
assert ( int2 : m - ( m - k ) = k ) . exact ( natminusmmk ge2 ) .
rewrite int2 in int1 . exact ( ! int1 ) .
Defined.
(** [ nateq ] *)
Notation nateqandplusrinv := natplusrcan .
Notation nateqandpluslinv := natpluslcan .
Definition nateqandplusr ( n m k : nat ) : n = m -> n + k = m + k :=
maponpaths ( fun x => x + k ) .
Definition nateqandplusl ( n m k : nat ) : n = m -> k + n = k + m :=
maponpaths ( fun x => k + x ) .
(** **** Cancellation properties of minus on nat *)
Lemma natminusrcan { n m k : nat } ( ge1 : n >= k ) ( ge2 : m >= k ) ( is : n - k = m - k ) :
n = m .
Proof .
intros .
assert ( is' := nateqandplusr _ _ k is ) .
rewrite ( minusplusnmm _ _ ge1 ) in is' .
rewrite ( minusplusnmm _ _ ge2 ) in is' .
exact is' .
Defined.
(* **** Greater and minus *)
Definition natgthrightminus ( n m k : nat ) ( ge : k >= m ) : n + m > k -> n > k - m .
Proof . intros n m k ge gt . apply ( natgthandplusrinv _ _ m ) .
rewrite ( minusplusnmm _ _ ge ) .
exact gt .
Defined.
Definition natgthrightplus ( n m k : nat ) : n - m > k -> n > k + m .
Proof .
intros n m k gt . assert ( ge : n >= m ) . apply natgthtogeh .
apply minusgth0inv . apply ( natgthgehtrans _ _ _ gt ( natgehn0 k ) ) .
assert ( gt' : ( n - m ) + m > k + m ) . apply ( natgthandplusr _ _ _ gt ) .
rewrite ( minusplusnmm _ _ ge ) in gt' . exact gt' .
Defined.
Definition natgthleftminus ( n m k : nat ) : n > m + k -> n - k > m .
Proof.
intros n m k gt . apply ( natgthandplusrinv _ _ k ) .
assert ( ge' : n >= k ) . apply natgthtogeh .
exact ( natgthgehtrans _ _ _ gt ( natgehplusnmm _ _ ) ) .
rewrite ( minusplusnmm _ _ ge' ) .
exact gt .
Defined.
Definition natgthleftplus ( n m k : nat ) : n > m - k -> n + k > m .
Proof .
intros n m k gt .
assert ( gt' : n + k > m - k + k ) . exact ( natgthandplusr _ _ k gt ) .
exact ( natgthgehtrans _ _ _ gt' ( minusplusnmmineq _ _ ) ) .
Defined .
Definition natgthleftminusminus ( n m k : nat ) : n - m > k -> n - k > m .
Proof .
intros n m k gt . assert ( gt' : n > k + m ) . exact ( natgthrightplus _ _ _ gt ) .
rewrite ( natpluscomm _ _ ) in gt' . exact ( natgthleftminus _ _ _ gt' ) .
Defined.
(* **** Greater or equal and minus *)
Definition natgehrightminus ( n m k : nat ) ( ge : k >= m ) : n + m >= k -> n >= k - m .
Proof.
intros n m k ge ge' . apply ( natgehandplusrinv _ _ m ) . rewrite ( minusplusnmm _ _ ge ) .
exact ge' .
Defined.
Definition natgehrightplus ( n m k : nat ) ( ge : n >= m ) : n - m >= k -> n >= k + m .
Proof.
intros n m k ge ge' . rewrite ( ! minusplusnmm _ _ ge ) . apply ( natgehandplusr _ _ _ ) .
exact ge' .
Defined.
Definition natgehleftminus ( n m k : nat ) : n >= m + k -> n - k >= m .
Proof.
intros n m k ge . apply ( natgehandplusrinv _ _ k ) .
assert ( ge' : n >= k ) . exact ( istransnatgeh _ _ _ ge ( natgehplusnmm _ _ ) ) .
rewrite ( minusplusnmm _ _ ge' ) .
exact ge .
Defined.
Definition natgehleftplus ( n m k : nat ) : n >= m - k -> n + k >= m .
Proof.
intros n m k ge .
assert ( ge' : n + k >= m - k + k ) . exact ( natgehandplusr _ _ k ge ) .
exact ( istransnatgeh _ _ _ ge' ( minusplusnmmineq _ _ ) ) .
Defined .
Definition natgehleftminusminus ( n m k : nat ) ( ge : n >= m ) : n - m >= k -> n - k >= m .
Proof .
intros n m k ge ge' . assert ( ge'' : n >= k + m ) . exact ( natgehrightplus _ _ _ ge ge' ) .
rewrite ( natpluscomm _ _ ) in ge'' . exact ( natgehleftminus _ _ _ ge'' ) .
Defined.
(* Two-sided minus and greater *)
Definition natgthandminusinvr { n m k : nat } ( is : n > m ) ( is' : m >= k ) :
n - k > m - k .
Proof .
intro n. induction n as [ | n IHn ] .
intros . destruct ( negnatgth0n _ is ) .
intro m . induction k as [ | k ] . intros .
repeat rewrite natminuseqn .
exact is .
intros . induction m as [ | m ] .
destruct ( negnatgeh0sn _ is' ) .
exact ( IHn m k is is' ) .
Defined.
(* Two-sided minus and greater or equal *)
Definition natgehandminusl ( n m k : nat ) ( ge : n >= m ) : k - m >= k - n .
Proof.
intro n. induction n as [ | n IHn ] .
intros . rewrite ( nat0gehtois0 _ ge ) . apply isreflnatleh .
intro m . induction m as [ | m ] . intros . induction k as [ | k ] .
apply natminuslehn .
apply natminuslehn .
intro k . induction k as [ | k ] . intro is .
apply isreflnatleh .
intro is . apply ( IHn m k ) . apply is .
Defined.
Definition natgthandminuslinv { n m k : nat } ( gt : n - k > n - m ) : m > k .
Proof .
intros .
apply ( negnatlehtogth _ _ ) .
intro ge .
assert ( ge' : n - m >= n - k ) .
exact ( natgehandminusl _ _ _ ge ) .
exact ( ge' gt ) .
Defined.
(* Decrement function on nat *)
Definition dec ( n : nat ) : nat :=
match n with
O => O |
S n' => n'
end .
(* End of the file lBsystems_prelim.v *)