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Proofs.v
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Proofs.v
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Require Import QuickSort.
Require Import Prelude.
Require Import GHC.DeferredFix.
Require Import Coq.Lists.List.
Import ListNotations.
Inductive AlreadySorted {a} `{Ord a} : list a -> Prop :=
| ASNil: AlreadySorted []
| ASCons: forall x xs,
AlreadySorted xs ->
Forall (fun y => (op_zl__ y x) = false) xs ->
AlreadySorted (x::xs).
Axiom unroll_deferred_fix: forall a r `{Default r} (f : (a -> r) -> (a -> r)),
deferredFix1 f = f (deferredFix1 f).
Lemma Forall_partition:
forall a p (xs : list a),
Forall (fun y => (p y) = false) xs ->
OldList.partition p xs = ([], xs).
Proof.
intros.
induction H.
* reflexivity.
* unfold OldList.partition in *.
simpl in *. rewrite IHForall.
unfold OldList.select.
rewrite H. reflexivity.
Qed.
Theorem quicksort_already_sorted:
forall a `(Ord a) (xs : list a),
AlreadySorted xs -> quicksort xs = xs.
Proof.
intros.
induction H1.
* unfold quicksort.
rewrite unroll_deferred_fix. reflexivity.
* change (quicksort (x :: xs) = [] ++ [x] ++ xs).
unfold quicksort.
rewrite unroll_deferred_fix.
rewrite Forall_partition by assumption.
f_equal;[|f_equal].
+ rewrite unroll_deferred_fix. reflexivity.
+ apply IHAlreadySorted.
Qed.
Require Import Coq.Sorting.Permutation.
Lemma partition_permutation:
forall a p (xs ys zs : list a),
OldList.partition p xs = (ys, zs) ->
Permutation xs (ys ++ zs).
Proof.
induction xs; intros.
* simpl in *. inversion_clear H. constructor.
* simpl in *. unfold OldList.select in *.
destruct (p a0).
- destruct (OldList.partition p xs) eqn:?.
inversion H.
apply Permutation_cons; [reflexivity|].
apply IHxs.
congruence.
- destruct (OldList.partition p xs) eqn:?.
inversion H.
apply Permutation_cons_app.
apply IHxs.
congruence.
Qed.
Lemma Forall_partition_l:
forall a p (xs ys zs : list a),
OldList.partition p xs = (ys, zs) ->
Forall (fun y => (p y) = true) ys.
Proof.
induction xs; intros.
* simpl in H. inversion H. auto.
* simpl in *. unfold OldList.select in *.
destruct (p a0) eqn:?.
- destruct (OldList.partition p xs) eqn:?.
inversion H. simpl.
constructor; try assumption.
apply (IHxs l zs).
congruence.
- destruct (OldList.partition p xs) eqn:?.
inversion H. simpl.
destruct zs. try congruence.
apply (IHxs ys zs).
congruence.
Qed.
Lemma Forall_partition_r:
forall a p (xs ys zs : list a),
OldList.partition p xs = (ys, zs) ->
Forall (fun y => (p y) = false) zs.
Proof.
induction xs; intros.
* simpl in H. inversion H. auto.
* simpl in *. unfold OldList.select in *.
destruct (p a0) eqn:?.
- destruct (OldList.partition p xs) eqn:?.
inversion H. simpl.
destruct ys. try congruence.
apply (IHxs ys zs).
congruence.
- destruct (OldList.partition p xs) eqn:?.
inversion H. simpl.
constructor; try assumption.
apply (IHxs ys l0).
congruence.
Qed.
Lemma partition_length_l:
forall a p (xs ys zs : list a),
OldList.partition p xs = (ys, zs) ->
Peano.le (length ys) (length xs).
Proof.
induction xs; intros.
* simpl in *. inversion_clear H. constructor.
* simpl in *. unfold OldList.select in *.
destruct (p a0).
- destruct (OldList.partition p xs) eqn:?.
inversion H.
simpl.
apply le_n_S.
apply (IHxs l zs).
congruence.
- destruct (OldList.partition p xs) eqn:?.
inversion H.
destruct zs. try congruence.
inversion H2.
simpl.
rewrite (IHxs ys zs).
omega.
congruence.
Qed.
Lemma partition_length_r:
forall a p (xs ys zs : list a),
OldList.partition p xs = (ys, zs) ->
Peano.le (length zs) (length xs).
Proof.
induction xs; intros.
* simpl in *. inversion_clear H. constructor.
* simpl in *. unfold OldList.select in *.
destruct (p a0).
- destruct (OldList.partition p xs) eqn:?.
inversion H.
destruct ys. try congruence.
inversion H2.
simpl.
rewrite (IHxs ys zs).
omega.
congruence.
- destruct (OldList.partition p xs) eqn:?.
inversion H.
simpl.
apply le_n_S.
apply (IHxs ys l0).
congruence.
Qed.
Theorem quicksort_permutation:
forall a `(Ord a) (xs : list a),
Permutation (quicksort xs) xs.
Proof.
intros.
remember (length xs) as n.
generalize dependent xs.
induction n using lt_wf_ind.
intros.
unfold quicksort.
rewrite unroll_deferred_fix.
destruct xs.
* apply perm_nil.
* destruct (OldList.partition (fun arg_1__ : a => _<_ arg_1__ a0) xs) eqn:?.
simpl app.
rewrite <- Permutation_middle.
apply Permutation_cons; [reflexivity|].
rewrite (partition_permutation _ _ _ _ _ Heqp).
apply Permutation_app.
- apply (H1 (length l)).
+ pose (partition_length_l _ _ _ _ _ Heqp).
subst n. simpl.
omega.
+ reflexivity.
- apply (H1 (length l0)).
+ pose (partition_length_r _ _ _ _ _ Heqp).
subst n. simpl.
omega.
+ reflexivity.
Qed.
Require Import Coq.Sorting.Sorted.
Require Import Coq.Sets.Relations_1.
Require Import Coq.Lists.List.
Lemma Forall_app:
forall a P (xs ys : list a),
Forall P xs -> Forall P ys ->
Forall P (xs ++ ys).
Proof.
intros. induction xs; try assumption. constructor.
inversion H; assumption.
apply IHxs. inversion H; assumption.
Qed.
Lemma Forall_Permutation:
forall a P (xs ys : list a),
Forall P ys -> Permutation xs ys ->
Forall P xs.
Proof.
intros.
induction H0.
* auto.
* constructor.
+ inversion_clear H. assumption.
+ apply IHPermutation. inversion_clear H. assumption.
* inversion_clear H. inversion_clear H1.
repeat (constructor; try assumption).
* auto.
Qed.
Section sorted.
Variable a : Type.
Variable eq : Eq_ a.
Variable ord : Ord a.
Definition R x y := x < y = true.
Variable trans : Transitive R.
Variable total : forall a b, R a b \/ R b a.
Lemma StronglySorted_app_cons:
forall (xs ys : list a) p,
StronglySorted R xs ->
StronglySorted R ys ->
Forall (fun x => R x p) xs ->
Forall (fun y => R p y) ys ->
StronglySorted R (xs ++ p :: ys).
Proof.
intros.
induction xs.
* simpl.
apply SSorted_cons.
- assumption.
- assumption.
* simpl. apply SSorted_cons.
- apply IHxs.
+ inversion H; assumption.
+ inversion H1; assumption.
- apply Forall_app.
inversion_clear H; assumption.
constructor.
inversion_clear H1; assumption.
refine (Forall_impl _ _ H2). intros.
refine (trans _ _ _ _ H3).
inversion_clear H1; assumption.
Qed.
Theorem quicksort_sorted:
forall (xs : list a), StronglySorted R (quicksort xs).
Proof.
intros.
remember (length xs) as n.
generalize dependent xs.
induction n using lt_wf_ind.
intros.
unfold quicksort.
rewrite unroll_deferred_fix.
destruct xs.
* apply SSorted_nil.
* destruct (OldList.partition (fun arg_1__ : a => _<_ arg_1__ a0) xs) eqn:?.
simpl app.
apply StronglySorted_app_cons.
- apply (H (length l)).
+ pose (partition_length_l _ _ _ _ _ Heqp).
subst n. simpl.
omega.
+ reflexivity.
- apply (H (length l0)).
+ pose (partition_length_r _ _ _ _ _ Heqp).
subst n. simpl.
omega.
+ reflexivity.
- refine (Forall_Permutation _ _ _ _ _ (quicksort_permutation _ _ _)).
apply (Forall_partition_l _ _ _ _ _ Heqp).
- refine (Forall_Permutation _ _ _ _ _ (quicksort_permutation _ _ _)).
specialize (Forall_partition_r _ _ _ _ _ Heqp).
apply Forall_impl.
intros.
destruct (total a0 a1); [assumption|congruence].
Qed.
End sorted.
Print Assumptions quicksort_sorted.