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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(*i $Id: PermutSetoid.v 8823 2006-05-16 16:17:43Z letouzey $ i*)
Require Import Omega.
Require Import Relations.
Require Import List.
Require Import Multiset.
Require Import Permutation.
Require Import SetoidList.
Set Implicit Arguments.
(** This file contains additional results about permutations
with respect to an setoid equality (i.e. an equivalence relation).
*)
Section Perm.
Variable A : Set.
Variable eqA : relation A.
Hypothesis eqA_dec : forall x y:A, {eqA x y} + {~ eqA x y}.
Notation permutation := (permutation _ eqA_dec).
Notation list_contents := (list_contents _ eqA_dec).
(** The following lemmas need some knowledge on [eqA] *)
Variable eqA_refl : forall x, eqA x x.
Variable eqA_sym : forall x y, eqA x y -> eqA y x.
Variable eqA_trans : forall x y z, eqA x y -> eqA y z -> eqA x z.
(** we can use [multiplicity] to define [InA] and [NoDupA]. *)
Lemma multiplicity_InA :
forall l a, InA eqA a l <-> 0 < multiplicity (list_contents l) a.
Proof.
induction l.
simpl.
split; inversion 1.
simpl.
split; intros.
inversion_clear H.
destruct (eqA_dec a a0) as [_|H1]; auto with arith.
destruct H1; auto.
destruct (eqA_dec a a0); auto with arith.
simpl; rewrite <- IHl; auto.
destruct (eqA_dec a a0) as [H0|H0]; auto.
simpl in H.
constructor 2; rewrite IHl; auto.
Qed.
Lemma multiplicity_InA_O :
forall l a, ~ InA eqA a l -> multiplicity (list_contents l) a = 0.
Proof.
intros l a; rewrite multiplicity_InA;
destruct (multiplicity (list_contents l) a); auto with arith.
destruct 1; auto with arith.
Qed.
Lemma multiplicity_InA_S :
forall l a, InA eqA a l -> multiplicity (list_contents l) a >= 1.
Proof.
intros l a; rewrite multiplicity_InA; auto with arith.
Qed.
Lemma multiplicity_NoDupA : forall l,
NoDupA eqA l <-> (forall a, multiplicity (list_contents l) a <= 1).
Proof.
induction l.
simpl.
split; auto with arith.
split; simpl.
inversion_clear 1.
rewrite IHl in H1.
intros; destruct (eqA_dec a a0) as [H2|H2]; simpl; auto.
rewrite multiplicity_InA_O; auto.
swap H0.
apply InA_eqA with a0; auto.
intros; constructor.
rewrite multiplicity_InA.
generalize (H a).
destruct (eqA_dec a a) as [H0|H0].
destruct (multiplicity (list_contents l) a); auto with arith.
simpl; inversion 1.
inversion H3.
destruct H0; auto.
rewrite IHl; intros.
generalize (H a0); auto with arith.
destruct (eqA_dec a a0); simpl; auto with arith.
Qed.
(** Permutation is compatible with InA. *)
Lemma permut_InA_InA :
forall l1 l2 e, permutation l1 l2 -> InA eqA e l1 -> InA eqA e l2.
Proof.
intros l1 l2 e.
do 2 rewrite multiplicity_InA.
unfold Permutation.permutation, meq.
intros H;rewrite H; auto.
Qed.
Lemma permut_cons_InA :
forall l1 l2 e, permutation (e :: l1) l2 -> InA eqA e l2.
Proof.
intros; apply (permut_InA_InA (e:=e) H); auto.
Qed.
(** Permutation of an empty list. *)
Lemma permut_nil :
forall l, permutation l nil -> l = nil.
Proof.
intro l; destruct l as [ | e l ]; trivial.
assert (InA eqA e (e::l)) by auto.
intro Abs; generalize (permut_InA_InA Abs H).
inversion 1.
Qed.
(** Permutation for short lists. *)
Lemma permut_length_1:
forall a b, permutation (a :: nil) (b :: nil) -> eqA a b.
Proof.
intros a b; unfold Permutation.permutation, meq; intro P;
generalize (P b); clear P; simpl.
destruct (eqA_dec b b) as [H|H]; [ | destruct H; auto].
destruct (eqA_dec a b); simpl; auto; intros; discriminate.
Qed.
Lemma permut_length_2 :
forall a1 b1 a2 b2, permutation (a1 :: b1 :: nil) (a2 :: b2 :: nil) ->
(eqA a1 a2) /\ (eqA b1 b2) \/ (eqA a1 b2) /\ (eqA a2 b1).
Proof.
intros a1 b1 a2 b2 P.
assert (H:=permut_cons_InA P).
inversion_clear H.
left; split; auto.
apply permut_length_1.
red; red; intros.
generalize (P a); clear P; simpl.
destruct (eqA_dec a1 a) as [H2|H2];
destruct (eqA_dec a2 a) as [H3|H3]; auto.
destruct H3; apply eqA_trans with a1; auto.
destruct H2; apply eqA_trans with a2; auto.
right.
inversion_clear H0; [|inversion H].
split; auto.
apply permut_length_1.
red; red; intros.
generalize (P a); clear P; simpl.
destruct (eqA_dec a1 a) as [H2|H2];
destruct (eqA_dec b2 a) as [H3|H3]; auto.
simpl; rewrite <- plus_n_Sm; inversion 1; auto.
destruct H3; apply eqA_trans with a1; auto.
destruct H2; apply eqA_trans with b2; auto.
Qed.
(** Permutation is compatible with length. *)
Lemma permut_length :
forall l1 l2, permutation l1 l2 -> length l1 = length l2.
Proof.
induction l1; intros l2 H.
rewrite (permut_nil (permut_sym H)); auto.
assert (H0:=permut_cons_InA H).
destruct (InA_split H0) as (h2,(b,(t2,(H1,H2)))).
subst l2.
rewrite app_length.
simpl; rewrite <- plus_n_Sm; f_equal.
rewrite <- app_length.
apply IHl1.
apply permut_remove_hd with b.
apply permut_tran with (a::l1); auto.
revert H1; unfold Permutation.permutation, meq; simpl.
intros; f_equal; auto.
destruct (eqA_dec b a0) as [H2|H2];
destruct (eqA_dec a a0) as [H3|H3]; auto.
destruct H3; apply eqA_trans with b; auto.
destruct H2; apply eqA_trans with a; auto.
Qed.
Lemma NoDupA_eqlistA_permut :
forall l l', NoDupA eqA l -> NoDupA eqA l' ->
eqlistA eqA l l' -> permutation l l'.
Proof.
intros.
red; unfold meq; intros.
rewrite multiplicity_NoDupA in H, H0.
generalize (H a) (H0 a) (H1 a); clear H H0 H1.
do 2 rewrite multiplicity_InA.
destruct 3; omega.
Qed.
Variable B : Set.
Variable eqB : B->B->Prop.
Variable eqB_dec : forall x y:B, { eqB x y }+{ ~eqB x y }.
Variable eqB_trans : forall x y z, eqB x y -> eqB y z -> eqB x z.
(** Permutation is compatible with map. *)
Lemma permut_map :
forall f,
(forall x y, eqA x y -> eqB (f x) (f y)) ->
forall l1 l2, permutation l1 l2 ->
Permutation.permutation _ eqB_dec (map f l1) (map f l2).
Proof.
intros f; induction l1.
intros l2 P; rewrite (permut_nil (permut_sym P)); apply permut_refl.
intros l2 P.
simpl.
assert (H0:=permut_cons_InA P).
destruct (InA_split H0) as (h2,(b,(t2,(H1,H2)))).
subst l2.
rewrite map_app.
simpl.
apply permut_tran with (f b :: map f l1).
revert H1; unfold Permutation.permutation, meq; simpl.
intros; f_equal; auto.
destruct (eqB_dec (f b) a0) as [H2|H2];
destruct (eqB_dec (f a) a0) as [H3|H3]; auto.
destruct H3; apply eqB_trans with (f b); auto.
destruct H2; apply eqB_trans with (f a); auto.
apply permut_add_cons_inside.
rewrite <- map_app.
apply IHl1; auto.
apply permut_remove_hd with b.
apply permut_tran with (a::l1); auto.
revert H1; unfold Permutation.permutation, meq; simpl.
intros; f_equal; auto.
destruct (eqA_dec b a0) as [H2|H2];
destruct (eqA_dec a a0) as [H3|H3]; auto.
destruct H3; apply eqA_trans with b; auto.
destruct H2; apply eqA_trans with a; auto.
Qed.
End Perm.
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