Imported Upstream version 8.0pl3+8.1betaupstream/8.0pl3+8.1beta

3 files changed, 601 insertions, 31 deletions

diff --git a/theories/Sorting/PermutEq.v b/theories/Sorting/PermutEq.v
new file mode 100644
index 00000000..e56ff27d
--- /dev/null
+++ b/theories/Sorting/PermutEq.v
@@ -0,0 +1,241 @@
+(************************************************************************)
+(*  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: PermutEq.v 8853 2006-05-23 18:17:38Z herbelin $ i*)
+
+Require Import Omega.
+Require Import Relations.
+Require Import Setoid.
+Require Import List.
+Require Import Multiset.
+Require Import Permutation. 
+
+Set Implicit Arguments.
+
+(** This file is similar to [PermutSetoid], except that the equality used here
+      is Coq usual one instead of a setoid equality. In particular, we can then 
+      prove the equivalence between [List.Permutation] and 
+      [Permutation.permutation].
+*)
+
+Section Perm.
+
+Variable A : Set.
+Hypothesis eq_dec : forall x y:A, {x=y} + {~ x=y}.
+
+Notation permutation := (permutation _ eq_dec).
+Notation list_contents := (list_contents _ eq_dec).
+
+(** we can use [multiplicity] to define [In] and [NoDup]. *)
+
+Lemma multiplicity_In : 
+  forall l a, In a l <-> 0 < multiplicity (list_contents l) a.
+Proof.
+induction l.
+simpl.
+split; inversion 1.
+simpl.
+split; intros.
+inversion_clear H.
+subst a0.
+destruct (eq_dec a a) as [_|H]; auto with arith; destruct H; auto.
+destruct (eq_dec a a0) as [H1|H1]; auto with arith; simpl.
+rewrite <- IHl; auto.
+destruct (eq_dec a a0); auto.
+simpl in H.
+right; rewrite IHl; auto.
+Qed.
+
+Lemma multiplicity_In_O :
+  forall l a, ~ In a l -> multiplicity (list_contents l) a = 0.
+Proof.
+intros l a; rewrite multiplicity_In; 
+ destruct (multiplicity (list_contents l) a); auto.
+destruct 1; auto with arith.
+Qed.
+
+Lemma multiplicity_In_S :
+ forall l a, In a l -> multiplicity (list_contents l) a >= 1.
+Proof.
+intros l a; rewrite multiplicity_In; auto.
+Qed.
+
+Lemma multiplicity_NoDup : 
+  forall l, NoDup l <-> (forall a, multiplicity (list_contents l) a <= 1).
+Proof.
+induction l.
+simpl.
+split; auto with arith.
+intros; apply NoDup_nil.
+split; simpl.
+inversion_clear 1.
+rewrite IHl in H1.
+intros; destruct (eq_dec a a0) as [H2|H2]; simpl; auto.
+subst a0.
+rewrite multiplicity_In_O; auto.
+intros; constructor.
+rewrite multiplicity_In.
+generalize (H a).
+destruct (eq_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 (eq_dec a a0); simpl; auto with arith.
+Qed.
+
+Lemma NoDup_permut : 
+  forall l l', NoDup l -> NoDup l' -> 
+     (forall x, In x l <-> In x l') -> permutation l l'.
+Proof.
+intros.
+red; unfold meq; intros.
+rewrite multiplicity_NoDup in H, H0. 
+generalize (H a) (H0 a) (H1 a); clear H H0 H1.
+do 2 rewrite multiplicity_In.
+destruct 3; omega.
+Qed.
+
+(** Permutation is compatible with In. *)
+Lemma permut_In_In :
+  forall l1 l2 e, permutation l1 l2 -> In e l1 -> In e l2.
+Proof.
+unfold Permutation.permutation, meq; intros l1 l2 e P IN.
+generalize (P e); clear P.
+destruct (In_dec eq_dec e l2) as [H|H]; auto.
+rewrite (multiplicity_In_O _ _ H).
+intros.
+generalize (multiplicity_In_S _ _ IN).
+rewrite H0.
+inversion 1.
+Qed.
+
+Lemma permut_cons_In :
+  forall l1 l2 e, permutation (e :: l1) l2 -> In e l2.
+Proof.
+intros; eapply permut_In_In; eauto.
+red; 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 (In e (e::l)) by (red; auto).
+intro Abs; generalize (permut_In_In _ Abs H).
+inversion 1.
+Qed.
+
+(** When used with [eq], this permutation notion is equivalent to 
+  the one defined in [List.v]. *)
+
+Lemma permutation_Permutation : 
+ forall l l', Permutation l l' <-> permutation l l'.
+Proof.
+split.
+induction 1.
+apply permut_refl.
+apply permut_cons; auto.
+change (permutation (y::x::l) ((x::nil)++y::l)).
+apply permut_add_cons_inside; simpl; apply permut_refl.
+apply permut_tran with l'; auto.
+revert l'.
+induction l.
+intros.
+rewrite (permut_nil (permut_sym H)).
+apply Permutation_refl.
+intros.
+destruct (In_split _ _ (permut_cons_In H)) as (h2,(t2,H1)).
+subst l'.
+apply Permutation_cons_app.
+apply IHl.
+apply permut_remove_hd with a; auto.
+Qed.
+
+(** Permutation for short lists. *)
+
+Lemma permut_length_1:
+ forall a b, permutation (a :: nil) (b :: nil)  -> a=b.
+Proof.
+intros a b; unfold Permutation.permutation, meq; intro P;
+generalize (P b); clear P; simpl.
+destruct (eq_dec b b) as [H|H]; [ | destruct H; auto].
+destruct (eq_dec a b); simpl; auto; intros; discriminate.
+Qed.
+
+Lemma permut_length_2 :
+ forall a1 b1 a2 b2, permutation (a1 :: b1 :: nil) (a2 :: b2 :: nil) ->
+ (a1=a2) /\ (b1=b2) \/ (a1=b2) /\ (a2=b1).
+Proof.
+intros a1 b1 a2 b2 P.
+assert (H:=permut_cons_In P).
+inversion_clear H.
+left; split; auto.
+apply permut_length_1.
+red; red; intros.
+generalize (P a); clear P; simpl.
+destruct (eq_dec a1 a) as [H2|H2]; 
+ destruct (eq_dec a2 a) as [H3|H3]; auto.
+destruct H3; transitivity a1; auto.
+destruct H2; transitivity a2; auto.
+right.
+inversion_clear H0; [|inversion H].
+split; auto.
+apply permut_length_1.
+red; red; intros.
+generalize (P a); clear P; simpl.
+destruct (eq_dec a1 a) as [H2|H2]; 
+ destruct (eq_dec b2 a) as [H3|H3]; auto.
+simpl; rewrite <- plus_n_Sm; inversion 1; auto.
+destruct H3; transitivity a1; auto.
+destruct H2; transitivity 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.
+destruct (In_split _ _ (permut_cons_In H)) as (h2,(t2,H1)).
+subst l2.
+rewrite app_length.
+simpl; rewrite <- plus_n_Sm; f_equal.
+rewrite <- app_length.
+apply IHl1.
+apply permut_remove_hd with a; auto.
+Qed.
+
+Variable B : Set.
+Variable eqB_dec : forall x y:B, { x=y }+{ ~x=y }. 
+
+(** Permutation is compatible with map. *)
+
+Lemma permutation_map :
+  forall f 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.
+destruct (In_split _ _ (permut_cons_In P)) as (h2,(t2,H1)).
+subst l2.
+rewrite map_app.
+simpl.
+apply permut_add_cons_inside.
+rewrite <- map_app.
+apply IHl1; auto.
+apply permut_remove_hd with a; auto.
+Qed.
+
+End Perm.
+
diff --git a/theories/Sorting/PermutSetoid.v b/theories/Sorting/PermutSetoid.v
new file mode 100644
index 00000000..46ea088f
--- /dev/null
+++ b/theories/Sorting/PermutSetoid.v
@@ -0,0 +1,243 @@
+(************************************************************************)
+(*  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.
diff --git a/theories/Sorting/Permutation.v b/theories/Sorting/Permutation.v
index b3287cd1..0f2e02b5 100644
--- a/theories/Sorting/Permutation.v
+++ b/theories/Sorting/Permutation.v
@@ -6,30 +6,39 @@
 (*         *       GNU Lesser General Public License Version 2.1        *)
 (************************************************************************)
 
-(*i $Id: Permutation.v 5920 2004-07-16 20:01:26Z herbelin $ i*)
+(*i $Id: Permutation.v 8823 2006-05-16 16:17:43Z letouzey $ i*)
 
 Require Import Relations.
 Require Import List.
 Require Import Multiset.
+Require Import Arith.
+
+(** This file define a notion of permutation for lists, based on multisets: 
+     there exists a permutation between two lists iff every elements have 
+     the same multiplicities in the two lists. 
+
+     Unlike [List.Permutation], the present notion of permutation requires
+     a decidable equality. At the same time, this definition can be used 
+     with a non-standard equality, whereas [List.Permutation] cannot.
+      
+     The present file contains basic results, obtained without any particular
+     assumption on the decidable equality used.
+	
+     File [PermutSetoid] contains additional results about permutations 
+     with respect to an setoid equality (i.e. an equivalence relation). 
+
+     Finally, file [PermutEq] concerns Coq equality : this file is similar 
+     to the previous one, but proves in addition that [List.Permutation]
+     and [permutation] are equivalent in this context.
+*)
 
 Set Implicit Arguments.
 
 Section defs.
 
 Variable A : Set.
-Variable leA : relation A.
 Variable eqA : relation A.
-
-Let gtA (x y:A) := ~ leA x y.
-
-Hypothesis leA_dec : forall x y:A, {leA x y} + {~ leA x y}.
 Hypothesis eqA_dec : forall x y:A, {eqA x y} + {~ eqA x y}.
-Hypothesis leA_refl : forall x y:A, eqA x y -> leA x y.
-Hypothesis leA_trans : forall x y z:A, leA x y -> leA y z -> leA x z.
-Hypothesis leA_antisym : forall x y:A, leA x y -> leA y x -> eqA x y.
-
-Hint Resolve leA_refl: default.
-Hint Immediate eqA_dec leA_dec leA_antisym: default.
 
 Let emptyBag := EmptyBag A.
 Let singletonBag := SingletonBag _ eqA_dec.
@@ -63,6 +72,12 @@ unfold permutation in |- *; auto with datatypes.
 Qed.
 Hint Resolve permut_refl.
 
+Lemma permut_sym :
+ forall l1 l2 : list A, permutation l1 l2 -> permutation l2 l1.
+Proof.
+unfold permutation, meq; intros; apply sym_eq; trivial.
+Qed.
+
 Lemma permut_tran :
  forall l m n:list A, permutation l m -> permutation m n -> permutation l n.
 Proof.
@@ -70,51 +85,122 @@ unfold permutation in |- *; intros.
 apply meq_trans with (list_contents m); auto with datatypes.
 Qed.
 
-Lemma permut_right :
+Lemma permut_cons :
  forall l m:list A,
    permutation l m -> forall a:A, permutation (a :: l) (a :: m).
 Proof.
 unfold permutation in |- *; simpl in |- *; auto with datatypes.
 Qed.
-Hint Resolve permut_right.
+Hint Resolve permut_cons.
 
 Lemma permut_app :
  forall l l' m m':list A,
    permutation l l' -> permutation m m' -> permutation (l ++ m) (l' ++ m').
 Proof.
 unfold permutation in |- *; intros.
-apply meq_trans with (munion (list_contents l) (list_contents m));
+apply meq_trans with (munion (list_contents l) (list_contents m)); 
  auto with datatypes.
-apply meq_trans with (munion (list_contents l') (list_contents m'));
+apply meq_trans with (munion (list_contents l') (list_contents m')); 
  auto with datatypes.
 apply meq_trans with (munion (list_contents l') (list_contents m));
  auto with datatypes.
 Qed.
 Hint Resolve permut_app.
 
-Lemma permut_cons :
- forall l m:list A,
-   permutation l m -> forall a:A, permutation (a :: l) (a :: m).
+Lemma permut_add_inside :
+ forall a l1 l2 l3 l4, 
+  permutation (l1 ++ l2) (l3 ++ l4) ->
+  permutation (l1 ++ a :: l2) (l3 ++ a :: l4).
 Proof.
-intros l m H a.
-change (permutation ((a :: nil) ++ l) ((a :: nil) ++ m)) in |- *.
-apply permut_app; auto with datatypes.
+unfold permutation, meq in *; intros.
+generalize (H a0); clear H.
+do 4 rewrite list_contents_app.
+simpl.
+destruct (eqA_dec a a0); simpl; auto with arith.
+do 2 rewrite <- plus_n_Sm; f_equal; auto.
+Qed.
+
+Lemma permut_add_cons_inside :
+ forall a l l1 l2, 
+  permutation l (l1 ++ l2) ->
+  permutation (a :: l) (l1 ++ a :: l2).
+Proof.
+intros;
+replace (a :: l) with (nil ++ a :: l); trivial;
+apply permut_add_inside; trivial.
 Qed.
-Hint Resolve permut_cons.
 
 Lemma permut_middle :
  forall (l m:list A) (a:A), permutation (a :: l ++ m) (l ++ a :: m).
 Proof.
-unfold permutation in |- *.
-simple induction l; simpl in |- *; auto with datatypes.
-intros.
-apply meq_trans with
- (munion (singletonBag a)
-    (munion (singletonBag a0) (list_contents (l0 ++ m))));
- auto with datatypes.
-apply munion_perm_left; auto with datatypes.
+intros; apply permut_add_cons_inside; auto.
 Qed.
 Hint Resolve permut_middle.
 
+Lemma permut_sym_app :
+ forall l1 l2, permutation (l1 ++ l2) (l2 ++ l1).
+Proof.
+intros l1 l2;
+unfold permutation, meq; 
+intro a; do 2 rewrite list_contents_app; simpl; 
+auto with arith.
+Qed.
+
+Lemma permut_rev : 
+ forall l, permutation l (rev l).
+Proof.
+induction l.
+simpl; auto.
+simpl.
+apply permut_add_cons_inside.
+rewrite <- app_nil_end; auto.
+Qed.
+
+(** Some inversion results. *)
+Lemma permut_conv_inv :
+  forall e l1 l2, permutation (e :: l1) (e :: l2) -> permutation l1 l2.
+Proof.
+intros e l1 l2; unfold permutation, meq; simpl; intros H a;
+generalize (H a); apply plus_reg_l.
+Qed.
+
+Lemma permut_app_inv1 : 
+  forall l l1 l2, permutation (l1 ++ l) (l2 ++ l) -> permutation l1 l2.
+Proof.
+intros l l1 l2; unfold permutation, meq; simpl;
+intros H a; generalize (H a); clear H.
+do 2 rewrite list_contents_app.
+simpl.
+intros; apply plus_reg_l with (multiplicity (list_contents l) a).
+rewrite plus_comm; rewrite H; rewrite plus_comm.
+trivial.
+Qed.
+
+Lemma permut_app_inv2 : 
+  forall l l1 l2, permutation (l ++ l1) (l ++ l2) -> permutation l1 l2.
+Proof.
+intros l l1 l2; unfold permutation, meq; simpl;
+intros H a; generalize (H a); clear H.
+do 2 rewrite list_contents_app.
+simpl.
+intros; apply plus_reg_l with (multiplicity (list_contents l) a).
+trivial.
+Qed.
+
+Lemma permut_remove_hd :
+  forall l l1 l2 a,   
+  permutation (a :: l) (l1 ++ a :: l2) -> permutation l (l1 ++ l2).
+Proof.
+intros l l1 l2 a; unfold permutation, meq; simpl; intros H a0; generalize (H a0); clear H.
+do 2 rewrite list_contents_app; simpl; intro H.
+apply plus_reg_l with (if eqA_dec a a0 then 1 else 0).
+rewrite H; clear H.
+symmetry; rewrite plus_comm.
+repeat rewrite <- plus_assoc; f_equal.
+apply plus_comm.
+Qed.
+
 End defs.
+(* For compatibilty *) 
+Notation permut_right := permut_cons.
 Unset Implicit Arguments.