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(************************************************************************)
(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
Require Import Omega Relations Multiset SetoidList.
(** This file is deprecated, use [Permutation.v] instead.
Indeed, this file defines a notion of permutation based on
multisets (there exists a permutation between two lists iff every
elements have the same multiplicity in the two lists) which
requires a more complex apparatus (the equipment of the domain
with a decidable equality) than [Permutation] in [Permutation.v].
The relation between the two relations are in lemma
[permutation_Permutation].
File [Permutation] concerns Leibniz equality : it shows in particular
that [List.Permutation] and [permutation] are equivalent in this context.
*)
Set Implicit Arguments.
Local Notation "[ ]" := nil.
Local Notation "[ a ; .. ; b ]" := (a :: .. (b :: []) ..).
Section Permut.
(** * From lists to multisets *)
Variable A : Type.
Variable eqA : relation A.
Hypothesis eqA_equiv : Equivalence eqA.
Hypothesis eqA_dec : forall x y:A, {eqA x y} + {~ eqA x y}.
Let emptyBag := EmptyBag A.
Let singletonBag := SingletonBag _ eqA_dec.
(** contents of a list *)
Fixpoint list_contents (l:list A) : multiset A :=
match l with
| [] => emptyBag
| a :: l => munion (singletonBag a) (list_contents l)
end.
Lemma list_contents_app :
forall l m:list A,
meq (list_contents (l ++ m)) (munion (list_contents l) (list_contents m)).
Proof.
simple induction l; simpl; auto with datatypes.
intros.
apply meq_trans with
(munion (singletonBag a) (munion (list_contents l0) (list_contents m)));
auto with datatypes.
Qed.
(** * [permutation]: definition and basic properties *)
Definition permutation (l m:list A) := meq (list_contents l) (list_contents m).
Lemma permut_refl : forall l:list A, permutation l l.
Proof.
unfold permutation; auto with datatypes.
Qed.
Lemma permut_sym :
forall l1 l2 : list A, permutation l1 l2 -> permutation l2 l1.
Proof.
unfold permutation, meq; intros; symmetry; trivial.
Qed.
Lemma permut_trans :
forall l m n:list A, permutation l m -> permutation m n -> permutation l n.
Proof.
unfold permutation; intros.
apply meq_trans with (list_contents m); auto with datatypes.
Qed.
Lemma permut_cons_eq :
forall l m:list A,
permutation l m -> forall a a', eqA a a' -> permutation (a :: l) (a' :: m).
Proof.
unfold permutation; simpl; intros.
apply meq_trans with (munion (singletonBag a') (list_contents l)).
apply meq_left, meq_singleton; auto.
auto with datatypes.
Qed.
Lemma permut_cons :
forall l m:list A,
permutation l m -> forall a:A, permutation (a :: l) (a :: m).
Proof.
unfold permutation; simpl; auto with datatypes.
Qed.
Lemma permut_app :
forall l l' m m':list A,
permutation l l' -> permutation m m' -> permutation (l ++ m) (l' ++ m').
Proof.
unfold permutation; intros.
apply meq_trans with (munion (list_contents l) (list_contents m));
auto using permut_cons, list_contents_app with datatypes.
apply meq_trans with (munion (list_contents l') (list_contents m'));
auto using permut_cons, list_contents_app with datatypes.
apply meq_trans with (munion (list_contents l') (list_contents m));
auto using permut_cons, list_contents_app with datatypes.
Qed.
Lemma permut_add_inside_eq :
forall a a' l1 l2 l3 l4, eqA a a' ->
permutation (l1 ++ l2) (l3 ++ l4) ->
permutation (l1 ++ a :: l2) (l3 ++ a' :: l4).
Proof.
unfold permutation, meq in *; intros.
specialize H0 with a0.
repeat rewrite list_contents_app in *; simpl in *.
destruct (eqA_dec a a0) as [Ha|Ha]; rewrite H in Ha;
decide (eqA_dec a' a0) with Ha; simpl; auto with arith.
do 2 rewrite <- plus_n_Sm; f_equal; auto.
Qed.
Lemma permut_add_inside :
forall a l1 l2 l3 l4,
permutation (l1 ++ l2) (l3 ++ l4) ->
permutation (l1 ++ a :: l2) (l3 ++ a :: l4).
Proof.
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_eq :
forall a a' l l1 l2, eqA a a' ->
permutation l (l1 ++ l2) ->
permutation (a :: l) (l1 ++ a' :: l2).
Proof.
intros;
replace (a :: l) with ([] ++ a :: l); trivial;
apply permut_add_inside_eq; trivial.
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 ([] ++ a :: l); trivial;
apply permut_add_inside; trivial.
Qed.
Lemma permut_middle :
forall (l m:list A) (a:A), permutation (a :: l ++ m) (l ++ a :: m).
Proof.
intros; apply permut_add_cons_inside; auto using permut_sym, permut_refl.
Qed.
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; trivial using permut_refl.
simpl.
apply permut_add_cons_inside.
rewrite app_nil_r. trivial.
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.
(** we can use [multiplicity] to define [InA] and [NoDupA]. *)
Fact if_eqA_then : forall a a' (B:Type)(b b':B),
eqA a a' -> (if eqA_dec a a' then b else b') = b.
Proof.
intros. destruct eqA_dec as [_|NEQ]; auto.
contradict NEQ; auto.
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_eq :
forall l l1 l2 a b, eqA a b ->
permutation (a :: l) (l1 ++ b :: l2) -> permutation l (l1 ++ l2).
Proof.
unfold permutation, meq; simpl; intros l l1 l2 a b Heq H a0.
specialize H with a0.
rewrite list_contents_app in *; simpl in *.
apply plus_reg_l with (if eqA_dec a a0 then 1 else 0).
rewrite H; clear H.
symmetry; rewrite plus_comm, <- ! plus_assoc; f_equal.
rewrite plus_comm.
destruct (eqA_dec a a0) as [Ha|Ha]; rewrite Heq in Ha;
decide (eqA_dec b a0) with Ha; reflexivity.
Qed.
Lemma permut_remove_hd :
forall l l1 l2 a,
permutation (a :: l) (l1 ++ a :: l2) -> permutation l (l1 ++ l2).
Proof.
eauto using permut_remove_hd_eq, Equivalence_Reflexive.
Qed.
Fact if_eqA_else : forall a a' (B:Type)(b b':B),
~eqA a a' -> (if eqA_dec a a' then b else b') = b'.
Proof.
intros. decide (eqA_dec a a') with H; auto.
Qed.
Fact if_eqA_refl : forall a (B:Type)(b b':B),
(if eqA_dec a a then b else b') = b.
Proof.
intros; apply (decide_left (eqA_dec a a)); auto with *.
Qed.
(** PL: Inutilisable dans un rewrite sans un change prealable. *)
Global Instance if_eqA (B:Type)(b b':B) :
Proper (eqA==>eqA==>@eq _) (fun x y => if eqA_dec x y then b else b').
Proof.
intros x x' Hxx' y y' Hyy'.
intros; destruct (eqA_dec x y) as [H|H];
destruct (eqA_dec x' y') as [H'|H']; auto.
contradict H'; transitivity x; auto with *; transitivity y; auto with *.
contradict H; transitivity x'; auto with *; transitivity y'; auto with *.
Qed.
Fact if_eqA_rewrite_l : forall a1 a1' a2 (B:Type)(b b':B),
eqA a1 a1' -> (if eqA_dec a1 a2 then b else b') =
(if eqA_dec a1' a2 then b else b').
Proof.
intros; destruct (eqA_dec a1 a2) as [A1|A1];
destruct (eqA_dec a1' a2) as [A1'|A1']; auto.
contradict A1'; transitivity a1; eauto with *.
contradict A1; transitivity a1'; eauto with *.
Qed.
Fact if_eqA_rewrite_r : forall a1 a2 a2' (B:Type)(b b':B),
eqA a2 a2' -> (if eqA_dec a1 a2 then b else b') =
(if eqA_dec a1 a2' then b else b').
Proof.
intros; destruct (eqA_dec a1 a2) as [A2|A2];
destruct (eqA_dec a1 a2') as [A2'|A2']; auto.
contradict A2'; transitivity a2; eauto with *.
contradict A2; transitivity a2'; eauto with *.
Qed.
Global Instance multiplicity_eqA (l:list A) :
Proper (eqA==>@eq _) (multiplicity (list_contents l)).
Proof.
intros x x' Hxx'.
induction l as [|y l Hl]; simpl; auto.
rewrite (@if_eqA_rewrite_r y x x'); auto.
Qed.
Lemma multiplicity_InA :
forall l a, InA eqA a l <-> 0 < multiplicity (list_contents l) a.
Proof.
induction l.
simpl.
split; inversion 1.
simpl.
intros a'; split; intros H. inversion_clear H.
apply (decide_left (eqA_dec a a')); auto with *.
destruct (eqA_dec a a'); auto with *. simpl; rewrite <- IHl; auto.
destruct (eqA_dec a a'); auto with *. right. 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 [EQ|NEQ]; simpl; auto with *.
rewrite <- EQ.
rewrite multiplicity_InA_O; auto.
intros; constructor.
rewrite multiplicity_InA.
specialize (H a).
rewrite if_eqA_refl in H.
clear IHl; omega.
rewrite IHl; intros.
specialize (H a0). omega.
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, 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 with *.
Qed.
(** Permutation of an empty list. *)
Lemma permut_nil :
forall l, permutation l [] -> l = [].
Proof.
intro l; destruct l as [ | e l ]; trivial.
assert (InA eqA e (e::l)) by (auto with *).
intro Abs; generalize (permut_InA_InA Abs H).
inversion 1.
Qed.
(** Permutation for short lists. *)
Lemma permut_length_1:
forall a b, permutation [a] [b] -> eqA a b.
Proof.
intros a b; unfold permutation, meq.
intro P; specialize (P b); simpl in *.
rewrite if_eqA_refl in *.
destruct (eqA_dec a b); simpl; auto; discriminate.
Qed.
Lemma permut_length_2 :
forall a1 b1 a2 b2, permutation [a1; b1] [a2; b2] ->
(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.
specialize (P a). simpl in *.
rewrite (@if_eqA_rewrite_l a1 a2 a) in P by auto. omega.
right.
inversion_clear H0; [|inversion H].
split; auto.
apply permut_length_1.
red; red; intros.
specialize (P a); simpl in *.
rewrite (@if_eqA_rewrite_l a1 b2 a) in P by auto. omega.
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_trans with (a::l1); auto.
revert H1; unfold permutation, meq; simpl.
intros; f_equal; auto.
rewrite (@if_eqA_rewrite_l a b a0); auto.
Qed.
Lemma NoDupA_equivlistA_permut :
forall l l', NoDupA eqA l -> NoDupA eqA l' ->
equivlistA 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.
End Permut.
Section Permut_map.
Variables A B : Type.
Variable eqA : relation A.
Hypothesis eqA_dec : forall x y:A, {eqA x y} + {~ eqA x y}.
Hypothesis eqA_equiv : Equivalence eqA.
Variable eqB : B->B->Prop.
Hypothesis eqB_dec : forall x y:B, { eqB x y }+{ ~eqB x y }.
Hypothesis eqB_trans : Transitive eqB.
(** Permutation is compatible with map. *)
Lemma permut_map :
forall f,
(Proper (eqA==>eqB) f) ->
forall l1 l2, permutation _ eqA_dec l1 l2 ->
permutation _ eqB_dec (map f l1) (map f l2).
Proof.
intros f; induction l1.
intros l2 P; rewrite (permut_nil eqA_equiv (permut_sym P)); apply permut_refl.
intros l2 P.
simpl.
assert (H0:=permut_cons_InA eqA_equiv P).
destruct (InA_split H0) as (h2,(b,(t2,(H1,H2)))).
subst l2.
rewrite map_app.
simpl.
apply permut_trans with (f b :: map f l1).
revert H1; unfold 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; transitivity (f b); auto with *.
destruct H2; transitivity (f a); auto with *.
apply permut_add_cons_inside.
rewrite <- map_app.
apply IHl1; auto.
apply permut_remove_hd with b; trivial.
apply permut_trans with (a::l1); auto.
revert H1; unfold permutation, meq; simpl.
intros; f_equal; auto.
rewrite (@if_eqA_rewrite_l _ _ eqA_equiv eqA_dec a b a0); auto.
Qed.
End Permut_map.
Require Import Permutation.
Section Permut_permut.
Variable A : Type.
Variable eqA : relation A.
Hypothesis eqA_dec : forall x y:A, {eqA x y} + {~ eqA x y}.
Hypothesis eqA_equiv : Equivalence eqA.
Lemma Permutation_impl_permutation : forall l l',
Permutation l l' -> permutation _ eqA_dec l l'.
Proof.
induction 1.
apply permut_refl.
apply permut_cons; auto using Equivalence_Reflexive.
change (x :: y :: l) with ([x] ++ y :: l);
apply permut_add_cons_inside; simpl;
apply permut_cons_eq; auto using Equivalence_Reflexive, permut_refl.
apply permut_trans with l'; trivial.
Qed.
Lemma permut_eqA : forall l l', Forall2 eqA l l' -> permutation _ eqA_dec l l'.
Proof.
induction 1.
apply permut_refl.
apply permut_cons_eq; trivial.
Qed.
Lemma permutation_Permutation : forall l l',
permutation _ eqA_dec l l' <->
exists l'', Permutation l l'' /\ Forall2 eqA l'' l'.
Proof.
split; intro H.
(* -> *)
induction l in l', H |- *.
exists []; apply permut_sym, permut_nil in H as ->; auto using Forall2.
pose proof H as H'.
apply permut_cons_InA, InA_split in H
as (l1 & y & l2 & Heq & ->); trivial.
apply permut_remove_hd_eq, IHl in H'
as (l'' & IHP & IHA); clear IHl; trivial.
apply Forall2_app_inv_r in IHA as (l1'' & l2'' & Hl1 & Hl2 & ->).
exists (l1'' ++ a :: l2''); split.
apply Permutation_cons_app; trivial.
apply Forall2_app, Forall2_cons; trivial.
(* <- *)
destruct H as (l'' & H & Heq).
apply permut_trans with l''.
apply Permutation_impl_permutation; trivial.
apply permut_eqA; trivial.
Qed.
End Permut_permut.
(* begin hide *)
(** For compatibility *)
Notation permut_right := permut_cons (only parsing).
Notation permut_tran := permut_trans (only parsing).
(* end hide *)
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