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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: Permutation.v 9245 2006-10-17 12:53:34Z notin $ 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.
x*)
Set Implicit Arguments.
Section defs.
(** * From lists to multisets *)
Variable A : Set.
Variable eqA : relation A.
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
| nil => 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 in |- *; 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 in |- *; auto with datatypes.
Qed.
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.
unfold permutation in |- *; intros.
apply meq_trans with (list_contents m); 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 in |- *; simpl in |- *; 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 in |- *; 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 :
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 :
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.
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_end. 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.
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.
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