From 6b649aba925b6f7462da07599fe67ebb12a3460e Mon Sep 17 00:00:00 2001
From: Samuel Mimram <samuel.mimram@ens-lyon.org>
Date: Wed, 28 Jul 2004 21:54:47 +0000
Subject: Imported Upstream version 8.0pl1

---
 theories7/Sorting/Permutation.v | 111 ++++++++++++++++++++++++++++++++++++++++
 1 file changed, 111 insertions(+)
 create mode 100644 theories7/Sorting/Permutation.v

(limited to 'theories7/Sorting/Permutation.v')

diff --git a/theories7/Sorting/Permutation.v b/theories7/Sorting/Permutation.v
new file mode 100644
index 00000000..46b8da00
--- /dev/null
+++ b/theories7/Sorting/Permutation.v
@@ -0,0 +1,111 @@
+(************************************************************************)
+(*  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,v 1.1.2.1 2004/07/16 19:31:41 herbelin Exp $ i*)
+
+Require Relations.
+Require PolyList.
+Require Multiset.
+
+Set Implicit Arguments.
+
+Section defs.
+
+Variable A : Set.
+Variable leA : (relation A).
+Variable eqA : (relation A).
+
+Local gtA := [x,y:A]~(leA x y).
+
+Hypothesis leA_dec : (x,y:A){(leA x y)}+{~(leA x y)}.
+Hypothesis eqA_dec : (x,y:A){(eqA x y)}+{~(eqA x y)}.
+Hypothesis leA_refl : (x,y:A) (eqA x y) -> (leA x y).
+Hypothesis leA_trans : (x,y,z:A) (leA x y) -> (leA y z) -> (leA x z).
+Hypothesis leA_antisym : (x,y:A)(leA x y) -> (leA y x) -> (eqA x y).
+
+Hints Resolve leA_refl : default.
+Hints Immediate eqA_dec leA_dec leA_antisym : default.
+
+Local emptyBag := (EmptyBag A).
+Local singletonBag := (SingletonBag eqA_dec).
+
+(** contents of a list *)
+
+Fixpoint list_contents [l:(list A)] : (multiset A) :=
+ Cases l of
+   nil        => emptyBag
+ | (cons a l) => (munion (singletonBag a) (list_contents l))
+ end.
+
+Lemma list_contents_app : (l,m:(list A))
+  (meq (list_contents (app l m)) (munion (list_contents l) (list_contents m))).
+Proof.
+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.
+Hints Resolve list_contents_app.
+
+Definition permutation := [l,m:(list A)](meq (list_contents l) (list_contents m)).
+
+Lemma permut_refl : (l:(list A))(permutation l l).
+Proof.
+Unfold permutation; Auto with datatypes.
+Qed.
+Hints Resolve permut_refl.
+
+Lemma permut_tran : (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_right : (l,m:(list A))
+   (permutation l m) -> (a:A)(permutation (cons a l) (cons a m)).
+Proof.
+Unfold permutation; Simpl; Auto with datatypes.
+Qed.
+Hints Resolve permut_right.
+
+Lemma permut_app : (l,l',m,m':(list A))
+   (permutation l l') -> (permutation m m') ->
+   (permutation (app l m) (app l' m')).
+Proof.
+Unfold permutation; Intros.
+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.
+Apply meq_trans with (munion (list_contents l') (list_contents m)); Auto with datatypes.
+Qed.
+Hints Resolve permut_app.
+
+Lemma permut_cons : (l,m:(list A))(permutation l m) ->
+    (a:A)(permutation (cons a l) (cons a m)).
+Proof.
+Intros l m H a.
+Change (permutation (app (cons a (nil A)) l) (app (cons a (nil A)) m)).
+Apply permut_app; Auto with datatypes.
+Qed.
+Hints Resolve permut_cons.
+
+Lemma permut_middle : (l,m:(list A))
+   (a:A)(permutation (cons a (app l m)) (app l (cons a m))).
+Proof.
+Unfold permutation.
+Induction l; Simpl; Auto with datatypes.
+Intros.
+Apply meq_trans with (munion (singletonBag a)
+          (munion (singletonBag a0) (list_contents (app l0 m)))); Auto with datatypes.
+Apply munion_perm_left; Auto with datatypes.
+Qed.
+Hints Resolve permut_middle.
+
+End defs.
+Unset Implicit Arguments.
+
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