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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        *)
-(************************************************************************)
-
-(* Certification of Imperative Programs / Jean-Christophe Filliātre *)
-
-(* $Id: ArrayPermut.v,v 1.1.2.1 2004/07/16 19:30:16 herbelin Exp $ *)
-
-(****************************************************************************)
-(*                    Permutations of elements in arrays                    *)
-(*                        Definition and properties                         *)
-(****************************************************************************)
-
-Require ProgInt.
-Require Arrays.
-Require Export Exchange.
-
-Require Omega.
-
-Set Implicit Arguments.
-
-(* We define "permut" as the smallest equivalence relation which contains
- * transpositions i.e. exchange of two elements.
- *)
-
-Inductive permut [n:Z; A:Set] : (array n A)->(array n A)->Prop :=
-    exchange_is_permut : 
-      (t,t':(array n A))(i,j:Z)(exchange t t' i j) -> (permut t t')
-  | permut_refl : 
-      (t:(array n A))(permut t t)
-  | permut_sym : 
-      (t,t':(array n A))(permut t t') -> (permut t' t)
-  | permut_trans : 
-      (t,t',t'':(array n A))
-      (permut t t') -> (permut t' t'') -> (permut t t'').
-
-Hints Resolve exchange_is_permut permut_refl permut_sym permut_trans : v62 datatypes.
-
-(* We also define the permutation on a segment of an array, "sub_permut",
- * the other parts of the array being unchanged
- *
- * One again we define it as the smallest equivalence relation containing
- * transpositions on the given segment.
- *)
-
-Inductive sub_permut [n:Z; A:Set; g,d:Z] : (array n A)->(array n A)->Prop :=
-    exchange_is_sub_permut : 
-      (t,t':(array n A))(i,j:Z)`g <= i <= d` -> `g <= j <= d`
-      -> (exchange t t' i j) -> (sub_permut g d t t')
-  | sub_permut_refl : 
-      (t:(array n A))(sub_permut g d t t)
-  | sub_permut_sym : 
-      (t,t':(array n A))(sub_permut g d t t') -> (sub_permut g d t' t)
-  | sub_permut_trans : 
-      (t,t',t'':(array n A))
-      (sub_permut g d t t') -> (sub_permut g d t' t'') 
-      -> (sub_permut g d t t'').
-
-Hints Resolve exchange_is_sub_permut sub_permut_refl sub_permut_sym sub_permut_trans
-  : v62 datatypes.
-
-(* To express that some parts of arrays are equal we introduce the
- * property "array_id" which says that a segment is the same on two
- * arrays.
- *)
-
-Definition array_id := [n:Z][A:Set][t,t':(array n A)][g,d:Z]
-  (i:Z) `g <= i <= d` -> #t[i] = #t'[i].
-
-(* array_id is an equivalence relation *)
-
-Lemma array_id_refl : 
-  (n:Z)(A:Set)(t:(array n A))(g,d:Z)
-  (array_id t t g d).
-Proof.
-Unfold array_id.
-Auto with datatypes.
-Save.
-
-Hints Resolve array_id_refl : v62 datatypes.
-
-Lemma array_id_sym :
-  (n:Z)(A:Set)(t,t':(array n A))(g,d:Z)
-  (array_id t t' g d)
-  -> (array_id t' t g d).
-Proof.
-Unfold array_id. Intros.
-Symmetry; Auto with datatypes.
-Save.
-
-Hints Resolve  array_id_sym : v62 datatypes.
-
-Lemma array_id_trans :
-  (n:Z)(A:Set)(t,t',t'':(array n A))(g,d:Z)
-  (array_id t t' g d)
-  -> (array_id t' t'' g d)
-    -> (array_id t t'' g d).
-Proof.
-Unfold array_id. Intros.
-Apply trans_eq with y:=#t'[i]; Auto with datatypes.
-Save.
-
-Hints Resolve array_id_trans: v62 datatypes.
-
-(* Outside the segment [g,d] the elements are equal *)
-
-Lemma sub_permut_id :
-  (n:Z)(A:Set)(t,t':(array n A))(g,d:Z)
-  (sub_permut g d t t') ->
-  (array_id t t' `0` `g-1`) /\ (array_id t t' `d+1` `n-1`).
-Proof.
-Intros n A t t' g d. Induction 1; Intros.
-Elim H2; Intros.
-Unfold array_id; Split; Intros.
-Apply H7; Omega.
-Apply H7; Omega.
-Auto with datatypes.
-Decompose [and] H1; Auto with datatypes.
-Decompose [and] H1; Decompose [and] H3; EAuto with datatypes.
-Save.
-
-Hints Resolve sub_permut_id.
-
-Lemma sub_permut_eq :
-  (n:Z)(A:Set)(t,t':(array n A))(g,d:Z)
-  (sub_permut g d t t') ->
-  (i:Z) (`0<=i<g` \/ `d<i<n`) -> #t[i]=#t'[i].
-Proof.
-Intros n A t t' g d Htt' i Hi.
-Elim (sub_permut_id Htt'). Unfold array_id. 
-Intros.
-Elim Hi; [ Intro; Apply H; Omega | Intro; Apply H0; Omega ].
-Save.
-
-(* sub_permut is a particular case of permutation *)
-
-Lemma sub_permut_is_permut :
-  (n:Z)(A:Set)(t,t':(array n A))(g,d:Z)
-  (sub_permut g d t t') ->
-  (permut t t').
-Proof.
-Intros n A t t' g d. Induction 1; Intros; EAuto with datatypes.
-Save.
-
-Hints Resolve sub_permut_is_permut.
-
-(* If we have a sub-permutation on an empty segment, then we have a 
- * sub-permutation on any segment.
- *)
-
-Lemma sub_permut_void :
-  (N:Z)(A:Set)(t,t':(array N A))
-  (g,g',d,d':Z) `d < g`
-   -> (sub_permut g d t t') -> (sub_permut g' d' t t').
-Proof.
-Intros N A t t' g g' d d' Hdg.
-(Induction 1; Intros).
-(Absurd `g <= d`; Omega).
-Auto with datatypes.
-Auto with datatypes.
-EAuto with datatypes.
-Save.
-
-(* A sub-permutation on a segment may be extended to any segment that
- * contains the first one.
- *)
-
-Lemma sub_permut_extension :
-  (N:Z)(A:Set)(t,t':(array N A))
-  (g,g',d,d':Z) `g' <= g` -> `d <= d'`
-   -> (sub_permut g d t t') -> (sub_permut g' d' t t').
-Proof.
-Intros N A t t' g g' d d' Hgg' Hdd'.
-(Induction 1; Intros).
-Apply exchange_is_sub_permut with i:=i j:=j; [ Omega | Omega | Assumption ].
-Auto with datatypes.
-Auto with datatypes.
-EAuto with datatypes.
-Save.