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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        *)
-(************************************************************************)
-
-(* $Id: Zwf.v,v 1.1.2.1 2004/07/16 19:31:44 herbelin Exp $ *)
-
-Require ZArith_base.
-Require Export Wf_nat.
-Require Omega.
-V7only [Import Z_scope.].
-Open Local Scope Z_scope.
-
-(** Well-founded relations on Z. *)
-
-(** We define the following family of relations on [Z x Z]: 
-
-    [x (Zwf c) y]   iff   [x < y & c <= y]
- *)
-
-Definition Zwf := [c:Z][x,y:Z] `c <= y` /\ `x < y`.
-
-(** and we prove that [(Zwf c)] is well founded *)
-
-Section wf_proof.
-
-Variable c : Z.
-
-(** The proof of well-foundness is classic: we do the proof by induction
-    on a measure in nat, which is here [|x-c|] *)
-
-Local f := [z:Z](absolu (Zminus z c)).
-
-Lemma Zwf_well_founded : (well_founded Z (Zwf c)).
-Red; Intros.
-Assert (n:nat)(a:Z)(lt (f a) n)\/(`a<c`) -> (Acc Z (Zwf c) a).
-Clear a; Induction n; Intros.
-(** n= 0 *)
-Case H; Intros.
-Case (lt_n_O (f a)); Auto.
-Apply Acc_intro; Unfold Zwf; Intros.
-Assert False;Omega Orelse Contradiction.
-(** inductive case *)
-Case H0; Clear H0; Intro; Auto.
-Apply Acc_intro; Intros.
-Apply H.
-Unfold Zwf in H1.
-Case (Zle_or_lt c y); Intro; Auto with zarith.
-Left.
-Red in H0.
-Apply lt_le_trans with (f a); Auto with arith.
-Unfold f.
-Apply absolu_lt; Omega.
-Apply (H (S (f a))); Auto.
-Save.
-
-End wf_proof.
-
-Hints Resolve Zwf_well_founded : datatypes v62.
-
-
-(** We also define the other family of relations:
-
-    [x (Zwf_up c) y]   iff   [y < x <= c]
- *)
-
-Definition Zwf_up := [c:Z][x,y:Z] `y < x <= c`.
-
-(** and we prove that [(Zwf_up c)] is well founded *)
-
-Section wf_proof_up.
-
-Variable c : Z.
-
-(** The proof of well-foundness is classic: we do the proof by induction
-    on a measure in nat, which is here [|c-x|] *)
-
-Local f := [z:Z](absolu (Zminus c z)).
-
-Lemma Zwf_up_well_founded : (well_founded Z (Zwf_up c)).
-Proof.
-Apply well_founded_lt_compat with f:=f.
-Unfold Zwf_up f.
-Intros.
-Apply absolu_lt.
-Unfold Zminus. Split.
-Apply Zle_left; Intuition.
-Apply Zlt_reg_l; Unfold Zlt; Rewrite <- Zcompare_Zopp; Intuition.
-Save.
-
-End wf_proof_up.
-
-Hints Resolve Zwf_up_well_founded : datatypes v62.