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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: Euclid.v,v 1.1.2.1 2004/07/16 19:31:24 herbelin Exp $ i*)
+
+Require Mult.
+Require Compare_dec.
+Require Wf_nat.
+
+V7only [Import nat_scope.].
+Open Local Scope nat_scope.
+
+Implicit Variables Type a,b,n,q,r:nat.
+
+Inductive diveucl [a,b:nat] : Set 
+      := divex : (q,r:nat)(gt b r)->(a=(plus (mult q b) r))->(diveucl a b).
+
+
+Lemma eucl_dev : (b:nat)(gt b O)->(a:nat)(diveucl a b).
+Intros b H a; Pattern a; Apply gt_wf_rec; Intros n H0.
+Elim (le_gt_dec b n).
+Intro lebn.
+Elim (H0 (minus n b)); Auto with arith.
+Intros q r g e.
+Apply divex with (S q) r; Simpl; Auto with arith.
+Elim plus_assoc_l.
+Elim e; Auto with arith.
+Intros gtbn.
+Apply divex with O n; Simpl; Auto with arith.
+Qed.
+
+Lemma quotient : (b:nat)(gt b O)->
+     (a:nat){q:nat|(EX r:nat | (a=(plus (mult q b) r))/\(gt b r))}.
+Intros b H a; Pattern a; Apply gt_wf_rec; Intros n H0.
+Elim (le_gt_dec b n).
+Intro lebn.
+Elim (H0 (minus n b)); Auto with arith.
+Intros q Hq; Exists (S q).
+Elim Hq; Intros r Hr.
+Exists r; Simpl; Elim Hr; Intros.
+Elim plus_assoc_l.
+Elim H1; Auto with arith.
+Intros gtbn.
+Exists O; Exists n; Simpl; Auto with arith.
+Qed.
+
+Lemma modulo : (b:nat)(gt b O)->
+     (a:nat){r:nat|(EX q:nat | (a=(plus (mult q b) r))/\(gt b r))}.
+Intros b H a; Pattern a; Apply gt_wf_rec; Intros n H0.
+Elim (le_gt_dec b n).
+Intro lebn.
+Elim (H0 (minus n b)); Auto with arith.
+Intros r Hr; Exists r.
+Elim Hr; Intros q Hq.
+Elim Hq; Intros; Exists (S q); Simpl.
+Elim plus_assoc_l.
+Elim H1; Auto with arith.
+Intros gtbn.
+Exists n; Exists O; Simpl; Auto with arith.
+Qed.