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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: Compare.v,v 1.1.2.1 2004/07/16 19:31:23 herbelin Exp $ i*)
+
+(** Equality is decidable on [nat] *)
+V7only [Import nat_scope.].
+Open Local Scope nat_scope.
+
+(*
+Lemma not_eq_sym : (A:Set)(p,q:A)(~p=q) -> ~(q=p).
+Proof sym_not_eq.
+Hints Immediate not_eq_sym : arith.
+*)
+Notation not_eq_sym := sym_not_eq.
+
+Implicit Variables Type m,n,p,q:nat.
+
+Require Arith.
+Require Peano_dec.
+Require Compare_dec.
+
+Definition le_or_le_S := le_le_S_dec.
+
+Definition compare := gt_eq_gt_dec.
+
+Lemma le_dec : (n,m:nat) {le n m} + {le m n}.
+Proof le_ge_dec.
+
+Definition lt_or_eq := [n,m:nat]{(gt m n)}+{n=m}.
+
+Lemma le_decide : (n,m:nat)(le n m)->(lt_or_eq n m).
+Proof le_lt_eq_dec.
+
+Lemma le_le_S_eq : (p,q:nat)(le p q)->((le (S p) q)\/(p=q)).
+Proof le_lt_or_eq.
+
+(* By special request of G. Kahn - Used in Group Theory *)
+Lemma discrete_nat : (m, n: nat) (lt m n) ->
+   (S m) = n \/ (EX r: nat | n = (S (S (plus m r)))).
+Proof.
+Intros m n H.
+LApply (lt_le_S m n); Auto with arith.
+Intro H'; LApply (le_lt_or_eq (S m) n); Auto with arith.
+NewInduction 1; Auto with arith.
+Right; Exists (minus n (S (S m))); Simpl.
+Rewrite (plus_sym m (minus n (S (S m)))).
+Rewrite (plus_n_Sm (minus n (S (S m))) m).
+Rewrite (plus_n_Sm (minus n (S (S m))) (S m)).
+Rewrite (plus_sym (minus n (S (S m))) (S (S m))); Auto with arith.
+Qed.
+
+Require Export Wf_nat.
+
+Require Export Min.