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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 1.12.2.1 2004/07/16 19:31:00 herbelin Exp $ i*)
+
+(** Equality is decidable on [nat] *)
+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 Types m n p q : nat.
+
+Require Import Arith.
+Require Import Peano_dec.
+Require Import Compare_dec.
+
+Definition le_or_le_S := le_le_S_dec.
+
+Definition Pcompare := gt_eq_gt_dec.
+
+Lemma le_dec : forall n m, {n <= m} + {m <= n}.
+Proof le_ge_dec.
+
+Definition lt_or_eq n m := {m > n} + {n = m}.
+
+Lemma le_decide : forall n m, n <= m -> lt_or_eq n m.
+Proof le_lt_eq_dec.
+
+Lemma le_le_S_eq : forall n m, n <= m -> S n <= m \/ n = m.
+Proof le_lt_or_eq.
+
+(* By special request of G. Kahn - Used in Group Theory *)
+Lemma discrete_nat :
+ forall n m, n < m -> S n = m \/ (exists r : nat, m = S (S (n + 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.
+induction 1; auto with arith.
+right; exists (n - S (S m)); simpl in |- *.
+rewrite (plus_comm m (n - S (S m))).
+rewrite (plus_n_Sm (n - S (S m)) m).
+rewrite (plus_n_Sm (n - S (S m)) (S m)).
+rewrite (plus_comm (n - S (S m)) (S (S m))); auto with arith.
+Qed.
+
+Require Export Wf_nat.
+
+Require Export Min.