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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: Peano_dec.v,v 1.10.2.1 2004/07/16 19:31:00 herbelin Exp $ i*)
+
+Require Import Decidable.
+
+Open Local Scope nat_scope.
+
+Implicit Types m n x y : nat.
+
+Theorem O_or_S : forall n, {m : nat | S m = n} + {0 = n}.
+Proof.
+induction n.
+auto.
+left; exists n; auto.
+Defined.
+
+Theorem eq_nat_dec : forall n m, {n = m} + {n <> m}.
+Proof.
+induction n; induction m; auto.
+elim (IHn m); auto.
+Defined.
+
+Hint Resolve O_or_S eq_nat_dec: arith.
+
+Theorem dec_eq_nat : forall n m, decidable (n = m).
+intros x y; unfold decidable in |- *; elim (eq_nat_dec x y); auto with arith.
+Defined.