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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 1.1.2.1 2004/07/16 19:31:25 herbelin Exp $ i*)
+
+Require Decidable.
+
+V7only [Import nat_scope.].
+Open Local Scope nat_scope.
+
+Implicit Variables Type m,n,x,y:nat.
+
+Theorem O_or_S : (n:nat)({m:nat|(S m)=n})+{O=n}.
+Proof.
+NewInduction n.
+Auto.
+Left; Exists n; Auto.
+Defined.
+
+Theorem eq_nat_dec : (n,m:nat){n=m}+{~(n=m)}.
+Proof.
+NewInduction n; NewInduction m; Auto.
+Elim (IHn m); Auto.
+Defined.
+
+Hints Resolve O_or_S eq_nat_dec : arith.
+
+Theorem dec_eq_nat:(x,y:nat)(decidable (x=y)).
+Intros x y; Unfold decidable; Elim (eq_nat_dec x y); Auto with arith.
+Defined.
+