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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.
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