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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(*i $Id: Peano_dec.v 14641 2011-11-06 11:59:10Z herbelin $ 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; destruct 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.
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