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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: Zerob.v,v 1.1.2.1 2004/07/16 19:31:25 herbelin Exp $ i*)
Require Arith.
Require Bool.
V7only [Import nat_scope.].
Open Local Scope nat_scope.
Definition zerob : nat->bool
:= [n:nat]Cases n of O => true | (S _) => false end.
Lemma zerob_true_intro : (n:nat)(n=O)->(zerob n)=true.
NewDestruct n; [Trivial with bool | Inversion 1].
Qed.
Hints Resolve zerob_true_intro : bool.
Lemma zerob_true_elim : (n:nat)(zerob n)=true->(n=O).
NewDestruct n; [Trivial with bool | Inversion 1].
Qed.
Lemma zerob_false_intro : (n:nat)~(n=O)->(zerob n)=false.
NewDestruct n; [NewDestruct 1; Auto with bool | Trivial with bool].
Qed.
Hints Resolve zerob_false_intro : bool.
Lemma zerob_false_elim : (n:nat)(zerob n)=false -> ~(n=O).
NewDestruct n; [Intro H; Inversion H | Auto with bool].
Qed.
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