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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.