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(************************************************************************)
(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
Require Import Arith.
Require Import Bool.
Local Open Scope nat_scope.
Definition zerob (n:nat) : bool :=
match n with
| O => true
| S _ => false
end.
Lemma zerob_true_intro : forall n:nat, n = 0 -> zerob n = true.
Proof.
destruct n; [ trivial with bool | inversion 1 ].
Qed.
Hint Resolve zerob_true_intro: bool.
Lemma zerob_true_elim : forall n:nat, zerob n = true -> n = 0.
Proof.
destruct n; [ trivial with bool | inversion 1 ].
Qed.
Lemma zerob_false_intro : forall n:nat, n <> 0 -> zerob n = false.
Proof.
destruct n; [ destruct 1; auto with bool | trivial with bool ].
Qed.
Hint Resolve zerob_false_intro: bool.
Lemma zerob_false_elim : forall n:nat, zerob n = false -> n <> 0.
Proof.
destruct n; [ inversion 1 | auto with bool ].
Qed.
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