Library Coq.Bool.Zerob

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.
Hint Resolve zerob_true_intro: bool.

Lemma zerob_true_elim : forall n:nat, zerob n = true -> n = 0.

Lemma zerob_false_intro : forall n:nat, n <> 0 -> zerob n = false.
Hint Resolve zerob_false_intro: bool.

Lemma zerob_false_elim : forall n:nat, zerob n = false -> n <> 0.