Zerob.v

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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.forall n : nat, n = 0 -> zerob n = true
Proof.forall n : nat, n = 0 -> zerob n = true
  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.forall n : nat, zerob n = true -> n = 0
Proof.forall n : nat, zerob n = true -> n = 0
  destruct n; [ trivial with bool | inversion 1 ].
Qed.

Lemma zerob_false_intro : forall n:nat, n <> 0 -> zerob n = false.forall n : nat, n <> 0 -> zerob n = false
Proof.forall n : nat, n <> 0 -> zerob n = false
  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.forall n : nat, zerob n = false -> n <> 0
Proof.forall n : nat, zerob n = false -> n <> 0
  destruct n; [ inversion 1 | auto with bool ].
Qed.