Definition zerob (n:nat) : bool :=
  match n with
    | O => true
    | S _ => false
  end.

(* Specifications: *)
Lemma zerob_true_intro (n : nat) : n = 0 -> zerob n = true.
Lemma zerob_true_elim (n : nat) : zerob n = true -> n = 0.
Lemma zerob_false_intro (n : nat) : n <> 0 -> zerob n = false.
Lemma zerob_false_elim (n : nat) : zerob n = false -> n <> 0.