simple induction b; simpl in |- *; intros; auto.
absurd (x <> 0).
apply no_no_A; apply zerob_true_elim; auto.
trivial.
Qed.

Lemma lt_no_zerob : forall n : nat, 0 < n -> zerob n <> true.
simple induction n; [ intros; inversion H | auto ].