intros p q; case p; case q; simpl in |- *; auto;
try (intros; red in |- *; intros; discriminate; fail).
intros p' q'; generalize (pos_eq_bool_correct q' p');
case (pos_eq_bool q' p'); simpl in |- *; auto.
intros H1; rewrite H1; auto.
intros H1; Contradict H1; injection H1; auto.
intros p' q'; generalize (pos_eq_bool_correct q' p');
case (pos_eq_bool q' p'); simpl in |- *; auto.
intros H1; rewrite H1; auto.
intros H1; Contradict H1; injection H1; auto.
Qed.

Theorem Zlt_mult_ZERO :
forall x y : Z, (0 < x)%Z -> (0 < y)%Z -> (0 < x * y)%Z.