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

Theorem Z_O_1 : (0 < 1)%Z.
red in |- *; simpl in |- *; auto; intros; red in |- *; intros; discriminate.