apply Zmult_lt_O_compat; auto with zarith.
Qed.

(* Specifications: *)
Theorem Zlt_pos_neg: forall x, (0 < -x -> x < 0)%Z.
auto with zarith.
Theorem Zlt_sign_neg_pos: forall x y: Z, (x < 0 -> 0 < y  -> x * y < 0)%Z.
intros x y H1 H2; apply Zlt_pos_neg; replace (- (x * y))%Z with (-x * y)%Z; auto with zarith; try ring.
Theorem Zge_sign_neg_neg: forall x y: Z, (0 >= x -> 0 >= y  -> x * y >= 0)%Z.
intros; apply Z.le_ge; apply Zle_sign_neg_neg; auto with zarith.