Definition Z_eq_bool a b :=
  match a, b with
  | Z0, Z0 => true
  | Zpos a', Zpos b' => pos_eq_bool a' b'
  | Zneg a', Zneg b' => pos_eq_bool a' b'
  | _, _ => false
  end.

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