intros n m p H1 H2; replace (p * n)%Z with (-(-p * n))%Z; auto with zarith; try ring.
replace (p * m)%Z with (-(-p * m))%Z; auto with zarith; try ring.
apply Zopp_lt; apply Zmult_lt_compat_l; auto with zarith.
Qed.

Theorem Zopp_ge: forall n m, (m >= n -> -n >= -m)%Z.
auto with zarith.