auto with arith.
intros n0; case n0.
simpl in |- *; auto with arith.
intros n1 H' q r H'0 H'1.
apply lt_trans with (m := q + S n1 * r); auto with arith.
Qed.
Hint Resolve lt_comp_mult_l: arith.

Theorem lt_comp_mult_r : forall p q r : nat, 0 < p -> q < r -> q * p < r * p.
intros; repeat rewrite (fun x : nat => mult_comm x p); auto with arith.