proposition: Lemma mod_small_iff : forall a b, 0<=a -> 0<b -> (a mod b == a <-> a<b).
proof: Proof. intros a b Ha Hb. split; intros H; auto using mod_small. rewrite <- div_small_iff; auto. rewrite <- (mul_cancel_l _ _ b) by order. rewrite <- (add_cancel_r _ _ (a mod b)). rewrite <- div_mod, H by order. now nzsimpl. Qed.
imports: NZAxioms NZMulOrder