replace ((x / m * m + x mod m) mod m) with (x mod m).
rewrite Zdiv_small_0; auto. rewrite Zdivmod_split. trivial. omega. omega. Qed.

Lemma Zdiv_a_b_b_Zmod : forall a b : Z, 0 < b -> a / b * b = a - a mod b.
intros a b H. cut (a / b * b + a mod b = a). omega. apply Zdivmod_split. assumption. Qed.