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 Zminus_Zdiv_Zmult : forall a b : Z, 0 < b -> a - a / b * b = a mod b.
intros. rewrite Zdiv_a_b_b_Zmod. ring. assumption. Qed.