rewrite Zmod_mult_distr_l. rewrite Zplus_comm. rewrite Zmod_plus_distr_l. rewrite Zplus_comm. rewrite <- Zmod_mult_distr_l.
trivial. assumption. assumption. assumption. omega. omega. Qed.

Lemma sum_r'_i : forall a b m : Z, 0 < m -> (a mod m + b mod m) / m = 0 \/ (a mod m + b mod m) / m = 1.
intros a b m H. cut (0 <= (a mod m + b mod m) / m). cut ((a mod m + b mod m) / m <= 1). omega.