apply Zle_trans with ((m - 1 + m) / m). apply Zdiv_le. omega. apply Zplus_le_compat. cut (a mod m < m). omega.
apply Zmod_lt_z_m. omega. cut (b mod m < m). omega. apply Zmod_lt_z_m. omega. replace (m - 1 + m) with (2 * m - 1).
rewrite Zdiv_mult_minus. omega. omega. omega. omega. ring. apply Zle_0_div. fold (0 + 0). apply Zplus_le_compat.
apply Zmod_le_0_z. omega. apply Zmod_le_0_z. omega. omega. Qed.

Lemma mod_plus : forall a b m : Z, 0 < m -> a mod m + b mod m = (a + b) mod m \/ a mod m + b mod m = (a + b) mod m + m.
intros a b m H. rewrite Z_div_mod_eq with (a := a mod m + b mod m) (b := m). elim sum_r'_i with (a := a) (b := b) (m := m).