intros y a b x lt_0_y lt_0_a le_a_y lt_0_b le_0_x. replace (x / y) with ((a * (x / a) + x mod a) / y).
unfold Zminus. rewrite <- Z_div_plus. apply Zdiv_le. assumption. rewrite Zmult_comm.
cut (x mod a <= 1 * y). omega. apply Zle_trans with a. apply Zlt_le_weak. apply Zmod_lt_z_m. assumption.
omega. omega. rewrite <- Z_div_mod_eq. trivial. omega. Qed.

Lemma approx_m2 : forall y a b x : Z, 0 < y -> 0 < a -> a <= y -> 0 < b -> 0 <= x ->
x < a * b + a -> x / y - 2 <= x / a * (a * b / y) / b.