intros y a b x lt_0_y lt_0_a lt_0_b le_0_x. rewrite <- Zdiv_simpl with (z := y). rewrite <- Zmult_assoc.
apply Zle_trans with (x / a * (a * b) / (b * y)). apply Zdiv_le. apply Zmult_lt_0_compat; omega.
apply Zmult_le_compat_l. rewrite Zmult_comm. apply Z_mult_div_ge. omega. apply Zle_0_div;
omega. rewrite Zmult_comm with (m := y). rewrite Zmult_assoc. rewrite Zdiv_simpl. omega. assumption.
assumption. assumption. assumption. Qed.

Lemma approx_0_two : forall y a b x : Z, 0 < y -> 0 < a -> 0 < b -> 0 <= x ->
x / a * a / y <= x / y.