unfold ratDistance, maxRat, minRat in *.
intuition.
case_eq (bleRat r1 r2); intuition;
rewrite H0 in H.
rewrite ratAdd_0_r.
eapply ratAdd_leRat_compat.
trivial.
eapply rat0_le_all.
eapply ratSubtract_le_sum.
eapply bleRat_total.
trivial.
trivial.
Qed.

Lemma ratSubtract_ratDistance_le : forall r1 r2,
ratSubtract r1 r2 <= ratDistance r1 r2.