intuition.
unfold ratDistance, maxRat, minRat.
case_eq (bleRat r3 r1); intuition.
assert (r2 + r3 <= r2 + r1).
eapply ratAdd_leRat_compat; trivial.
eapply leRat_refl.
rewrite H0.
apply ratSubtract_add_same_l; eauto.
apply bleRat_total in H.
assert (r2 + r1 <= r2 + r3).
eapply ratAdd_leRat_compat; trivial.
apply leRat_refl.
case_eq (bleRat (r2 + r3) (r2 + r1)); intuition.
assert (r2 + r3 == r2 + r1).
eapply leRat_antisymm; eauto.
repeat rewrite ratSubtract_0.
intuition.
eapply eqRat_impl_leRat.
eapply ratAdd_add_same_l.
eauto.
eapply eqRat_impl_leRat.
eapply eqRat_symm.
trivial.
apply ratSubtract_add_same_l; eauto.
Qed.

Theorem rat_distance_of_sum : forall r1 r2 r3 r4,
ratDistance (r1 + r2) (r3 + r4) <= (ratDistance r1 r3) + (ratDistance r2 r4).