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

Lemma ratDistance_add_same_l : forall r1 r2 r3,
(ratDistance (r2 + r3) (r2 + r1)) == (ratDistance r3 r1).