intuition.
unfold ratDistance, maxRat, minRat in *.
case_eq (bleRat r1 r3); intuition;
rewrite H5 in H.
case_eq (bleRat r2 r4); intuition;
rewrite H6 in H0.
assert (r1 * r2 <= r3 * r4).
apply ratMult_leRat_compat; eauto.
rewrite H7.
rewrite ratSubtract_ratMult_le; eauto.
repeat rewrite ratMult_small_le.
rewrite H.
rewrite H0.
eapply eqRat_impl_leRat.
symmetry.
eapply ratMult_3_ratAdd.
eapply ratSubtract_le.
trivial.
trivial.
trivial.
apply bleRat_total in H6.
case_eq (bleRat (r1 * r2) (r3 * r4)); intuition.
rewrite ratSubtract_leRat.
2:{
eapply ratMult_leRat_compat.
eapply leRat_refl.
eapply H6.
}
2:{
eapply ratMult_leRat_compat.
eapply leRat_refl.
eapply H6.
}
rewrite ratSubtract_ratMult_le; eauto.
repeat rewrite ratMult_small_le.
rewrite H.
rewrite H0.
eapply eqRat_impl_leRat.
symmetry.
eapply ratMult_3_ratAdd.
eapply ratSubtract_le.
trivial.
trivial.
trivial.
apply bleRat_total in H7.
eapply leRat_trans.
eapply ratSubtract_leRat.
eapply ratMult_leRat_compat.
eapply H5.
eapply leRat_refl.
eapply ratMult_leRat_compat.
eapply H5.
eapply leRat_refl.
rewrite ratSubtract_ratMult_le; eauto.
repeat rewrite ratMult_small_le.
rewrite H.
rewrite H0.
eapply eqRat_impl_leRat.
symmetry.
eapply ratMult_3_ratAdd.
eapply ratSubtract_le.
trivial.
trivial.
trivial.
apply bleRat_total in H5.
case_eq (bleRat r2 r4); intuition.
rewrite H6 in H0.
case_eq (bleRat (r1 * r2) (r3 * r4)); intuition.
eapply leRat_trans.
eapply ratSubtract_leRat.
eapply ratMult_leRat_compat.
eapply H5.
eapply leRat_refl.
eapply ratMult_leRat_compat.
eapply H5.
eapply leRat_refl.
rewrite ratSubtract_ratMult_le; eauto.
repeat rewrite ratMult_small_le.
rewrite H.
rewrite H0.
eapply eqRat_impl_leRat.
symmetry.
eapply ratMult_3_ratAdd.
eapply ratSubtract_le.
trivial.
trivial.
trivial.
apply bleRat_total in H7.
eapply leRat_trans.
eapply ratSubtract_leRat.
eapply ratMult_leRat_compat.
eapply leRat_refl.
eapply H6.
eapply ratMult_leRat_compat.
eapply leRat_refl.
eapply H6.
rewrite ratSubtract_ratMult_le; eauto.
repeat rewrite ratMult_small_le.
rewrite H.
rewrite H0.
eapply eqRat_impl_leRat.
symmetry.
eapply ratMult_3_ratAdd.
eapply ratSubtract_le.
trivial.
trivial.
trivial.
rewrite H6 in H0.
apply bleRat_total in H6.
assert (r3 * r4 <= r1 * r2).
eapply ratMult_leRat_compat; eauto.
case_eq (bleRat (r1 * r2) (r3 * r4)); intuition.
rewrite ratSubtract_0; eauto.
eapply rat0_le_all.
apply bleRat_total in H8.
rewrite ratSubtract_ratMult_le; eauto.
repeat rewrite ratMult_small_le.
rewrite H.
rewrite H0.
eapply eqRat_impl_leRat.
symmetry.
eapply ratMult_3_ratAdd.
eapply ratSubtract_le.
trivial.
trivial.
trivial.
Qed.

Lemma ratAdd_any_le : forall r1 r2 r3,
r1 + r2 <= r3 ->
r1 <= r3.