intuition.
eapply leRat_trans.
eapply ratDistance_le_max.
eapply rat0_le_all.
eapply (@ratAdd_any_leRat_l r1 r2).
eapply leRat_refl.
eapply leRat_trans.
eapply eqRat_impl_leRat.
eapply maxRat_eqRat_compat.
eapply ratDistance_from_0.
eapply eqRat_trans.
eapply ratDistance_eqRat_compat.
eapply eqRat_refl.
eapply ratAdd_0_l.
rewrite ratDistance_add_same_r.
rewrite ratDistance_comm.
eapply ratDistance_from_0.
rewrite maxRat_comm.
intuition.
Qed.

Lemma ratAdd_2_ratMax :
forall r1 r2,
(r1 + r2 <= 2 / 1 * (maxRat r1 r2))%rat.