intuition.
unfold maxRat.
case_eq (bleRat r1 r2); intuition.
eapply leRat_trans.
eapply ratAdd_leRat_compat.
eapply H.
eapply leRat_refl.
rewrite ratMult_2.
rewrite ratMult_comm.
intuition.
apply bleRat_total in H.
eapply leRat_trans.
eapply ratAdd_leRat_compat.
eapply leRat_refl.
eapply H.
rewrite ratMult_2.
rewrite ratMult_comm.
intuition.
Qed.

(* Specifications: *)
Theorem rat_num_not_le :
forall (d1 d2 : posnat),
(RatIntro 1 d1 <= RatIntro 1 d2)%rat ->
d1 < d2 ->
False.