induction n; intuition; simpl in *.
eapply leRat_trans.
2:{
eapply eqRat_impl_leRat.
rewrite sumList_cons.
eapply eqRat_refl.
}
destruct n.
eapply leRat_trans.
rewrite ratAdd_0_r.
eapply leRat_refl.
eapply ratAdd_leRat_compat; intuition.
eapply rat0_le_all.
rewrite ratAdd_0_l.
eapply ratAdd_leRat_compat; intuition.
eapply rat0_le_all.
Qed.

(* Specifications: *)
Lemma sumList_forNats_distance :
forall (n : nat)(f : nat -> Rat),
(| sumList (forNats n) f - sumList (forNats n) (fun i => f (S i)) |) == (| (f O) - (f n) |).