induction l; intros.
simpl.
rewrite Rmult_0_l...
simpl @length.
simpl Rsum.
rewrite S_INR.
rewrite Rmult_plus_distr_r.
rewrite Rmult_1_l.
rewrite Rplus_comm.
apply Rplus_le_compat.
apply H.
left...
apply IHl.
intros.
apply H.
right...
Qed.

(* Specifications: *)
Lemma Rsum_map_plus X f g (l: list X):
Rsum (map (fun x => f x + g x) l) =
Rsum (map f l) + Rsum (map g l).