induction ls; intuition; simpl in *.
eapply eqRat_refl.
rewrite IHls.
eapply fold_add_body_eq.
repeat rewrite <- ratAdd_assoc.
apply ratAdd_eqRat_compat; intuition.
rewrite ratAdd_assoc.
rewrite ratAdd_assoc.
apply ratAdd_eqRat_compat; intuition.
eapply ratAdd_comm.
intuition.
Qed.

Lemma fold_add_comm : forall (A B : Set)(lsa : list A)(lsb : list B) f,
fold_left (fun r1 a => r1 + (fold_left (fun r2 b => r2 + (f a b)) lsb 0)) lsa 0  ==
fold_left (fun r1 b => r1 + (fold_left (fun r2 a => r2 + (f a b)) lsa 0)) lsb 0.