intuition.
eapply eqRat_trans.
apply fold_add_body_const.
rewrite <- ratAdd_0_r.
eapply eqRat_refl.
Qed.

Lemma fold_add_iter_le : forall (A : Set)(ls : list A) f r init,
fold_left (fun r a => r + (f a)) ls init <= r ->
init <= r /\
(forall a, In a ls -> f a <= r).