intuition.
eapply eqRat_trans.
eapply fold_add_body_eq.
apply eqRat_refl.
eapply H.
simpl.
eapply fold_add_eq_init.
Qed.

Lemma fold_add_eq : forall (A : Set)(ls : list A)(f1 f2 : A -> Rat) init1 init2,
fold_left (fun r a => r + (f1 a)) ls init1 +
fold_left (fun r a => r + (f2 a)) ls init2 ==
fold_left (fun r a => r + (f1 a + f2 a)) ls (init1 + init2).