induction ls; intuition; simpl in *.
eapply eqRat_refl.
eapply eqRat_trans.
eapply IHls.
eapply eqRat_trans.
eapply ratAdd_assoc.
eapply ratAdd_eqRat_compat.
eapply eqRat_refl.
eapply eqRat_symm.
eapply IHls.
Qed.

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