induction ls; intuition; simpl in *.
rewrite <- plus_assoc.
eapply IHls.
Qed.

(* Specifications: *)
Lemma fold_add_init_nat :
forall (A : Set)(f : A -> nat)(ls : list A)(init : nat),
(fold_left (fun acc a => acc + (f a))%nat ls init =
init + fold_left (fun acc a => acc + (f a))%nat ls O)%nat.