induction ls; intuition; simpl in *.
rewrite app_length.
rewrite fold_add_init_nat.
f_equal.
eauto.
Qed.

Lemma fold_left_map_eq :
forall (A B C : Set)(ls : list A)(f1 : A -> B)(f2 : C -> B -> C)(init : C),
fold_left (fun acc b => f2 acc b) (map f1 ls) init =
fold_left (fun acc a => f2 acc (f1 a)) ls init.