induction ls; intuition; simpl in *.
eauto.
Qed.

Lemma fold_add_nat_Permutation :
forall (A : Set)(f : A -> nat)(ls1 ls2 : list A),
Permutation ls1 ls2 ->
fold_left (fun acc a => acc + (f a))%nat ls1 O =
fold_left (fun acc a => acc + (f a))%nat ls2 O.