induction 1; intuition.
inversion H0; clear H0; subst.
econstructor.
intuition.
eapply H3.
eapply Permutation_in.
eapply Permutation_sym.
eapply H.
eauto.
eauto.
inversion H; clear H; subst.
inversion H3; clear H3; subst.
simpl in *.
econstructor.
simpl.
intuition.
econstructor;
intuition.
Qed.

(* Specifications: *)
Lemma fold_add_subset : forall (A : Set)(eqd : eq_dec A)(ls1 ls2 : list A)(f : A -> Rat) init1 init2,
init1 == init2 ->
NoDup ls1 ->
NoDup ls2 ->
(forall a, In a ls1 -> In a ls2) ->
(forall a, (~In a ls1) -> (f a) == 0) ->
fold_left (fun r a => r + (f a)) ls1 init1 == fold_left (fun r a => r + (f a)) ls2 init2.