induction ls; intuition; simpl in *.
case_eq (f a); intuition.
f_equal.
eauto.
rewrite H in H0.
discriminate.
intuition.
Qed.

(* Specifications: *)
Theorem sumList_subset_le :
forall (A : Set){eqd: EqDec A}(ls1 ls2 : list A)(f : A -> Rat),
NoDup ls1 ->
NoDup ls2 ->
(forall a, In a ls1 -> In a ls2) ->
sumList ls1 f <= sumList ls2 f.