induction ls; intuition; simpl in *.
inversion H.
intuition.
inversion H; clear H; subst.
rewrite H6.
rewrite (H0 a r2); eauto.
rewrite <- ratAdd_0_l.
eapply IHls; eauto.
Qed.

Lemma sumList_rel_only_one_inv : forall (A : Set)(rel : A -> Rat -> Prop)(ls : list A)(a : A) r r',
sumList_rel rel ls r' ->
In a ls ->
NoDup ls ->
(forall a', In a' ls -> a <> a' -> forall v, rel a' v -> v == 0) ->
(forall v, rel a v -> v == r) ->
r' == r.