induction ls; intuition; simpl in *.
intuition.
unfold sumList.
simpl.
eapply eqRat_refl.
intuition; subst.
unfold sumList in *; simpl in *.
eapply eqRat_trans.
eapply ratAdd_0.
eapply eqRat_trans.
eapply ratAdd_comm.
eapply eqRat_trans.
eapply eqRat_symm.
eapply fold_add_init.
eapply eqRat_trans.
eapply fold_add_body_eq.
eapply ratAdd_comm.
intuition.
eapply eqRat_refl.
eapply H.
eapply eqRat_refl.
eapply IHls.
unfold sumList in *; simpl in *.
eapply ratAdd_0.
eapply eqRat_trans.
eapply eqRat_symm.
eapply fold_add_init.
eapply eqRat_trans.
eapply fold_add_body_eq.
eapply ratAdd_comm.
intuition.
eapply eqRat_refl.
eapply H.
trivial.
unfold sumList.
simpl.
eapply eqRat_trans.
eapply fold_add_body_eq.
eapply ratAdd_comm.
intuition.
eapply eqRat_refl.
eapply eqRat_trans.
eapply fold_add_init.
eapply ratAdd_0.
intuition.
unfold sumList in *.
eapply IHls.
intuition.
Qed.

Lemma sumList_nz : forall (ls : list A) f,
(~sumList ls f == 0) <-> exists a : _, In a ls /\ (~f a == 0).