induction lsa; destruct lsb; intuition; simpl in *; intuition.
apply eqRat_symm.
eapply eqRat_trans.
eapply fold_add_body_eq; intuition.
rewrite <- ratAdd_0_r.
apply eqRat_refl.
apply eqRat_refl.
apply fold_add_eq_init.
eapply eqRat_trans.
eapply fold_add_body_eq; intuition.
rewrite <- ratAdd_0_r.
apply eqRat_refl.
apply eqRat_refl.
apply fold_add_eq_init.
eapply eqRat_trans.
eapply fold_add_body_eq; intuition.
rewrite <- ratAdd_0_l.
eapply fold_add_body_eq; intuition.
rewrite <- ratAdd_0_l.
apply eqRat_refl.
apply eqRat_refl.
eapply fold_add_body_eq; intuition.
rewrite <- ratAdd_0_l.
apply eqRat_refl.
apply eqRat_refl.
simpl.
eapply eqRat_trans.
eapply fold_add_body_eq; intuition.
eapply eqRat_refl.
eapply fold_add_init_0.
simpl.
eapply eqRat_trans.
eapply fold_add_init_0.
eapply eqRat_trans.
eapply ratAdd_eqRat_compat.
apply eqRat_refl.
eapply eqRat_trans.
eapply fold_add_body_eq; intuition.
apply ratAdd_0_r.
apply eqRat_refl.
apply eqRat_symm.
eapply fold_add_eq.
eapply eqRat_trans.
eapply ratAdd_eqRat_compat.
apply eqRat_refl.
apply ratAdd_eqRat_compat.
apply eqRat_refl.
eapply IHlsa.
eapply eqRat_symm.
eapply eqRat_trans.
eapply fold_add_body_eq; intuition.
rewrite <- ratAdd_0_l.
eapply fold_add_body_eq; intuition.
rewrite <- ratAdd_0_l.
apply eqRat_refl.
apply eqRat_refl.
eapply fold_add_body_eq; intuition.
rewrite <- ratAdd_0_l.
apply eqRat_refl.
apply eqRat_refl.
simpl.
eapply eqRat_trans.
eapply fold_add_body_eq; intuition.
eapply eqRat_refl.
eapply fold_add_init_0.
simpl.
eapply eqRat_trans.
eapply fold_add_init_0.
eapply eqRat_trans.
eapply ratAdd_eqRat_compat.
apply eqRat_refl.
eapply eqRat_trans.
eapply fold_add_body_eq; intuition.
apply ratAdd_0_r.
apply eqRat_refl.
apply eqRat_symm.
eapply fold_add_eq.
eapply eqRat_trans.
eapply ratAdd_eqRat_compat.
eapply fold_add_init_0.
eapply eqRat_refl.
eapply eqRat_symm.
eapply eqRat_trans.
eapply ratAdd_eqRat_compat.
eapply fold_add_init_0.
eapply eqRat_refl.
repeat rewrite <- ratAdd_assoc.
eapply ratAdd_eqRat_compat.
repeat rewrite (ratAdd_assoc (f a b)).
eapply ratAdd_eqRat_compat; intuition.
eapply ratAdd_comm.
eapply eqRat_refl.
Qed.

Lemma sumList_comm : forall (A B : Set)(lsa : list A)(lsb : list B) f,
sumList lsa (fun a => sumList lsb (fun b => (f a b))) ==
sumList lsb (fun b => sumList lsa (fun a => (f a b))).