intuition.
destruct (eq_nat_dec n O).
subst.
simpl.
unfold sumList.
simpl.
eapply eqRat_trans.
rewrite <- ratIdentityIndiscernables.
intuition.
symmetry.
rewrite <- ratIdentityIndiscernables.
intuition.
assert (sumList (forNats n) f == (ratSubtract (sumList (forNats n) f) (f O)) + (f O)).
induction n; intuition.
simpl.
repeat rewrite sumList_cons.
destruct n.
simpl.
unfold sumList.
simpl.
rewrite <- ratAdd_0_r.
rewrite ratSubtract_0.
rewrite <- ratAdd_0_l.
intuition.
intuition.
rewrite IHn.
rewrite ratSubtract_ratAdd_assoc.
rewrite ratSubtract_ratAdd_assoc; intuition.
rewrite (@ratSubtract_0 (f O)); intuition.
rewrite <-ratAdd_0_r.
symmetry.
eapply ratAdd_assoc.
eapply leRat_trans.
eapply eqRat_impl_leRat.
eapply ratAdd_0_l.
eapply ratAdd_leRat_compat; intuition.
eapply rat0_le_all.
intuition.
rewrite H.
clear H.
assert (sumList (forNats n) (fun i : nat => f (S i)) == (ratSubtract (sumList (forNats n) f) (f O)) + (f n)).
induction n; intuition.
simpl.
repeat rewrite sumList_cons.
destruct n.
simpl.
unfold sumList.
simpl.
repeat rewrite <- ratAdd_0_r.
rewrite ratSubtract_0; intuition.
rewrite <- ratAdd_0_l.
intuition.
rewrite IHn.
rewrite ratAdd_comm.
eapply ratAdd_eqRat_compat; intuition.
rewrite ratSubtract_ratAdd_assoc.
rewrite ratAdd_comm.
intuition.
eapply sumList_forNats_first_ls;
intuition.
intuition.
rewrite H.
eapply ratDistance_add_same_l.
Qed.

Lemma flatten_map_eq :
forall (A B : Set)(ls : list A)(f : A -> B),
flatten (map (fun a => (f a) :: nil) ls) =
map f ls.