Definition sum_n_m (a : nat -> G) n m :=
  iter_nat plus zero a n m.Definition sum_n (a : nat -> G) n :=
  sum_n_m a O n.

Lemma sum_n_m_plus : forall (u v : nat -> G) (n m : nat), sum_n_m (fun k => plus (u k) (v k)) n m = plus (sum_n_m u n m) (sum_n_m v n m).