End AbelianGroup1.
(** Sum *)
Section Sums.
Context {G : AbelianGroup}.
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_S (a : nat -> G) (n m : nat) :
sum_n_m (fun n => a (S n)) n m = sum_n_m a (S n) (S m).