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_O (a : nat -> G) : sum_n a 0 = a O.
Lemma sum_n_Sm (a : nat -> G) (n m : nat) :
(n <= S m)%nat -> sum_n_m a n (S m) = plus (sum_n_m a n m) (a (S m)).