Section PS_incr.
Context {K : AbsRing} {V : NormedModule K}.
Definition PS_incr_1 (a : nat -> V) (n : nat) : V :=
match n with
| 0 => zero
| S n => a n
end.
Fixpoint PS_incr_n (a : nat -> V) (n k : nat) : V :=
match n with
| O => a k
| S n => PS_incr_1 (PS_incr_n a n) k
end.
Definition PS_decr_1 (a : nat -> V) (n : nat) : V := a (S n).
Definition PS_decr_n (a : nat -> V) (n k : nat) : V := a (n + k)%nat.

Lemma is_pseries_decr_n (a : nat -> V) (n : nat) (x y:K) (l : V) :
mult y x = one -> (0 < n)%nat -> is_pseries a x l
-> is_pseries (PS_decr_n a n) x (scal (pow_n y n) (plus l (opp (sum_n (fun k => scal (pow_n x k) (a k)) (n-1)%nat)))).