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).

Lemma is_pseries_decr_1 (a : nat -> V) (x y : K) (l : V) :
mult y x = one -> is_pseries a x l
-> is_pseries (PS_decr_1 a) x (scal y (plus l (opp (a O)))).