(** Multiplication and division by a variable *)
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.

(* Specifications: *)
Lemma is_pseries_incr_1 (a : nat -> V) (x:K) (l : V) :
is_pseries a x l -> is_pseries (PS_incr_1 a) x (scal x l).
Lemma ex_pseries_incr_1 (a : nat -> V) (x : K) :
ex_pseries a x -> ex_pseries (PS_incr_1 a) x.