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.