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.

(* Specifications: *)
Lemma PS_incr_n_simplify (a : nat -> V) (n k : nat) :
PS_incr_n a n k =
match (le_lt_dec n k) with
| left _ => a (k-n)%nat
| right _ => zero
end.
Lemma is_pseries_incr_n (a : nat -> V) (n : nat) (x : K) (l : V) :
is_pseries a x l -> is_pseries (PS_incr_n a n) x (scal (pow_n x n) l).
Lemma ex_pseries_incr_n (a : nat -> V) (n : nat) (x : K) :
ex_pseries a x -> ex_pseries (PS_incr_n a n) x.