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.