Definition PS_decr_n (a : nat -> V) (n k : nat) : V := a (n + k)%nat.

(* Specifications: *)
Lemma is_pseries_decr_n (a : nat -> V) (n : nat) (x y:K) (l : V) :
mult y x = one -> (0 < n)%nat -> is_pseries a x l
-> is_pseries (PS_decr_n a n) x (scal (pow_n y n) (plus l (opp (sum_n (fun k => scal (pow_n x k) (a k)) (n-1)%nat)))).
Lemma ex_pseries_decr_n (a : nat -> V) (n : nat) (x : K) :
(x = zero \/ exists y, mult y x = one) -> ex_pseries a x -> ex_pseries (PS_decr_n a n) x.