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.
Lemma PSeries_incr_1 a x : PSeries (PS_incr_1 a) x = x * PSeries a x.
Lemma PSeries_incr_n (a : nat -> R) (n : nat) (x : R) : PSeries (PS_incr_n a n) x = x^n * PSeries a x.
Lemma PSeries_decr_1 (a : nat -> R) (x : R) : ex_pseries a x -> PSeries a x = a O + x * PSeries (PS_decr_1 a) x.
Lemma PSeries_decr_1_aux (a : nat -> R) (x : R) : a O = 0 -> (PSeries a x) = x * PSeries (PS_decr_1 a) x.
Lemma PSeries_decr_n (a : nat -> R) (n : nat) (x : R) : ex_pseries a x -> PSeries a x = sum_f_R0 (fun k => a k * x^k) n + x^(S n) * PSeries (PS_decr_n a (S n)) x.
Lemma PSeries_decr_n_aux (a : nat -> R) (n : nat) (x : R) : (forall k : nat, (k < n)%nat -> a k = 0) -> PSeries a x = x^n * PSeries (PS_decr_n a n) x.
Lemma CV_radius_incr_1 (a : nat -> R) : CV_radius (PS_incr_1 a) = CV_radius a.
Lemma CV_radius_decr_1 (a : nat -> R) : CV_radius (PS_decr_1 a) = CV_radius a.