(** * Unicity *)

(* Specifications: *)
Lemma Bessel1_uniqueness_aux_0 (a : nat -> R) (n : nat) :
Rbar_lt 0 (CV_radius a) ->
(forall x : R, Rbar_lt (Rabs x) (CV_radius a) -> x^2 * Derive_n (PSeries a) 2 x + x * Derive (PSeries a) x + (x^2 - (INR n)^2) * PSeries a x = 0)
->
(a 0%nat = 0 \/ n = O) /\
(a 1%nat = 0 \/ n = 1%nat) /\
(forall k, (INR (S (S k)) ^ 2 - INR n ^ 2) * a (S (S k)) + a k = 0).
Lemma Bessel1_uniqueness_aux_1 (a : nat -> R) (n : nat) :
(a 0%nat = 0 \/ n = O) ->
(a 1%nat = 0 \/ n = 1%nat) ->
(forall k, (INR (S (S k)) ^ 2 - INR n ^ 2) * a (S (S k)) + a k = 0) ->
(forall k : nat, (k < n)%nat -> a k = 0)
/\ (forall p : nat, a (n + 2 * p + 1)%nat = 0)
/\ (forall p : nat, a (n + 2 * p)%nat = Bessel1_seq n p * / 2 ^ (2 * p) * INR (fact n) * a n).
Lemma Bessel1_uniqueness (a : nat -> R) (n : nat) :
(Rbar_lt 0 (CV_radius a)) ->
(forall x : R, x^2 * Derive_n (PSeries a) 2 x + x * Derive (PSeries a) x + (x^2 - (INR n)^2) * PSeries a x = 0)
-> {b : R | forall x, PSeries a x = b * Bessel1 n x}.