proposition: Lemma is_derive_2_Bessel1 (n : nat) (x : R) : is_derive_n (Bessel1 n) 2 x (((x/2)^(S (S n)) * PSeries (PS_derive (PS_derive (Bessel1_seq n))) ((x / 2) ^ 2)) + ((INR (2*n+1)/2) * (x/2)^n * PSeries (PS_derive (Bessel1_seq n)) ((x / 2) ^ 2)) + (INR (n * pred n) / 4 * (x / 2) ^ pred (pred n) * PSeries (Bessel1_seq n) ((x / 2) ^ 2))).
proof: Proof. rewrite plus_INR ?mult_INR ; simpl INR. eapply is_derive_ext. move => y ; by apply sym_eq, is_derive_unique, is_derive_Bessel1. auto_derive. repeat split. apply ex_derive_PSeries. by rewrite CV_radius_derive CV_Bessel1. apply ex_derive_PSeries. by rewrite CV_Bessel1. rewrite !Derive_PSeries. case: n => [ | n] ; rewrite ?S_INR /Rdiv /= ; field. by rewrite CV_Bessel1. by rewrite CV_radius_derive CV_Bessel1. Qed.
imports: Coq(Reals Psatz ssreflect) Coquelicot(Rcomplements Rbar Hierarchy Derive Series PSeries Lim_seq AutoDerive)