(** Equivalence with standard library Reals *)

(* Specifications: *)
Lemma is_lim_seq_Reals (u : nat -> R) (l : R) :
is_lim_seq u l <-> Un_cv u l.
Lemma is_lim_seq_p_infty_Reals (u : nat -> R) :
is_lim_seq u p_infty <-> cv_infty u.
Lemma is_lim_LimSup_seq (u : nat -> R) (l : Rbar) :
is_lim_seq u l -> is_LimSup_seq u l.
Lemma is_lim_LimInf_seq (u : nat -> R) (l : Rbar) :
is_lim_seq u l -> is_LimInf_seq u l.
Lemma is_LimSup_LimInf_lim_seq (u : nat -> R) (l : Rbar) :
is_LimSup_seq u l -> is_LimInf_seq u l -> is_lim_seq u l.
Lemma ex_lim_LimSup_LimInf_seq (u : nat -> R) :
ex_lim_seq u <-> LimSup_seq u = LimInf_seq u.
Lemma is_pseries_Reals (a : nat -> R) (x l : R) :
is_pseries a x l <-> Pser a x l.
Lemma CV_radius_Reals_0 (a : nat -> R) (r : posreal) :
Rbar_lt (Finite r) (CV_radius a) -> CVN_r (fun n x => a n * x ^ n) r.
Lemma CV_radius_Reals_1 (a : nat -> R) (r : posreal) :
CVN_r (fun n x => a n * x ^ n) r -> Rbar_le (Finite r) (CV_radius a).
Lemma CV_radius_Reals_2 (a : nat -> R) (x : R) :
Rbar_lt (Finite (Rabs x)) (CV_radius a)
-> exists r : posreal, CVU (fun n x => sum_f_R0 (fun k => a k * x ^ k) n) (PSeries a) x r.
Lemma ex_derive_Reals_0 (f : R -> R) (x : R) :
ex_derive f x -> derivable_pt f x.
Lemma ex_derive_Reals_1 (f : R -> R) (x : R) :
derivable_pt f x -> ex_derive f x.
Lemma Derive_Reals (f : R -> R) (x : R) (pr : derivable_pt f x) :
derive_pt f x pr = Derive f x.
Lemma is_lim_Reals_0 (f : R -> R) (x l : R) :
is_lim f x l -> limit1_in f (fun y => y <> x) l x.
Lemma is_lim_Reals_1 (f : R -> R) (x l : R) :
limit1_in f (fun y => y <> x) l x -> is_lim f x l.
Lemma is_lim_Reals (f : R -> R) (x l : R) :
is_lim f x l <-> limit1_in f (fun y => y <> x) l x.