Library Coq.Reals.Rtrigo_reg

Require Import Rbase.
Require Import Rfunctions.
Require Import SeqSeries.
Require Import Rtrigo1.
Require Import Ranalysis1.
Require Import PSeries_reg.
Local Open Scope nat_scope.
Local Open Scope R_scope.

Lemma continuity_sin : continuity sin.

Lemma CVN_R_sin :
  forall fn:nat -> R -> R,
    fn =
    (fun (N:nat) (x:R) => (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N)) ->
    CVN_R fn.

Lemma derivable_pt_lim_sin_0 : derivable_pt_lim sin 0 1.

Lemma derivable_pt_lim_cos_0 : derivable_pt_lim cos 0 0.

Theorem derivable_pt_lim_sin : forall x:R, derivable_pt_lim sin x (cos x).

Lemma derivable_pt_lim_cos : forall x:R, derivable_pt_lim cos x (- sin x).

Lemma derivable_pt_sin : forall x:R, derivable_pt sin x.

Lemma derivable_pt_cos : forall x:R, derivable_pt cos x.

Lemma derivable_sin : derivable sin.

Lemma derivable_cos : derivable cos.

Lemma derive_pt_sin :
  forall x:R, derive_pt sin x (derivable_pt_sin _) = cos x.

Lemma derive_pt_cos :
  forall x:R, derive_pt cos x (derivable_pt_cos _) = - sin x.