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(*         *      The Rocq Prover / The Rocq Development Team           *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
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Local Open Scope nat_scope.
Local Open Scope R_scope.

(* Specifications: *)
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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.
(** (sin h)/h -> 1 when h -> 0 *)
Lemma derivable_pt_lim_sin_0 : derivable_pt_lim sin 0 1.
(** ((cos h)-1)/h -> 0 when h -> 0 *)
Lemma derivable_pt_lim_cos_0 : derivable_pt_lim cos 0 0.
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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.