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(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \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 R_scope.
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(** * Definition of exponential *)
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Definition exp_in (x l:R) : Prop :=
infinite_sum (fun i:nat => / INR (fact i) * x ^ i) l.
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(** * Definition of hyperbolic functions *)
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Definition cosh (x:R) : R := (exp x + exp (- x)) / 2.
Definition sinh (x:R) : R := (exp x - exp (- x)) / 2.
Definition tanh (x:R) : R := sinh x / cosh x.
Definition cos_n (n:nat) : R := (-1) ^ n / INR (fact (2 * n)).
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Definition cos_in (x l:R) : Prop :=
infinite_sum (fun i:nat => cos_n i * x ^ i) l.
(** Definition of cosinus *)
Definition cos (x:R) : R := let (a,_) := exist_cos (Rsqr x) in a.
Definition sin_n (n:nat) : R := (-1) ^ n / INR (fact (2 * n + 1)).
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Definition sin_in (x l:R) : Prop :=
infinite_sum (fun i:nat => sin_n i * x ^ i) l.

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Lemma exist_sin : forall x:R, { l:R | sin_in x l }.