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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 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.

(* Specifications: *)
Lemma exp_cof_no_R0 : forall n:nat, / INR (fact n) <> 0.
Lemma exist_exp : forall x:R, { l:R | exp_in x l }.
(* Value of [exp 0] *)
Lemma exp_0 : exp 0 = 1.
Definition cosh (x:R) : R := (exp x + exp (- x)) / 2.