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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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(*i Due to L.Thery i*)
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(* Definitions of log and Rpower : R->R->R; main properties *)
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Local Open Scope R_scope.
Definition P_Rmin_stt (P:R -> Prop) x y := Rmin_case x y P.
#[deprecated(since="8.16", note="Use Rmin_case instead.")]
Notation P_Rmin := P_Rmin_stt.

(* Specifications: *)
Lemma exp_le_3 : exp 1 <= 3.
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(** *                     Properties of  Exp                      *)
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Lemma exp_neq_0 : forall x:R, exp x <> 0.
Theorem exp_increasing : forall x y:R, x < y -> exp x < exp y.
Theorem exp_lt_inv : forall x y:R, exp x < exp y -> x < y.
Lemma exp_ineq1 : forall x : R, x <> 0 -> 1 + x < exp x.
Lemma exp_ineq1_le (x : R) : 1 + x <= exp x.
Lemma ln_exists1 : forall y:R, 1 <= y -> { z:R | y = exp z }.
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Lemma ln_exists : forall y:R, 0 < y -> { z:R | y = exp z }.
Definition Rln (y:posreal) : R :=
let (a,_) := ln_exists (pos y) (cond_pos y) in a.