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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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(*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.
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(** *                     Properties of  Exp                      *)
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(* Definition of log R+* -> R *)
Definition Rln (y:posreal) : R :=
let (a,_) := ln_exists (pos y) (cond_pos y) in a.
(* Extension on R *)
Definition ln (x:R) : R :=
match Rlt_dec 0 x with
| left a => Rln (mkposreal x a)
| right a => 0
end.
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(** *                     Properties of  Ln                       *)
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Theorem ln_increasing : forall x y:R, 0 < x -> x < y -> ln x < ln y.