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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 Some properties about pow and sum have been made with John Harrison i*)
(*i Some Lemmas (about pow and powerRZ) have been done by Laurent Thery i*)
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(**          Definition of the sum functions            *)
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Require Export ArithRing.
Require Export Rpow_def.
Require Export R_Ifp.
Require Export Rbasic_fun.
Require Export R_sqr.
Require Export SplitAbsolu.
Require Export SplitRmult.
Require Export ArithProp.
Local Open Scope nat_scope.
Local Open Scope R_scope.
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(** *       Power              *)
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Infix "^" := pow : R_scope.
Hint Resolve pow_O pow_1 pow_add pow_nonzero: real.
Hint Resolve pow_lt: real.
Hint Resolve Rlt_pow_R1: real.
Hint Resolve Rlt_pow: real.

Lemma pow_lt_1_zero :
forall x:R,
Rabs x < 1 ->
forall y:R,
0 < y ->
exists N : nat, (forall n:nat, (n >= N)%nat -> Rabs (x ^ n) < y).