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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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(** * Basic lemmas for the classical real numbers        *)
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Require Export Raxioms.
Require Export ZArithRing.
Require Export RealField.
Local Open Scope Z_scope.
Local Open Scope R_scope.
Implicit Type r : R.
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(** ** Relation between orders and equality              *)
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(** Reflexivity of the large order *)
Hint Immediate Rle_refl: rorders.
Hint Resolve Rlt_irrefl: real.
Hint Resolve Rlt_dichotomy_converse: real.
(** Reasoning by case on equality and order *)
Hint Resolve Req_dec: real.
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(** ** Relating [<], [>], [<=] and [>=]                  *)
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(** ** Order                                             *)
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(** *** Relating strict and large orders *)
Hint Resolve Rlt_le: real.
Hint Immediate Rle_ge: real.
Hint Resolve Rle_ge: rorders.
Hint Resolve Rge_le: real.
Hint Immediate Rge_le: rorders.
Hint Resolve Rlt_gt: rorders.
Hint Immediate Rgt_lt: rorders.
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Hint Immediate Rnot_le_lt: real.
Hint Immediate Rlt_not_le: real.
Hint Immediate Req_le: real.
Hint Immediate Req_ge: real.
Hint Immediate Req_le_sym: real.
Hint Immediate Req_ge_sym: real.
(** *** Asymmetry *)
(** Remark: [Rlt_asym] is an axiom *)
Hint Resolve Rle_antisym: real.
(** *** Compatibility with equality *)

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Lemma inser_trans_R :
forall r1 r2 r3 r4, r1 <= r2 < r3 -> {r1 <= r2 < r4} + {r4 <= r2 < r3}.