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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) *)
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(*    //   *    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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(** This file provides several hundred basic lemmas about foundamental operations on R: - addition denoted by [Rplus] (notation: infix [+]), opposite denoted by [Ropp] (notation: unary prefix [-]) and subtraction denoted by [Rminus] (notation: infix [-]); - multiplication denoted by [Rmult] (notation: infix [*]), inverse denoted by [Rinv] (notation: prefix [/]), division denoted by [Rdiv] (notation: infix [/]) and square denoted by [Rsqr] (notation: suffix [²]) - orders [<], [>], [<=] and [>=] - injective morphisms: - [INR : nat -> R] - [IPR : positive -> R] - [IZR : Z -> R]  All those lemmas are proved using a set of 17 "primitive" lemmas in [Raxioms.v] (plus the convenient choice that the inverse of 0 is 0 in [Rdefinitions.v]). These "primitive" lemmas are: - [Rplus_comm], [Rplus_assoc], [Rplus_0_l], [Rplus_opp_l] - [Rmult_comm], [Rmult_assoc], [Rmult_1_l], [Rinv_l] - [Rmult_plus_distr_l], [R1_neq_R0] - [Rlt_trans], [Rlt_asym], [Rplus_lt_compat_l], [Rmult_lt_compat_l] - [total_order_T] - [completeness], [archimed]  This makes this file independent of the actual construction of the real numbers, since these 17 axioms characterize, up to isomorphism, the ordered field of real numbers. *)
Require Export Raxioms.
Require Export ZArithRing.
Require Export RealField.
Local Open Scope Z_scope.
Local Open Scope R_scope.
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(** ** Relation between orders and equality              *)
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(** Reflexivity of the large orders *)

Lemma Rle_refl : forall r, r <= r.
#[global]
Hint Resolve Rlt_dichotomy_converse: real.