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(*         *      The Rocq Prover / The Rocq Development Team           *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2019       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
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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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(** Proof that classical reals are constructive reals with extra properties, namely total order, existence of all least upper bounds and setoid equivalence simplifying to Leibniz equality.  From this point of view, the quotient Rabst and Rrepr between classical Dedekind reals and constructive Cauchy reals becomes an isomorphism for the ConstructiveReals structure.  This allows to apply results from constructive reals to classical reals. *)
Local Open Scope R_scope.

(* Specifications: *)
(* Rlt is the transport of CRealLt via Rasbt, Rrepr, so it is linear as CRealLt. *)
Lemma RisLinearOrder : isLinearOrder Rlt.
Lemma RdisjunctEpsilon
: forall a b c d : R, (a < b)%R \/ (c < d)%R -> (a < b)%R + (c < d)%R.