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Rings_R.v
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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) *)
(************************************************************************)
Require Export Cring.
Require Export Integral_domain.
(* Real numbers *)
Require Import Reals.
Require Import RealField.
Lemma Rsth : Setoid_Theory R (@eq R).
constructor;red;intros;subst;trivial.
Qed.
Instance Rops: (@Ring_ops R 0%R 1%R Rplus Rmult Rminus Ropp (@eq R)).
Defined.
Instance Rri : (Ring (Ro:=Rops)).
constructor;
try (try apply Rsth;
try (unfold respectful, Proper; unfold equality; unfold eq_notation in *;
intros; try rewrite H; try rewrite H0; reflexivity)).
exact Rplus_0_l. exact Rplus_comm. symmetry. apply Rplus_assoc.
exact Rmult_1_l. exact Rmult_1_r. symmetry. apply Rmult_assoc.
exact Rmult_plus_distr_r. intros; apply Rmult_plus_distr_l.
exact Rplus_opp_r.
Defined.
Instance Rcri: (Cring (Rr:=Rri)).
red. exact Rmult_comm. Defined.
Lemma R_one_zero: 1%R <> 0%R.
discrR.
Qed.
Instance Rdi : (Integral_domain (Rcr:=Rcri)).
constructor.
exact Rmult_integral. exact R_one_zero. Defined.
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