(************************************************************************)
(*         *   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.
(* Rational numbers *)
Instance Qops: (@Ring_ops Q 0%Q 1%Q Qplus Qmult Qminus Qopp Qeq).
Defined.
Instance Qri : (Ring (Ro:=Qops)).
constructor.
try apply Q_Setoid.
apply Qplus_comp.
apply Qmult_comp.
apply Qminus_comp.
apply Qopp_comp.
exact Qplus_0_l. exact Qplus_comm. apply Qplus_assoc.
exact Qmult_1_l.  exact Qmult_1_r. apply Qmult_assoc.
apply Qmult_plus_distr_l.  intros. apply Qmult_plus_distr_r.
reflexivity. exact Qplus_opp_r.
Defined.
Instance Qcri: (Cring (Rr:=Qri)).
red. exact Qmult_comm. Defined.

Lemma Q_one_zero: not (Qeq 1%Q 0%Q).