(************************************************************************)
(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(*   \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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Require Export Cring.
Require Export Integral_domain.

(* Rational numbers *)
Require Import QArith.

#[global]
Instance Qops: (@Ring_ops Q 0%Q 1%Q Qplus Qmult Qminus Qopp Qeq).
Defined.

#[global]
Instance Qri : (Ring (Ro:=Qops)).
constructor.
- 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.

#[global]
Instance Qcri: (Cring (Rr:=Qri)).
red. exact Qmult_comm. Defined.

Lemma Q_one_zero: not (Qeq 1%Q 0%Q).
unfold Qeq. simpl. auto with *. Qed.

#[global]
Instance Qdi : (Integral_domain (Rcr:=Qcri)). 
constructor. 
- exact Qmult_integral.
- exact Q_one_zero.
Defined.
