Library Coq.setoid_ring.Rings_Q

Require Export Cring.
Require Export Integral_domain.

Require Import QArith.

Instance Qops: (@Ring_ops Q 0%Q 1%Q Qplus Qmult Qminus Qopp Qeq).

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).
unfold Qeq. simpl. auto with *. Qed.

Instance Qdi : (Integral_domain (Rcr:=Qcri)).
constructor.
exact Qmult_integral. exact Q_one_zero. Defined.