Library mathcomp.algebra.ring_quotient
(* (c) Copyright 2006-2015 Microsoft Corporation and Inria.
Distributed under the terms of CeCILL-B. *)
Require Import mathcomp.ssreflect.ssreflect.
Distributed under the terms of CeCILL-B. *)
Require Import mathcomp.ssreflect.ssreflect.
This file describes quotients of algebraic structures.
It defines a join hierarchy mxing the structures defined in file ssralg
(up to unit ring type) and the quotType quotient structure defined in
file generic_quotient. Every structure in that (join) hierarchy is
parametrized by a base type T and the constants and operations on the
base type that will be used to confer its algebraic structure to the
quotient. Note that T itself is in general not an instance of an
algebraic structure. The canonical surjection from T onto its quotient
should be compatible with the parameter operations.
The second part of the file provides a definition of (non trivial)
decidable ideals (resp. prime ideals) of an arbitrary instance of ring
structure and a construction of the quotient of a ring by such an ideal.
These definitions extend the hierarchy of sub-structures defined in file
ssralg (see Module Pred in ssralg), following a similar methodology.
Although the definition of the (structure of) quotient of a ring by an
ideal is a general one, we do not provide infrastructure for the case of
non commutative ring and left or two-sided ideals.
The file defines the following Structures:
zmodQuotType T e z n a == Z-module obtained by quotienting type T
with the relation e and whose neutral,
opposite and addition are the images in the
quotient of the parameters z, n and a,
respectively.
ringQuotType T e z n a o m == ring obtained by quotienting type T with
the relation e and whose zero opposite,
addition, one, and multiplication are the
images in the quotient of the parameters
z, n, a, o, m, respectively.
unitRingQuotType ... u i == As in the previous cases, instance of unit
ring whose unit predicate is obtained from
u and the inverse from i.
idealr R S == (S : pred R) is a non-trivial, decidable,
right ideal of the ring R.
prime_idealr R S == (S : pred R) is a non-trivial, decidable,
right, prime ideal of the ring R.
The formalization of ideals features the following constructions:
proper_ideal S == the collective predicate (S : pred R) on the
ring R is stable by the ring product and does
contain R's one.
prime_idealr_closed S := u * v \in S -> (u \in S) || (v \in S)
idealr_closed S == the collective predicate (S : pred R) on the
ring R represents a (right) ideal. This
implies its being a proper_ideal.
MkIdeal idealS == packs idealS : proper_ideal S into an
idealr S interface structure associating the
idealr_closed property to the canonical
pred_key S (see ssrbool), which must already
be an zmodPred (see ssralg).
MkPrimeIdeal pidealS == packs pidealS : prime_idealr_closed S into a
prime_idealr S interface structure associating
the prime_idealr_closed property to the
canonical pred_key S (see ssrbool), which must
already be an idealr (see above).
{ideal_quot kI} == quotient by the keyed (right) ideal predicate
kI of a commutative ring R. Note that we indeed
only provide canonical structures of ring
quotients for the case of commutative rings,
for which a right ideal is obviously a
two-sided ideal.
Note :
if (I : pred R) is a predicate over a ring R and (ideal : idealr I) is an
instance of (right) ideal, in order to quantify over an arbitrary (keyed)
predicate describing ideal, use type (keyed_pred ideal), as in:
forall (kI : keyed_pred ideal),...
Import GRing.Theory.
Set Implicit Arguments.
Local Open Scope ring_scope.
Local Open Scope quotient_scope.
Reserved Notation "{ideal_quot I }" (at level 0, format "{ideal_quot I }").
Reserved Notation "m = n %[mod_ideal I ]" (at level 70, n at next level,
format "'[hv ' m '/' = n '/' %[mod_ideal I ] ']'").
Reserved Notation "m == n %[mod_ideal I ]" (at level 70, n at next level,
format "'[hv ' m '/' == n '/' %[mod_ideal I ] ']'").
Reserved Notation "m <> n %[mod_ideal I ]" (at level 70, n at next level,
format "'[hv ' m '/' <> n '/' %[mod_ideal I ] ']'").
Reserved Notation "m != n %[mod_ideal I ]" (at level 70, n at next level,
format "'[hv ' m '/' != n '/' %[mod_ideal I ] ']'").
Section ZmodQuot.
Variable (T : Type).
Variable eqT : rel T.
Variables (zeroT : T) (oppT : T → T) (addT : T → T → T).
Record zmod_quot_mixin_of (Q : Type) (qc : quot_class_of T Q)
(zc : GRing.Zmodule.class_of Q) := ZmodQuotMixinPack {
zmod_eq_quot_mixin :> eq_quot_mixin_of eqT qc zc;
_ : \pi_(QuotTypePack qc Q) zeroT = 0 :> GRing.Zmodule.Pack zc Q;
_ : {morph \pi_(QuotTypePack qc Q) : x /
oppT x >-> @GRing.opp (GRing.Zmodule.Pack zc Q) x};
_ : {morph \pi_(QuotTypePack qc Q) : x y /
addT x y >-> @GRing.add (GRing.Zmodule.Pack zc Q) x y}
}.
Record zmod_quot_class_of (Q : Type) : Type := ZmodQuotClass {
zmod_quot_quot_class :> quot_class_of T Q;
zmod_quot_zmod_class :> GRing.Zmodule.class_of Q;
zmod_quot_mixin :> zmod_quot_mixin_of
zmod_quot_quot_class zmod_quot_zmod_class
}.
Structure zmodQuotType : Type := ZmodQuotTypePack {
zmod_quot_sort :> Type;
_ : zmod_quot_class_of zmod_quot_sort;
_ : Type
}.
Implicit Type zqT : zmodQuotType.
Definition zmod_quot_class zqT : zmod_quot_class_of zqT :=
let: ZmodQuotTypePack _ cT _ as qT' := zqT return zmod_quot_class_of qT' in cT.
Definition zmod_eq_quot_class zqT (zqc : zmod_quot_class_of zqT) :
eq_quot_class_of eqT zqT := EqQuotClass zqc.
Canonical zmodQuotType_eqType zqT := Equality.Pack (zmod_quot_class zqT) zqT.
Canonical zmodQuotType_choiceType zqT :=
Choice.Pack (zmod_quot_class zqT) zqT.
Canonical zmodQuotType_zmodType zqT :=
GRing.Zmodule.Pack (zmod_quot_class zqT) zqT.
Canonical zmodQuotType_quotType zqT := QuotTypePack (zmod_quot_class zqT) zqT.
Canonical zmodQuotType_eqQuotType zqT := EqQuotTypePack
(zmod_eq_quot_class (zmod_quot_class zqT)) zqT.
Coercion zmodQuotType_eqType : zmodQuotType >-> eqType.
Coercion zmodQuotType_choiceType : zmodQuotType >-> choiceType.
Coercion zmodQuotType_zmodType : zmodQuotType >-> zmodType.
Coercion zmodQuotType_quotType : zmodQuotType >-> quotType.
Coercion zmodQuotType_eqQuotType : zmodQuotType >-> eqQuotType.
Definition ZmodQuotType_pack Q :=
fun (qT : quotType T) (zT : zmodType) qc zc
of phant_id (quot_class qT) qc & phant_id (GRing.Zmodule.class zT) zc ⇒
fun m ⇒ ZmodQuotTypePack (@ZmodQuotClass Q qc zc m) Q.
Definition ZmodQuotMixin_pack Q :=
fun (qT : eqQuotType eqT) (qc : eq_quot_class_of eqT Q)
of phant_id (eq_quot_class qT) qc ⇒
fun (zT : zmodType) zc of phant_id (GRing.Zmodule.class zT) zc ⇒
fun e m0 mN mD ⇒ @ZmodQuotMixinPack Q qc zc e m0 mN mD.
Definition ZmodQuotType_clone (Q : Type) qT cT
of phant_id (zmod_quot_class qT) cT := @ZmodQuotTypePack Q cT Q.
Lemma zmod_quot_mixinP zqT :
zmod_quot_mixin_of (zmod_quot_class zqT) (zmod_quot_class zqT).
Lemma pi_zeror zqT : \pi_zqT zeroT = 0.
Lemma pi_oppr zqT : {morph \pi_zqT : x / oppT x >-> - x}.
Lemma pi_addr zqT : {morph \pi_zqT : x y / addT x y >-> x + y}.
Canonical pi_zero_quot_morph zqT := PiMorph (pi_zeror zqT).
Canonical pi_opp_quot_morph zqT := PiMorph1 (pi_oppr zqT).
Canonical pi_add_quot_morph zqT := PiMorph2 (pi_addr zqT).
End ZmodQuot.
Notation ZmodQuotType z o a Q m :=
(@ZmodQuotType_pack _ _ z o a Q _ _ _ _ id id m).
Notation "[ 'zmodQuotType' z , o & a 'of' Q ]" :=
(@ZmodQuotType_clone _ _ z o a Q _ _ id)
(at level 0, format "[ 'zmodQuotType' z , o & a 'of' Q ]") : form_scope.
Notation ZmodQuotMixin Q m0 mN mD :=
(@ZmodQuotMixin_pack _ _ _ _ _ Q _ _ id _ _ id (pi_eq_quot _) m0 mN mD).
Section PiAdditive.
Variables (V : zmodType) (equivV : rel V) (zeroV : V).
Variable Q : @zmodQuotType V equivV zeroV -%R +%R.
Lemma pi_is_additive : additive \pi_Q.
Canonical pi_additive := Additive pi_is_additive.
End PiAdditive.
Section RingQuot.
Variable (T : Type).
Variable eqT : rel T.
Variables (zeroT : T) (oppT : T → T) (addT : T → T → T).
Variables (oneT : T) (mulT : T → T → T).
Record ring_quot_mixin_of (Q : Type) (qc : quot_class_of T Q)
(rc : GRing.Ring.class_of Q) := RingQuotMixinPack {
ring_zmod_quot_mixin :> zmod_quot_mixin_of eqT zeroT oppT addT qc rc;
_ : \pi_(QuotTypePack qc Q) oneT = 1 :> GRing.Ring.Pack rc Q;
_ : {morph \pi_(QuotTypePack qc Q) : x y /
mulT x y >-> @GRing.mul (GRing.Ring.Pack rc Q) x y}
}.
Record ring_quot_class_of (Q : Type) : Type := RingQuotClass {
ring_quot_quot_class :> quot_class_of T Q;
ring_quot_ring_class :> GRing.Ring.class_of Q;
ring_quot_mixin :> ring_quot_mixin_of
ring_quot_quot_class ring_quot_ring_class
}.
Structure ringQuotType : Type := RingQuotTypePack {
ring_quot_sort :> Type;
_ : ring_quot_class_of ring_quot_sort;
_ : Type
}.
Implicit Type rqT : ringQuotType.
Definition ring_quot_class rqT : ring_quot_class_of rqT :=
let: RingQuotTypePack _ cT _ as qT' := rqT return ring_quot_class_of qT' in cT.
Definition ring_zmod_quot_class rqT (rqc : ring_quot_class_of rqT) :
zmod_quot_class_of eqT zeroT oppT addT rqT := ZmodQuotClass rqc.
Definition ring_eq_quot_class rqT (rqc : ring_quot_class_of rqT) :
eq_quot_class_of eqT rqT := EqQuotClass rqc.
Canonical ringQuotType_eqType rqT := Equality.Pack (ring_quot_class rqT) rqT.
Canonical ringQuotType_choiceType rqT := Choice.Pack (ring_quot_class rqT) rqT.
Canonical ringQuotType_zmodType rqT :=
GRing.Zmodule.Pack (ring_quot_class rqT) rqT.
Canonical ringQuotType_ringType rqT :=
GRing.Ring.Pack (ring_quot_class rqT) rqT.
Canonical ringQuotType_quotType rqT := QuotTypePack (ring_quot_class rqT) rqT.
Canonical ringQuotType_eqQuotType rqT :=
EqQuotTypePack (ring_eq_quot_class (ring_quot_class rqT)) rqT.
Canonical ringQuotType_zmodQuotType rqT :=
ZmodQuotTypePack (ring_zmod_quot_class (ring_quot_class rqT)) rqT.
Coercion ringQuotType_eqType : ringQuotType >-> eqType.
Coercion ringQuotType_choiceType : ringQuotType >-> choiceType.
Coercion ringQuotType_zmodType : ringQuotType >-> zmodType.
Coercion ringQuotType_ringType : ringQuotType >-> ringType.
Coercion ringQuotType_quotType : ringQuotType >-> quotType.
Coercion ringQuotType_eqQuotType : ringQuotType >-> eqQuotType.
Coercion ringQuotType_zmodQuotType : ringQuotType >-> zmodQuotType.
Definition RingQuotType_pack Q :=
fun (qT : quotType T) (zT : ringType) qc rc
of phant_id (quot_class qT) qc & phant_id (GRing.Ring.class zT) rc ⇒
fun m ⇒ RingQuotTypePack (@RingQuotClass Q qc rc m) Q.
Definition RingQuotMixin_pack Q :=
fun (qT : zmodQuotType eqT zeroT oppT addT) ⇒
fun (qc : zmod_quot_class_of eqT zeroT oppT addT Q)
of phant_id (zmod_quot_class qT) qc ⇒
fun (rT : ringType) rc of phant_id (GRing.Ring.class rT) rc ⇒
fun mZ m1 mM ⇒ @RingQuotMixinPack Q qc rc mZ m1 mM.
Definition RingQuotType_clone (Q : Type) qT cT
of phant_id (ring_quot_class qT) cT := @RingQuotTypePack Q cT Q.
Lemma ring_quot_mixinP rqT :
ring_quot_mixin_of (ring_quot_class rqT) (ring_quot_class rqT).
Lemma pi_oner rqT : \pi_rqT oneT = 1.
Lemma pi_mulr rqT : {morph \pi_rqT : x y / mulT x y >-> x × y}.
Canonical pi_one_quot_morph rqT := PiMorph (pi_oner rqT).
Canonical pi_mul_quot_morph rqT := PiMorph2 (pi_mulr rqT).
End RingQuot.
Notation RingQuotType o mul Q mix :=
(@RingQuotType_pack _ _ _ _ _ o mul Q _ _ _ _ id id mix).
Notation "[ 'ringQuotType' o & m 'of' Q ]" :=
(@RingQuotType_clone _ _ _ _ _ o m Q _ _ id)
(at level 0, format "[ 'ringQuotType' o & m 'of' Q ]") : form_scope.
Notation RingQuotMixin Q m1 mM :=
(@RingQuotMixin_pack _ _ _ _ _ _ _ Q _ _ id _ _ id (zmod_quot_mixinP _) m1 mM).
Section PiRMorphism.
Variables (R : ringType) (equivR : rel R) (zeroR : R).
Variable Q : @ringQuotType R equivR zeroR -%R +%R 1 *%R.
Lemma pi_is_multiplicative : multiplicative \pi_Q.
Canonical pi_rmorphism := AddRMorphism pi_is_multiplicative.
End PiRMorphism.
Section UnitRingQuot.
Variable (T : Type).
Variable eqT : rel T.
Variables (zeroT : T) (oppT : T → T) (addT : T → T → T).
Variables (oneT : T) (mulT : T → T → T).
Variables (unitT : pred T) (invT : T → T).
Record unit_ring_quot_mixin_of (Q : Type) (qc : quot_class_of T Q)
(rc : GRing.UnitRing.class_of Q) := UnitRingQuotMixinPack {
unit_ring_zmod_quot_mixin :>
ring_quot_mixin_of eqT zeroT oppT addT oneT mulT qc rc;
_ : {mono \pi_(QuotTypePack qc Q) : x /
unitT x >-> x \in @GRing.unit (GRing.UnitRing.Pack rc Q)};
_ : {morph \pi_(QuotTypePack qc Q) : x /
invT x >-> @GRing.inv (GRing.UnitRing.Pack rc Q) x}
}.
Record unit_ring_quot_class_of (Q : Type) : Type := UnitRingQuotClass {
unit_ring_quot_quot_class :> quot_class_of T Q;
unit_ring_quot_ring_class :> GRing.UnitRing.class_of Q;
unit_ring_quot_mixin :> unit_ring_quot_mixin_of
unit_ring_quot_quot_class unit_ring_quot_ring_class
}.
Structure unitRingQuotType : Type := UnitRingQuotTypePack {
unit_ring_quot_sort :> Type;
_ : unit_ring_quot_class_of unit_ring_quot_sort;
_ : Type
}.
Implicit Type rqT : unitRingQuotType.
Definition unit_ring_quot_class rqT : unit_ring_quot_class_of rqT :=
let: UnitRingQuotTypePack _ cT _ as qT' := rqT
return unit_ring_quot_class_of qT' in cT.
Definition unit_ring_ring_quot_class rqT (rqc : unit_ring_quot_class_of rqT) :
ring_quot_class_of eqT zeroT oppT addT oneT mulT rqT := RingQuotClass rqc.
Definition unit_ring_zmod_quot_class rqT (rqc : unit_ring_quot_class_of rqT) :
zmod_quot_class_of eqT zeroT oppT addT rqT := ZmodQuotClass rqc.
Definition unit_ring_eq_quot_class rqT (rqc : unit_ring_quot_class_of rqT) :
eq_quot_class_of eqT rqT := EqQuotClass rqc.
Canonical unitRingQuotType_eqType rqT :=
Equality.Pack (unit_ring_quot_class rqT) rqT.
Canonical unitRingQuotType_choiceType rqT :=
Choice.Pack (unit_ring_quot_class rqT) rqT.
Canonical unitRingQuotType_zmodType rqT :=
GRing.Zmodule.Pack (unit_ring_quot_class rqT) rqT.
Canonical unitRingQuotType_ringType rqT :=
GRing.Ring.Pack (unit_ring_quot_class rqT) rqT.
Canonical unitRingQuotType_unitRingType rqT :=
GRing.UnitRing.Pack (unit_ring_quot_class rqT) rqT.
Canonical unitRingQuotType_quotType rqT :=
QuotTypePack (unit_ring_quot_class rqT) rqT.
Canonical unitRingQuotType_eqQuotType rqT :=
EqQuotTypePack (unit_ring_eq_quot_class (unit_ring_quot_class rqT)) rqT.
Canonical unitRingQuotType_zmodQuotType rqT :=
ZmodQuotTypePack (unit_ring_zmod_quot_class (unit_ring_quot_class rqT)) rqT.
Canonical unitRingQuotType_ringQuotType rqT :=
RingQuotTypePack (unit_ring_ring_quot_class (unit_ring_quot_class rqT)) rqT.
Coercion unitRingQuotType_eqType : unitRingQuotType >-> eqType.
Coercion unitRingQuotType_choiceType : unitRingQuotType >-> choiceType.
Coercion unitRingQuotType_zmodType : unitRingQuotType >-> zmodType.
Coercion unitRingQuotType_ringType : unitRingQuotType >-> ringType.
Coercion unitRingQuotType_unitRingType : unitRingQuotType >-> unitRingType.
Coercion unitRingQuotType_quotType : unitRingQuotType >-> quotType.
Coercion unitRingQuotType_eqQuotType : unitRingQuotType >-> eqQuotType.
Coercion unitRingQuotType_zmodQuotType : unitRingQuotType >-> zmodQuotType.
Coercion unitRingQuotType_ringQuotType : unitRingQuotType >-> ringQuotType.
Definition UnitRingQuotType_pack Q :=
fun (qT : quotType T) (rT : unitRingType) qc rc
of phant_id (quot_class qT) qc & phant_id (GRing.UnitRing.class rT) rc ⇒
fun m ⇒ UnitRingQuotTypePack (@UnitRingQuotClass Q qc rc m) Q.
Definition UnitRingQuotMixin_pack Q :=
fun (qT : ringQuotType eqT zeroT oppT addT oneT mulT) ⇒
fun (qc : ring_quot_class_of eqT zeroT oppT addT oneT mulT Q)
of phant_id (zmod_quot_class qT) qc ⇒
fun (rT : unitRingType) rc of phant_id (GRing.UnitRing.class rT) rc ⇒
fun mR mU mV ⇒ @UnitRingQuotMixinPack Q qc rc mR mU mV.
Definition UnitRingQuotType_clone (Q : Type) qT cT
of phant_id (unit_ring_quot_class qT) cT := @UnitRingQuotTypePack Q cT Q.
Lemma unit_ring_quot_mixinP rqT :
unit_ring_quot_mixin_of (unit_ring_quot_class rqT) (unit_ring_quot_class rqT).
Lemma pi_unitr rqT : {mono \pi_rqT : x / unitT x >-> x \in GRing.unit}.
Lemma pi_invr rqT : {morph \pi_rqT : x / invT x >-> x^-1}.
Canonical pi_unit_quot_morph rqT := PiMono1 (pi_unitr rqT).
Canonical pi_inv_quot_morph rqT := PiMorph1 (pi_invr rqT).
End UnitRingQuot.
Notation UnitRingQuotType u i Q mix :=
(@UnitRingQuotType_pack _ _ _ _ _ _ _ u i Q _ _ _ _ id id mix).
Notation "[ 'unitRingQuotType' u & i 'of' Q ]" :=
(@UnitRingQuotType_clone _ _ _ _ _ _ _ u i Q _ _ id)
(at level 0, format "[ 'unitRingQuotType' u & i 'of' Q ]") : form_scope.
Notation UnitRingQuotMixin Q mU mV :=
(@UnitRingQuotMixin_pack _ _ _ _ _ _ _ _ _ Q
_ _ id _ _ id (zmod_quot_mixinP _) mU mV).
Section IdealDef.
Definition proper_ideal (R : ringType) (S : predPredType R) : Prop :=
1 \notin S ∧ ∀ a, {in S, ∀ u, a × u \in S}.
Definition prime_idealr_closed (R : ringType) (S : predPredType R) : Prop :=
∀ u v, u × v \in S → (u \in S) || (v \in S).
Definition idealr_closed (R : ringType) (S : predPredType R) :=
[/\ 0 \in S, 1 \notin S & ∀ a, {in S &, ∀ u v, a × u + v \in S}].
Lemma idealr_closed_nontrivial R S : @idealr_closed R S → proper_ideal S.
Lemma idealr_closedB R S : @idealr_closed R S → zmod_closed S.
Coercion idealr_closedB : idealr_closed >-> zmod_closed.
Coercion idealr_closed_nontrivial : idealr_closed >-> proper_ideal.
Structure idealr (R : ringType) (S : predPredType R) := MkIdeal {
idealr_zmod :> zmodPred S;
_ : proper_ideal S
}.
Structure prime_idealr (R : ringType) (S : predPredType R) := MkPrimeIdeal {
prime_idealr_zmod :> idealr S;
_ : prime_idealr_closed S
}.
Definition Idealr (R : ringType) (I : predPredType R) (zmodI : zmodPred I)
(kI : keyed_pred zmodI) : proper_ideal I → idealr I.
Section IdealTheory.
Variables (R : ringType) (I : predPredType R)
(idealrI : idealr I) (kI : keyed_pred idealrI).
Lemma idealr1 : 1 \in kI = false.
Lemma idealMr a u : u \in kI → a × u \in kI.
Lemma idealr0 : 0 \in kI.
End IdealTheory.
Section PrimeIdealTheory.
Variables (R : comRingType) (I : predPredType R)
(pidealrI : prime_idealr I) (kI : keyed_pred pidealrI).
Lemma prime_idealrM u v : (u × v \in kI) = (u \in kI) || (v \in kI).
End PrimeIdealTheory.
End IdealDef.
Module Quotient.
Section ZmodQuotient.
Variables (R : zmodType) (I : predPredType R)
(zmodI : zmodPred I) (kI : keyed_pred zmodI).
Definition equiv (x y : R) := (x - y) \in kI.
Lemma equivE x y : (equiv x y) = (x - y \in kI).
Lemma equiv_is_equiv : equiv_class_of equiv.
Canonical equiv_equiv := EquivRelPack equiv_is_equiv.
Canonical equiv_encModRel := defaultEncModRel equiv.
Definition type := {eq_quot equiv}.
Definition type_of of phant R := type.
Canonical rquot_quotType := [quotType of type].
Canonical rquot_eqType := [eqType of type].
Canonical rquot_choiceType := [choiceType of type].
Canonical rquot_eqQuotType := [eqQuotType equiv of type].
Lemma idealrBE x y : (x - y) \in kI = (x == y %[mod type]).
Lemma idealrDE x y : (x + y) \in kI = (x == - y %[mod type]).
Definition zero : type := lift_cst type 0.
Definition add := lift_op2 type +%R.
Definition opp := lift_op1 type -%R.
Canonical pi_zero_morph := PiConst zero.
Lemma pi_opp : {morph \pi : x / - x >-> opp x}.
Canonical pi_opp_morph := PiMorph1 pi_opp.
Lemma pi_add : {morph \pi : x y / x + y >-> add x y}.
Canonical pi_add_morph := PiMorph2 pi_add.
Lemma addqA: associative add.
Lemma addqC: commutative add.
Lemma add0q: left_id zero add.
Lemma addNq: left_inverse zero opp add.
Definition rquot_zmodMixin := ZmodMixin addqA addqC add0q addNq.
Canonical rquot_zmodType := Eval hnf in ZmodType type rquot_zmodMixin.
Definition rquot_zmodQuotMixin := ZmodQuotMixin type (lock _) pi_opp pi_add.
Canonical rquot_zmodQuotType := ZmodQuotType 0 -%R +%R type rquot_zmodQuotMixin.
End ZmodQuotient.
Notation "{quot I }" := (@type_of _ _ _ I (Phant _)).
Section RingQuotient.
Variables (R : comRingType) (I : predPredType R)
(idealI : idealr I) (kI : keyed_pred idealI).
Definition one: type := lift_cst type 1.
Definition mul := lift_op2 type *%R.
Canonical pi_one_morph := PiConst one.
Lemma pi_mul: {morph \pi : x y / x × y >-> mul x y}.
Canonical pi_mul_morph := PiMorph2 pi_mul.
Lemma mulqA: associative mul.
Lemma mulqC: commutative mul.
Lemma mul1q: left_id one mul.
Lemma mulq_addl: left_distributive mul +%R.
Lemma nonzero1q: one != 0.
Definition rquot_comRingMixin :=
ComRingMixin mulqA mulqC mul1q mulq_addl nonzero1q.
Canonical rquot_ringType := Eval hnf in RingType type rquot_comRingMixin.
Canonical rquot_comRingType := Eval hnf in ComRingType type mulqC.
Definition rquot_ringQuotMixin := RingQuotMixin type (lock _) pi_mul.
Canonical rquot_ringQuotType := RingQuotType 1 *%R type rquot_ringQuotMixin.
End RingQuotient.
Section IDomainQuotient.
Variables (R : comRingType) (I : predPredType R)
(pidealI : prime_idealr I) (kI : keyed_pred pidealI).
Lemma rquot_IdomainAxiom (x y : {quot kI}): x × y = 0 → (x == 0) || (y == 0).
End IDomainQuotient.
End Quotient.
Notation "{ideal_quot I }" := (@Quotient.type_of _ _ _ I (Phant _)).
Notation "x == y %[mod_ideal I ]" :=
(x == y %[mod {ideal_quot I}]) : quotient_scope.
Notation "x = y %[mod_ideal I ]" :=
(x = y %[mod {ideal_quot I}]) : quotient_scope.
Notation "x != y %[mod_ideal I ]" :=
(x != y %[mod {ideal_quot I}]) : quotient_scope.
Notation "x <> y %[mod_ideal I ]" :=
(x ≠ y %[mod {ideal_quot I}]) : quotient_scope.
Canonical Quotient.rquot_eqType.
Canonical Quotient.rquot_choiceType.
Canonical Quotient.rquot_zmodType.
Canonical Quotient.rquot_ringType.
Canonical Quotient.rquot_quotType.
Canonical Quotient.rquot_eqQuotType.
Canonical Quotient.rquot_zmodQuotType.
Canonical Quotient.rquot_ringQuotType.