- -

Library mathcomp.algebra.ring_quotient

- -

-(* (c) Copyright 2006-2016 Microsoft Corporation and Inria.
- Distributed under the terms of CeCILL-B. *)
- -
-

- -

- 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 a 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 only - provide canonical structures of ring quotients - for commutative rings, in 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 ) zeroT = 0 :> GRing.Zmodule.Pack zc;
-    _ : {morph \pi_(QuotTypePack qc ) : x /
-         oppT x >-> @GRing.opp (GRing.Zmodule.Pack zc) x };
-    _ : {morph \pi_(QuotTypePack qc ) : x y /
-         addT x y >-> @GRing.add (GRing.Zmodule.Pack zc) 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;
-
-}.
- -
-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).
-Canonical zmodQuotType_choiceType zqT :=
-  Choice.Pack (zmod_quot_class zqT).
-Canonical zmodQuotType_zmodType zqT :=
-  GRing.Zmodule.Pack (zmod_quot_class zqT).
-Canonical zmodQuotType_quotType zqT := QuotTypePack (zmod_quot_class zqT).
-Canonical zmodQuotType_eqQuotType zqT := EqQuotTypePack
-  (zmod_eq_quot_class (zmod_quot_class 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).
- -
-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.
- -
-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 ) oneT = 1 :> GRing.Ring.Pack rc;
-    _ : {morph \pi_(QuotTypePack qc ) : x y /
-         mulT x y >-> @GRing.mul (GRing.Ring.Pack rc) 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;
-
-}.
- -
-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).
-Canonical ringQuotType_choiceType rqT := Choice.Pack (ring_quot_class rqT).
-Canonical ringQuotType_zmodType rqT :=
-  GRing.Zmodule.Pack (ring_quot_class rqT).
-Canonical ringQuotType_ringType rqT :=
-  GRing.Ring.Pack (ring_quot_class rqT).
-Canonical ringQuotType_quotType rqT := QuotTypePack (ring_quot_class rqT).
-Canonical ringQuotType_eqQuotType rqT :=
-  EqQuotTypePack (ring_eq_quot_class (ring_quot_class rqT)).
-Canonical ringQuotType_zmodQuotType rqT :=
-  ZmodQuotTypePack (ring_zmod_quot_class (ring_quot_class 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).
- -
-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.
- -
-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 ) : x /
-         unitT x >-> x \in @GRing.unit (GRing.UnitRing.Pack rc)};
-    _ : {morph \pi_(QuotTypePack qc ) : x /
-         invT x >-> @GRing.inv (GRing.UnitRing.Pack rc) 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;
-
-}.
- -
-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).
-Canonical unitRingQuotType_choiceType rqT :=
-  Choice.Pack (unit_ring_quot_class rqT).
-Canonical unitRingQuotType_zmodType rqT :=
-  GRing.Zmodule.Pack (unit_ring_quot_class rqT).
-Canonical unitRingQuotType_ringType rqT :=
-  GRing.Ring.Pack (unit_ring_quot_class rqT).
-Canonical unitRingQuotType_unitRingType rqT :=
-  GRing.UnitRing.Pack (unit_ring_quot_class rqT).
-Canonical unitRingQuotType_quotType rqT :=
-  QuotTypePack (unit_ring_quot_class rqT).
-Canonical unitRingQuotType_eqQuotType rqT :=
-  EqQuotTypePack (unit_ring_eq_quot_class (unit_ring_quot_class rqT)).
-Canonical unitRingQuotType_zmodQuotType rqT :=
-  ZmodQuotTypePack (unit_ring_zmod_quot_class (unit_ring_quot_class rqT)).
-Canonical unitRingQuotType_ringQuotType rqT :=
-  RingQuotTypePack (unit_ring_ring_quot_class (unit_ring_quot_class 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).
- -
-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.
- -
-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 : {pred R }) : Prop :=
-  1 \notin S ∧ ∀ a, {in S , ∀ u, a × u \in S }.
- -
-Definition prime_idealr_closed (R : ringType) (S : {pred R }) : Prop :=
-  ∀ u v, u × v \in S → (u \in S ) || (v \in S ).
- -
-Definition idealr_closed (R : ringType) (S : {pred 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 : {pred R }) := MkIdeal {
-  idealr_zmod :> zmodPred S;
-  _ : proper_ideal S
-}.
- -
-Structure prime_idealr (R : ringType) (S : {pred R }) := MkPrimeIdeal {
-  prime_idealr_zmod :> idealr S;
-  _ : prime_idealr_closed S
-}.
- -
-Definition Idealr (R : ringType) (I : {pred R }) (zmodI : zmodPred I)
-            (kI : keyed_pred zmodI) : proper_ideal I → idealr I.
- -
-Section IdealTheory.
-Variables (R : ringType) (I : {pred 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 : {pred 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 : {pred 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 : {pred 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 : {pred 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_comRingType.
-Canonical Quotient.rquot_quotType.
-Canonical Quotient.rquot_eqQuotType.
-Canonical Quotient.rquot_zmodQuotType.
-Canonical Quotient.rquot_ringQuotType.
-