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Quotient ring (#81)

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Patrick Stevens
2019-11-21 07:28:25 +00:00
committed by GitHub
parent d4f51a3efe
commit b33baa5fb7
12 changed files with 125 additions and 64 deletions
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{-# OPTIONS --warning=error --safe --without-K #-}
open import Functions
open import Sets.FinSet
open import LogicalFormulae
open import Groups.Definition
open import Groups.Groups
open import Groups.FiniteGroups.Definition
open import Rings.Homomorphisms.Definition
open import Groups.Homomorphisms.Definition
open import Groups.Abelian.Definition
open import Setoids.Setoids
open import Rings.Definition
open import Fields.FieldOfFractions.Setoid
open import Sets.EquivalenceRelations
open import Groups.Lemmas
open import Groups.Subgroups.Definition
open import Groups.QuotientGroup.Definition
open import Rings.Ideals.Definition
module Rings.Cosets {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ _*_ : A → A → A} (R : Ring S _+_ _*_) {pred : A → Set c} (ideal : Ideal R pred) where
open Ring R
open import Rings.Lemmas R
open import Groups.Subgroups.Normal.Lemmas
open import Groups.Cosets additiveGroup (Ideal.isSubgroup ideal)
open Setoid S
open Equivalence eq
open Ideal ideal
open Group additiveGroup
cosetRing : Ring cosetSetoid _+_ _*_
Ring.additiveGroup cosetRing = cosetGroup (abelianGroupSubgroupIsNormal (Ideal.isSubgroup ideal) abelianUnderlyingGroup)
Ring.*WellDefined cosetRing {r} {s} {t} {u} r=t s=u = need
where
r=t' : pred ((inverse t + r) * u)
r=t' = accumulatesTimes r=t
s=u' : pred ((inverse u + s) * r)
s=u' = accumulatesTimes s=u
need : pred (inverse (t * u) + (r * s))
need = isSubset (transitive (+WellDefined (*DistributesOver+') *DistributesOver+') (transitive +Associative (+WellDefined (transitive (symmetric +Associative) (transitive (+WellDefined ringMinusExtracts' (transitive (+WellDefined reflexive (transitive ringMinusExtracts' (inverseWellDefined additiveGroup *Commutative))) invRight)) identRight)) *Commutative))) (closedUnderPlus r=t' s=u')
Ring.1R cosetRing = 1R
Ring.groupIsAbelian cosetRing = isSubset (transitive (symmetric invLeft) (+WellDefined (inverseWellDefined additiveGroup groupIsAbelian) reflexive)) containsIdentity
Ring.*Associative cosetRing = isSubset (transitive (symmetric invLeft) (+WellDefined (inverseWellDefined additiveGroup *Associative) reflexive)) containsIdentity
Ring.*Commutative cosetRing {a} {b} = isSubset (transitive (symmetric invLeft) (+WellDefined (inverseWellDefined additiveGroup *Commutative) reflexive)) containsIdentity
Ring.*DistributesOver+ cosetRing = isSubset (symmetric (transitive (+WellDefined (inverseWellDefined additiveGroup (symmetric *DistributesOver+)) reflexive) invLeft)) containsIdentity
Ring.identIsIdent cosetRing = isSubset (symmetric (transitive (Group.+WellDefined additiveGroup reflexive identIsIdent) invLeft)) containsIdentity