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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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29
Groups/Cosets.agda
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@@ -10,21 +10,18 @@ open import Groups.Definition
open import Groups.Subgroups.Definition
open import Groups.Subgroups.Normal.Definition
module Groups.Cosets {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} (G : Group S _+_) {c : _} {pred : A → Set c} (subgrp : subgroup G pred) where
module Groups.Cosets {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} (G : Group S _+_) {c : _} {pred : A → Set c} (subgrp : Subgroup G pred) where
open Equivalence (Setoid.eq S)
open import Groups.Lemmas G
open _&_&_ (_&&_.snd subgrp)
open Group G
private
subs = _&&_.fst subgrp
open Subgroup subgrp
cosetSetoid : Setoid A
Setoid._∼_ cosetSetoid g h = pred ((Group.inverse G h) + g)
Equivalence.reflexive (Setoid.eq cosetSetoid) = subs (symmetric (Group.invLeft G)) two
Equivalence.symmetric (Setoid.eq cosetSetoid) yx = subs (transitive invContravariant (+WellDefined reflexive invInv)) (three yx)
Equivalence.transitive (Setoid.eq cosetSetoid) yx zy = subs (transitive +Associative (+WellDefined (transitive (symmetric +Associative) (transitive (+WellDefined reflexive invRight) identRight)) reflexive)) (one zy yx)
Equivalence.reflexive (Setoid.eq cosetSetoid) = isSubset (symmetric (Group.invLeft G)) containsIdentity
Equivalence.symmetric (Setoid.eq cosetSetoid) yx = isSubset (transitive invContravariant (+WellDefined reflexive invInv)) (closedUnderInverse yx)
Equivalence.transitive (Setoid.eq cosetSetoid) yx zy = isSubset (transitive +Associative (+WellDefined (transitive (symmetric +Associative) (transitive (+WellDefined reflexive invRight) identRight)) reflexive)) (closedUnderPlus zy yx)
cosetGroup : normalSubgroup G subgrp → Group cosetSetoid _+_
Group.+WellDefined (cosetGroup norm) {m} {n} {x} {y} m=x n=y = ans
@@ -34,17 +31,17 @@ Group.+WellDefined (cosetGroup norm) {m} {n} {x} {y} m=x n=y = ans
u : pred (inverse x + m)
u = m=x
v : pred (m + inverse x)
v = subs (+WellDefined reflexive (transitive (symmetric +Associative) (transitive (+WellDefined reflexive invRight) identRight))) (norm u)
v = isSubset (+WellDefined reflexive (transitive (symmetric +Associative) (transitive (+WellDefined reflexive invRight) identRight))) (norm u)
ans' : pred ((inverse y) + ((inverse x + m) + inverse (inverse y)))
ans' = norm u
ans'' : pred ((inverse y) + ((inverse x + m) + y))
ans'' = subs (+WellDefined reflexive (+WellDefined reflexive (invTwice y))) ans'
ans'' = isSubset (+WellDefined reflexive (+WellDefined reflexive (invTwice y))) ans'
ans : pred (inverse (x + y) + (m + n))
ans = subs (transitive (transitive +Associative (transitive (+WellDefined (transitive (symmetric +Associative) (transitive (+WellDefined reflexive (transitive (symmetric +Associative) (transitive (+WellDefined reflexive invRight) identRight))) +Associative)) reflexive) (symmetric +Associative))) (symmetric (+WellDefined invContravariant reflexive))) (one ans'' t)
ans = isSubset (transitive (transitive +Associative (transitive (+WellDefined (transitive (symmetric +Associative) (transitive (+WellDefined reflexive (transitive (symmetric +Associative) (transitive (+WellDefined reflexive invRight) identRight))) +Associative)) reflexive) (symmetric +Associative))) (symmetric (+WellDefined invContravariant reflexive))) (closedUnderPlus ans'' t)
Group.0G (cosetGroup norm) = 0G
Group.inverse (cosetGroup norm) = inverse
Group.+Associative (cosetGroup norm) {a} {b} {c} = subs (symmetric (transitive (+WellDefined (inverseWellDefined (symmetric +Associative)) reflexive) (invLeft {a + (b + c)}))) two
Group.identRight (cosetGroup norm) = subs (symmetric (transitive +Associative (transitive (+WellDefined invLeft reflexive) identRight))) two
Group.identLeft (cosetGroup norm) = subs (symmetric (transitive (+WellDefined reflexive identLeft) invLeft)) two
Group.invLeft (cosetGroup norm) = subs (symmetric (transitive (+WellDefined reflexive invLeft) invLeft)) two
Group.invRight (cosetGroup norm) = subs (symmetric (transitive (+WellDefined reflexive invRight) invLeft)) two
Group.+Associative (cosetGroup norm) {a} {b} {c} = isSubset (symmetric (transitive (+WellDefined (inverseWellDefined (symmetric +Associative)) reflexive) (invLeft {a + (b + c)}))) containsIdentity
Group.identRight (cosetGroup norm) = isSubset (symmetric (transitive +Associative (transitive (+WellDefined invLeft reflexive) identRight))) containsIdentity
Group.identLeft (cosetGroup norm) = isSubset (symmetric (transitive (+WellDefined reflexive identLeft) invLeft)) containsIdentity
Group.invLeft (cosetGroup norm) = isSubset (symmetric (transitive (+WellDefined reflexive invLeft) invLeft)) containsIdentity
Group.invRight (cosetGroup norm) = isSubset (symmetric (transitive (+WellDefined reflexive invRight) invLeft)) containsIdentity