patrick/agdaproofs
1
0
Fork 0
mirror of https://github.com/Smaug123/agdaproofs synced 2025-10-12 23:28:39 +00:00
Code Issues Packages Projects Releases Wiki Activity

Lots of rings (#82)

Browse Source
This commit is contained in:
Patrick Stevens
2019-11-22 19:52:57 +00:00
committed by GitHub
parent b33baa5fb7
commit 660d7aa27c
40 changed files with 1246 additions and 881 deletions
Download Patch File Download Diff File Expand all files Collapse all files

54
Groups/QuotientGroup/Definition.agda
View File

@@ -7,20 +7,48 @@ open import Groups.Definition
open import Groups.Groups
open import Groups.FiniteGroups.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.Homomorphisms.Lemmas
open import Groups.Subgroups.Definition
open import Groups.Subgroups.Normal.Definition
module Groups.QuotientGroup.Definition where
module Groups.QuotientGroup.Definition {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) {H : Group T _·B_} {f : A → B} (fHom : GroupHom G H f) where
quotientGroupByHom : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) {H : Group T _·B_} → {underf : A → B} → (f : GroupHom G H underf) → Group (quotientGroupSetoid G f) _·A_
Group.+WellDefined (quotientGroupByHom {S = S} {T = T} {_·A_ = _·A_} {_·B_ = _·B_} G {H = H} {underf = f} fHom) {x} {y} {m} {n} x~m y~n = ans
quotientGroupSetoid : (Setoid {a} {d} A)
quotientGroupSetoid = ansSetoid
where
open Setoid T
open Group H
open Equivalence eq
ansSetoid : Setoid A
Setoid._∼_ ansSetoid r s = (f (r ·A (Group.inverse G s))) ∼ 0G
Equivalence.reflexive (Setoid.eq ansSetoid) {b} = transitive (GroupHom.wellDefined fHom (Group.invRight G)) (imageOfIdentityIsIdentity fHom)
Equivalence.symmetric (Setoid.eq ansSetoid) {m} {n} pr = i
where
g : f (Group.inverse G (m ·A Group.inverse G n)) ∼ 0G
g = transitive (homRespectsInverse fHom {m ·A Group.inverse G n}) (transitive (inverseWellDefined H pr) (invIdent H))
h : f (Group.inverse G (Group.inverse G n) ·A Group.inverse G m) ∼ 0G
h = transitive (GroupHom.wellDefined fHom (Equivalence.symmetric (Setoid.eq S) (invContravariant G))) g
i : f (n ·A Group.inverse G m) ∼ 0G
i = transitive (GroupHom.wellDefined fHom (Group.+WellDefined G (Equivalence.symmetric (Setoid.eq S) (invTwice G n)) (Equivalence.reflexive (Setoid.eq S)))) h
Equivalence.transitive (Setoid.eq ansSetoid) {m} {n} {o} prmn prno = transitive (GroupHom.wellDefined fHom (Group.+WellDefined G (Equivalence.reflexive (Setoid.eq S)) (Equivalence.symmetric (Setoid.eq S) (Group.identLeft G)))) k
where
g : f (m ·A Group.inverse G n) ·B f (n ·A Group.inverse G o) ∼ 0G ·B 0G
g = replaceGroupOp H reflexive reflexive prmn prno reflexive
h : f (m ·A Group.inverse G n) ·B f (n ·A Group.inverse G o) ∼ 0G
h = transitive g identLeft
i : f ((m ·A Group.inverse G n) ·A (n ·A Group.inverse G o)) ∼ 0G
i = transitive (GroupHom.groupHom fHom) h
j : f (m ·A (((Group.inverse G n) ·A n) ·A Group.inverse G o)) ∼ 0G
j = transitive (GroupHom.wellDefined fHom (fourWay+Associative G)) i
k : f (m ·A ((Group.0G G) ·A Group.inverse G o)) ∼ 0G
k = transitive (GroupHom.wellDefined fHom (Group.+WellDefined G (Equivalence.reflexive (Setoid.eq S)) (Group.+WellDefined G (Equivalence.symmetric (Setoid.eq S) (Group.invLeft G)) (Equivalence.reflexive (Setoid.eq S))))) j
quotientGroupByHom : Group (quotientGroupSetoid) _·A_
Group.+WellDefined (quotientGroupByHom) {x} {y} {m} {n} x~m y~n = ans
where
open Setoid T
open Equivalence (Setoid.eq T)
@@ -42,15 +70,15 @@ Group.+WellDefined (quotientGroupByHom {S = S} {T = T} {_·A_ = _·A_} {_·B_ =
p8 = x~m
ans : f ((x ·A y) ·A (Group.inverse G (m ·A n))) ∼ Group.0G H
ans = transitive p1 (transitive p2 (transitive p3 (transitive p4 (transitive p5 (transitive p6 (transitive p7 p8))))))
Group.0G (quotientGroupByHom G fHom) = Group.0G G
Group.inverse (quotientGroupByHom G fHom) = Group.inverse G
Group.+Associative (quotientGroupByHom {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {a} {b} {c} = ans
Group.0G (quotientGroupByHom) = Group.0G G
Group.inverse (quotientGroupByHom) = Group.inverse G
Group.+Associative (quotientGroupByHom) {a} {b} {c} = ans
where
open Setoid T
open Equivalence (Setoid.eq T)
ans : f ((a ·A (b ·A c)) ·A (Group.inverse G ((a ·A b) ·A c))) ∼ Group.0G H
ans = transitive (GroupHom.wellDefined fHom (transferToRight'' G (Group.+Associative G))) (imageOfIdentityIsIdentity fHom)
Group.identRight (quotientGroupByHom {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {a} = ans
Group.identRight (quotientGroupByHom) {a} = ans
where
open Group G
open Setoid T
@@ -58,7 +86,7 @@ Group.identRight (quotientGroupByHom {S = S} {T = T} {_·A_ = _·A_} G {H = H} {
transitiveG = Equivalence.transitive (Setoid.eq S)
ans : f ((a ·A 0G) ·A inverse a) ∼ Group.0G H
ans = transitive (GroupHom.wellDefined fHom (transitiveG (Group.+WellDefined G (Group.identRight G) (Equivalence.reflexive (Setoid.eq S))) (Group.invRight G))) (imageOfIdentityIsIdentity fHom)
Group.identLeft (quotientGroupByHom {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {a} = ans
Group.identLeft (quotientGroupByHom) {a} = ans
where
open Group G
open Setoid T
@@ -66,14 +94,14 @@ Group.identLeft (quotientGroupByHom {S = S} {T = T} {_·A_ = _·A_} G {H = H} {u
transitiveG = Equivalence.transitive (Setoid.eq S)
ans : f ((0G ·A a) ·A (inverse a)) ∼ Group.0G H
ans = transitive (GroupHom.wellDefined fHom (transitiveG (Group.+WellDefined G (Group.identLeft G) (Equivalence.reflexive (Setoid.eq S))) (Group.invRight G))) (imageOfIdentityIsIdentity fHom)
Group.invLeft (quotientGroupByHom {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {x} = ans
Group.invLeft (quotientGroupByHom) {x} = ans
where
open Group G
open Setoid T
open Equivalence eq
ans : f ((inverse x ·A x) ·A (inverse 0G)) ∼ (Group.0G H)
ans = transitive (GroupHom.wellDefined fHom (Equivalence.transitive (Setoid.eq S) (replaceGroupOp G (Equivalence.symmetric (Setoid.eq S) (Group.invLeft G)) (Equivalence.symmetric (Setoid.eq S) (invIdent G)) (Equivalence.reflexive (Setoid.eq S)) ((Equivalence.reflexive (Setoid.eq S))) ((Equivalence.reflexive (Setoid.eq S)))) (identRight {0G}))) (imageOfIdentityIsIdentity fHom)
Group.invRight (quotientGroupByHom {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {x} = ans
Group.invRight (quotientGroupByHom) {x} = ans
where
open Group G
open Setoid T