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Move quotient group (#66)
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Patrick Stevens
GitHub
@@ -73,68 +73,6 @@ quotientGroupSetoid {A = A} {S = S} {T = T} {_·A_ = _·A_} {_·B_ = _·B_} G {H
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k : f (m ·A ((Group.0G G) ·A Group.inverse G o)) ∼ 0G
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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
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quotientGroup : {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_
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Group.+WellDefined (quotientGroup {S = S} {T = T} {_·A_ = _·A_} {_·B_ = _·B_} G {H = H} {underf = f} fHom) {x} {y} {m} {n} x~m y~n = ans
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where
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open Setoid T
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open Equivalence (Setoid.eq T)
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p1 : f ((x ·A y) ·A (Group.inverse G (m ·A n))) ∼ f ((x ·A y) ·A ((Group.inverse G n) ·A (Group.inverse G m)))
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p2 : f ((x ·A y) ·A ((Group.inverse G n) ·A (Group.inverse G m))) ∼ f (x ·A ((y ·A (Group.inverse G n)) ·A (Group.inverse G m)))
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p3 : f (x ·A ((y ·A (Group.inverse G n)) ·A (Group.inverse G m))) ∼ (f x) ·B f ((y ·A (Group.inverse G n)) ·A (Group.inverse G m))
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p4 : (f x) ·B f ((y ·A (Group.inverse G n)) ·A (Group.inverse G m)) ∼ (f x) ·B (f (y ·A (Group.inverse G n)) ·B f (Group.inverse G m))
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p5 : (f x) ·B (f (y ·A (Group.inverse G n)) ·B f (Group.inverse G m)) ∼ (f x) ·B (Group.0G H ·B f (Group.inverse G m))
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p6 : (f x) ·B (Group.0G H ·B f (Group.inverse G m)) ∼ (f x) ·B f (Group.inverse G m)
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p7 : (f x) ·B f (Group.inverse G m) ∼ f (x ·A (Group.inverse G m))
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p8 : f (x ·A (Group.inverse G m)) ∼ Group.0G H
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p1 = GroupHom.wellDefined fHom (Group.+WellDefined G (Equivalence.reflexive (Setoid.eq S)) (invContravariant G))
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p2 = GroupHom.wellDefined fHom (Equivalence.symmetric (Setoid.eq S) (fourWay+Associative G))
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p3 = GroupHom.groupHom fHom
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p4 = Group.+WellDefined H reflexive (GroupHom.groupHom fHom)
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p5 = Group.+WellDefined H reflexive (replaceGroupOp H (symmetric y~n) reflexive reflexive reflexive reflexive)
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p6 = Group.+WellDefined H reflexive (Group.identLeft H)
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p7 = symmetric (GroupHom.groupHom fHom)
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p8 = x~m
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ans : f ((x ·A y) ·A (Group.inverse G (m ·A n))) ∼ Group.0G H
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ans = transitive p1 (transitive p2 (transitive p3 (transitive p4 (transitive p5 (transitive p6 (transitive p7 p8))))))
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Group.0G (quotientGroup G fHom) = Group.0G G
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Group.inverse (quotientGroup G fHom) = Group.inverse G
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Group.+Associative (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {a} {b} {c} = ans
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where
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open Setoid T
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open Equivalence (Setoid.eq T)
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ans : f ((a ·A (b ·A c)) ·A (Group.inverse G ((a ·A b) ·A c))) ∼ Group.0G H
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ans = transitive (GroupHom.wellDefined fHom (transferToRight'' G (Group.+Associative G))) (imageOfIdentityIsIdentity fHom)
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Group.identRight (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {a} = ans
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where
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open Group G
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open Setoid T
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open Equivalence eq
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transitiveG = Equivalence.transitive (Setoid.eq S)
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ans : f ((a ·A 0G) ·A inverse a) ∼ Group.0G H
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ans = transitive (GroupHom.wellDefined fHom (transitiveG (Group.+WellDefined G (Group.identRight G) (Equivalence.reflexive (Setoid.eq S))) (Group.invRight G))) (imageOfIdentityIsIdentity fHom)
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Group.identLeft (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {a} = ans
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where
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open Group G
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open Setoid T
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open Equivalence eq
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transitiveG = Equivalence.transitive (Setoid.eq S)
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ans : f ((0G ·A a) ·A (inverse a)) ∼ Group.0G H
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ans = transitive (GroupHom.wellDefined fHom (transitiveG (Group.+WellDefined G (Group.identLeft G) (Equivalence.reflexive (Setoid.eq S))) (Group.invRight G))) (imageOfIdentityIsIdentity fHom)
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Group.invLeft (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {x} = ans
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where
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open Group G
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open Setoid T
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open Equivalence eq
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ans : f ((inverse x ·A x) ·A (inverse 0G)) ∼ (Group.0G H)
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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)
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Group.invRight (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {x} = ans
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where
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open Group G
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open Setoid T
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open Equivalence eq
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ans : f ((x ·A inverse x) ·A (inverse 0G)) ∼ (Group.0G H)
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ans = transitive (GroupHom.wellDefined fHom (Equivalence.transitive (Setoid.eq S) (replaceGroupOp G (Equivalence.symmetric (Setoid.eq S) (Group.invRight 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)
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{-
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quotientHom : {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) → A → A
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quotientHom {S = S} {T = T} {_·A_ = _·A_} {_·B_ = _·B_} G {f = f} fHom a = {!!}
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@@ -16,6 +16,7 @@ open import Groups.Isomorphisms.Definition
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open import Groups.Subgroups.Definition
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open import Groups.Lemmas
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open import Groups.Abelian.Definition
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open import Groups.QuotientGroup.Definition
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open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
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80
Groups/QuotientGroup/Definition.agda
Normal file
80
Groups/QuotientGroup/Definition.agda
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@@ -0,0 +1,80 @@
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{-# OPTIONS --warning=error --safe --without-K #-}
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open import Functions
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open import Sets.FinSet
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open import LogicalFormulae
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open import Groups.Definition
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open import Groups.Groups
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open import Groups.FiniteGroups.Definition
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open import Groups.Homomorphisms.Definition
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open import Groups.Abelian.Definition
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open import Setoids.Setoids
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open import Rings.Definition
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open import Fields.FieldOfFractions.Setoid
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open import Sets.EquivalenceRelations
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open import Groups.Lemmas
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open import Groups.Homomorphisms.Lemmas
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module Groups.QuotientGroup.Definition where
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quotientGroup : {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_
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Group.+WellDefined (quotientGroup {S = S} {T = T} {_·A_ = _·A_} {_·B_ = _·B_} G {H = H} {underf = f} fHom) {x} {y} {m} {n} x~m y~n = ans
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where
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open Setoid T
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open Equivalence (Setoid.eq T)
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p1 : f ((x ·A y) ·A (Group.inverse G (m ·A n))) ∼ f ((x ·A y) ·A ((Group.inverse G n) ·A (Group.inverse G m)))
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p2 : f ((x ·A y) ·A ((Group.inverse G n) ·A (Group.inverse G m))) ∼ f (x ·A ((y ·A (Group.inverse G n)) ·A (Group.inverse G m)))
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p3 : f (x ·A ((y ·A (Group.inverse G n)) ·A (Group.inverse G m))) ∼ (f x) ·B f ((y ·A (Group.inverse G n)) ·A (Group.inverse G m))
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p4 : (f x) ·B f ((y ·A (Group.inverse G n)) ·A (Group.inverse G m)) ∼ (f x) ·B (f (y ·A (Group.inverse G n)) ·B f (Group.inverse G m))
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p5 : (f x) ·B (f (y ·A (Group.inverse G n)) ·B f (Group.inverse G m)) ∼ (f x) ·B (Group.0G H ·B f (Group.inverse G m))
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p6 : (f x) ·B (Group.0G H ·B f (Group.inverse G m)) ∼ (f x) ·B f (Group.inverse G m)
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p7 : (f x) ·B f (Group.inverse G m) ∼ f (x ·A (Group.inverse G m))
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p8 : f (x ·A (Group.inverse G m)) ∼ Group.0G H
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p1 = GroupHom.wellDefined fHom (Group.+WellDefined G (Equivalence.reflexive (Setoid.eq S)) (invContravariant G))
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p2 = GroupHom.wellDefined fHom (Equivalence.symmetric (Setoid.eq S) (fourWay+Associative G))
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p3 = GroupHom.groupHom fHom
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p4 = Group.+WellDefined H reflexive (GroupHom.groupHom fHom)
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p5 = Group.+WellDefined H reflexive (replaceGroupOp H (symmetric y~n) reflexive reflexive reflexive reflexive)
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p6 = Group.+WellDefined H reflexive (Group.identLeft H)
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p7 = symmetric (GroupHom.groupHom fHom)
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p8 = x~m
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ans : f ((x ·A y) ·A (Group.inverse G (m ·A n))) ∼ Group.0G H
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ans = transitive p1 (transitive p2 (transitive p3 (transitive p4 (transitive p5 (transitive p6 (transitive p7 p8))))))
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Group.0G (quotientGroup G fHom) = Group.0G G
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Group.inverse (quotientGroup G fHom) = Group.inverse G
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Group.+Associative (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {a} {b} {c} = ans
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where
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open Setoid T
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open Equivalence (Setoid.eq T)
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ans : f ((a ·A (b ·A c)) ·A (Group.inverse G ((a ·A b) ·A c))) ∼ Group.0G H
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ans = transitive (GroupHom.wellDefined fHom (transferToRight'' G (Group.+Associative G))) (imageOfIdentityIsIdentity fHom)
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Group.identRight (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {a} = ans
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where
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open Group G
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open Setoid T
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open Equivalence eq
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transitiveG = Equivalence.transitive (Setoid.eq S)
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ans : f ((a ·A 0G) ·A inverse a) ∼ Group.0G H
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ans = transitive (GroupHom.wellDefined fHom (transitiveG (Group.+WellDefined G (Group.identRight G) (Equivalence.reflexive (Setoid.eq S))) (Group.invRight G))) (imageOfIdentityIsIdentity fHom)
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Group.identLeft (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {a} = ans
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where
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open Group G
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open Setoid T
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open Equivalence eq
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transitiveG = Equivalence.transitive (Setoid.eq S)
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ans : f ((0G ·A a) ·A (inverse a)) ∼ Group.0G H
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ans = transitive (GroupHom.wellDefined fHom (transitiveG (Group.+WellDefined G (Group.identLeft G) (Equivalence.reflexive (Setoid.eq S))) (Group.invRight G))) (imageOfIdentityIsIdentity fHom)
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Group.invLeft (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {x} = ans
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where
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open Group G
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open Setoid T
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open Equivalence eq
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ans : f ((inverse x ·A x) ·A (inverse 0G)) ∼ (Group.0G H)
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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)
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Group.invRight (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {x} = ans
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where
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open Group G
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open Setoid T
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open Equivalence eq
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ans : f ((x ·A inverse x) ·A (inverse 0G)) ∼ (Group.0G H)
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ans = transitive (GroupHom.wellDefined fHom (Equivalence.transitive (Setoid.eq S) (replaceGroupOp G (Equivalence.symmetric (Setoid.eq S) (Group.invRight 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)
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