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Restructure towards ideals

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Smaug123
2019-11-20 21:20:03 +00:00
parent f2f4e867fc
commit b03a5279bc
22 changed files with 417 additions and 296 deletions
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18
Groups/Homomorphisms/Examples.agda Normal file
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@@ -0,0 +1,18 @@
{-# OPTIONS --safe --warning=error --without-K #-}
open import LogicalFormulae
open import Setoids.Setoids
open import Functions
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
open import Numbers.Naturals.Naturals
open import Sets.FinSet
open import Groups.Definition
open import Sets.EquivalenceRelations
open import Groups.Homomorphisms.Definition
open import Groups.Lemmas
module Groups.Homomorphisms.Examples where
identityHom : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+A_ : A → A → A} (G : Group S _+A_) → GroupHom G G id
GroupHom.groupHom (identityHom {S = S} G) = Equivalence.reflexive (Setoid.eq S)
GroupHom.wellDefined (identityHom G) = id

51
Groups/Homomorphisms/Image.agda
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@@ -5,6 +5,7 @@ open import Groups.Definition
open import Orders
open import Numbers.Integers.Integers
open import Setoids.Setoids
open import Setoids.Subset
open import LogicalFormulae
open import Sets.FinSet
open import Functions
@@ -20,45 +21,25 @@ open import Groups.QuotientGroup.Definition
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
module Groups.Homomorphisms.Image where
module Groups.Homomorphisms.Image {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
data GroupHomImageElement {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) : Set (a ⊔ b ⊔ c ⊔ d) where
ofElt : (x : A) → GroupHomImageElement fHom
imageGroupPred : B → Set (a ⊔ d)
imageGroupPred b = Sg A (λ a → Setoid._∼_ T (f a) b)
imageGroupSetoid : {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) → Setoid (GroupHomImageElement fHom)
(imageGroupSetoid {T = T} {f = f} fHom Setoid.∼ ofElt b1) (ofElt b2) = Setoid._∼_ T (f b1) (f b2)
Equivalence.reflexive (Setoid.eq (imageGroupSetoid {T = T} fHom)) {ofElt b1} = Equivalence.reflexive (Setoid.eq T)
Equivalence.symmetric (Setoid.eq (imageGroupSetoid {T = T} fHom)) {ofElt b1} {ofElt b2} = Equivalence.symmetric (Setoid.eq T)
Equivalence.transitive (Setoid.eq (imageGroupSetoid {T = T} fHom)) {ofElt b1} {ofElt b2} {ofElt b3} = Equivalence.transitive (Setoid.eq T)
imageGroupOp : {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) → GroupHomImageElement fHom → GroupHomImageElement fHom → GroupHomImageElement fHom
imageGroupOp {_+A_ = _+A_} fHom (ofElt a) (ofElt b) = ofElt (a +A b)
imageGroup : {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) → Group (imageGroupSetoid fHom) (imageGroupOp fHom)
Group.+WellDefined (imageGroup {T = T} {_+A_ = _+A_} {H = H} {f = f} fHom) {ofElt x} {ofElt y} {ofElt a} {ofElt b} x~a y~b = ans
imageGroupSubset : subset T imageGroupPred
imageGroupSubset {x} {y} x=y (a , fa=x) = a , transitive fa=x x=y
where
open Setoid T
open Equivalence eq
ans : f (x +A y) ∼ f (a +A b)
ans = transitive (GroupHom.groupHom fHom) (transitive (Group.+WellDefined H x~a y~b) (symmetric (GroupHom.groupHom fHom)))
Group.0G (imageGroup {G = G} fHom) = ofElt (Group.0G G)
Group.inverse (imageGroup {G = G} fHom) (ofElt a) = ofElt (Group.inverse G a)
Group.+Associative (imageGroup {G = G} fHom) {ofElt a} {ofElt b} {ofElt c} = GroupHom.wellDefined fHom (Group.+Associative G)
Group.identRight (imageGroup {G = G} fHom) {ofElt a} = GroupHom.wellDefined fHom (Group.identRight G)
Group.identLeft (imageGroup {G = G} fHom) {ofElt a} = GroupHom.wellDefined fHom (Group.identLeft G)
Group.invLeft (imageGroup {G = G} fHom) {ofElt a} = GroupHom.wellDefined fHom (Group.invLeft G)
Group.invRight (imageGroup {G = G} fHom) {ofElt a} = GroupHom.wellDefined fHom (Group.invRight G)
groupHomImageInclusion : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+G_ : A → A → A} {_+H_ : B → B → B} {G : Group S _+G_} {H : Group T _+H_} {f : A → B} (fHom : GroupHom G H f) → GroupHomImageElement fHom → B
groupHomImageInclusion {f = f} fHom (ofElt x) = f x
groupHomImageIncludes : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+G_ : A → A → A} {_+H_ : B → B → B} {G : Group S _+G_} {H : Group T _+H_} {f : A → B} (fHom : GroupHom G H f) → GroupHom (imageGroup fHom) H (groupHomImageInclusion fHom)
GroupHom.groupHom (groupHomImageIncludes fHom) {ofElt x} {ofElt y} = GroupHom.groupHom fHom
GroupHom.wellDefined (groupHomImageIncludes fHom) {ofElt x} {ofElt y} x~y = x~y
groupHomImageInjects : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+G_ : A → A → A} {_+H_ : B → B → B} {G : Group S _+G_} {H : Group T _+H_} {f : A → B} (fHom : GroupHom G H f) → SetoidInjection (imageGroupSetoid fHom) T (groupHomImageInclusion fHom)
groupHomImageInjects {S = S} {T} {_+G_} {_+H_} {G} {H} {f} fHom = record { wellDefined = λ {x} {y} → GroupHom.wellDefined (groupHomImageIncludes fHom) {x} {y} ; injective = λ {x} {y} → inj {x} {y} }
imageGroupSubgroup : subgroup H imageGroupPred
_&&_.fst imageGroupSubgroup = imageGroupSubset
_&_&_.one (_&&_.snd imageGroupSubgroup) {x} {y} (a , fa=x) (b , fb=y) = (a +A b) , transitive (GroupHom.groupHom fHom) (Group.+WellDefined H fa=x fb=y)
where
inj : {x y : GroupHomImageElement fHom} → (Setoid._∼_ T (groupHomImageInclusion fHom x) (groupHomImageInclusion fHom y)) → Setoid._∼_ (imageGroupSetoid fHom) x y
inj {ofElt x} {ofElt y} x~y = x~y
open Setoid T
open Equivalence eq
_&_&_.two (_&&_.snd imageGroupSubgroup) = Group.0G G , imageOfIdentityIsIdentity fHom
_&_&_.three (_&&_.snd imageGroupSubgroup) {x} (a , fa=x) = Group.inverse G a , transitive (homRespectsInverse fHom) (inverseWellDefined H fa=x)
where
open Setoid T
open Equivalence eq

44
Groups/Homomorphisms/Kernel.agda Normal file
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{-# OPTIONS --safe --warning=error --without-K #-}
open import Groups.Groups
open import Groups.Definition
open import Orders
open import Numbers.Integers.Integers
open import Setoids.Setoids
open import LogicalFormulae
open import Sets.FinSet
open import Functions
open import Sets.EquivalenceRelations
open import Numbers.Naturals.Naturals
open import Groups.Homomorphisms.Definition
open import Groups.Homomorphisms.Lemmas
open import Groups.Isomorphisms.Definition
open import Groups.Subgroups.Definition
open import Groups.Subgroups.Normal.Definition
open import Groups.Subgroups.Normal.Lemmas
open import Groups.Lemmas
open import Groups.Abelian.Definition
open import Groups.QuotientGroup.Definition
open import Groups.Cosets
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
module Groups.Homomorphisms.Kernel {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+G_ : A → A → A} {_+H_ : B → B → B} {G : Group S _+G_} {H : Group T _+H_} {f : A → B} (fHom : GroupHom G H f) where
open Setoid T
open Equivalence eq
groupKernelPred : A → Set d
groupKernelPred a = Setoid._∼_ T (f a) (Group.0G H)
groupKernelPredWd : {x y : A} → (Setoid._∼_ S x y) → groupKernelPred x → groupKernelPred y
groupKernelPredWd x=y fx=0 = transitive (GroupHom.wellDefined fHom (Equivalence.symmetric (Setoid.eq S) x=y)) fx=0
groupKernelIsSubgroup : subgroup G groupKernelPred
_&_&_.one (_&&_.snd groupKernelIsSubgroup) fg=0 fh=0 = transitive (transitive (GroupHom.groupHom fHom) (Group.+WellDefined H fg=0 fh=0)) (Group.identLeft H)
_&_&_.two (_&&_.snd groupKernelIsSubgroup) = imageOfIdentityIsIdentity fHom
_&_&_.three (_&&_.snd groupKernelIsSubgroup) fg=0 = transitive (homRespectsInverse fHom) (transitive (inverseWellDefined H fg=0) (invIdent H))
_&&_.fst groupKernelIsSubgroup = groupKernelPredWd
groupKernelIsNormalSubgroup : normalSubgroup G groupKernelIsSubgroup
groupKernelIsNormalSubgroup {g} fk=0 = transitive (transitive (transitive (GroupHom.groupHom fHom) (transitive (Group.+WellDefined H reflexive (transitive (GroupHom.groupHom fHom) (transitive (Group.+WellDefined H fk=0 reflexive) (Group.identLeft H)))) (symmetric (GroupHom.groupHom fHom)))) (GroupHom.wellDefined fHom (Group.invRight G {g}))) (imageOfIdentityIsIdentity fHom)

34
Groups/Homomorphisms/Lemmas.agda
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@@ -11,36 +11,38 @@ open import Sets.EquivalenceRelations
open import Groups.Homomorphisms.Definition
open import Groups.Lemmas
module Groups.Homomorphisms.Lemmas where
module Groups.Homomorphisms.Lemmas {a b c d : _} {A : Set a} {S : Setoid {a} {c} A} {B : Set b} {T : Setoid {b} {d} B} {_+A_ : A → A → A} {_+B_ : B → B → B} {G : Group S _+A_} {H : Group T _+B_} {f : A → B} (hom : GroupHom G H f) where
imageOfIdentityIsIdentity : {m n o p : _} {A : Set m} {S : Setoid {m} {o} A} {B : Set n} {T : Setoid {n} {p} B} {_·A_ : A → A → A} {_·B_ : B → B → B} {G : Group S _·A_} {H : Group T _·B_} {f : A → B} → (hom : GroupHom G H f) → Setoid._∼_ T (f (Group.0G G)) (Group.0G H)
imageOfIdentityIsIdentity {S = S} {T = T} {_·A_ = _·A_} {_·B_ = _·B_} {G = G} {H = H} {f = f} hom = Equivalence.symmetric (Setoid.eq T) t
imageOfIdentityIsIdentity : Setoid._∼_ T (f (Group.0G G)) (Group.0G H)
imageOfIdentityIsIdentity = Equivalence.symmetric (Setoid.eq T) t
where
open Group H
open Setoid T
id2 : Setoid._∼_ S (Group.0G G) ((Group.0G G) ·A (Group.0G G))
id2 : Setoid._∼_ S (Group.0G G) ((Group.0G G) +A (Group.0G G))
id2 = Equivalence.symmetric (Setoid.eq S) (Group.identRight G)
r : f (Group.0G G) ∼ f (Group.0G G) ·B f (Group.0G G)
s : 0G ·B f (Group.0G G) ∼ f (Group.0G G) ·B f (Group.0G G)
r : f (Group.0G G) ∼ f (Group.0G G) +B f (Group.0G G)
s : 0G +B f (Group.0G G) ∼ f (Group.0G G) +B f (Group.0G G)
t : 0G ∼ f (Group.0G G)
t = groupsHaveRightCancellation H (f (Group.0G G)) 0G (f (Group.0G G)) s
s = Equivalence.transitive (Setoid.eq T) identLeft r
r = Equivalence.transitive (Setoid.eq T) (GroupHom.wellDefined hom id2) (GroupHom.groupHom hom)
groupHomsCompose : {m n o r s t : _} {A : Set m} {S : Setoid {m} {r} A} {_+A_ : A → A → A} {B : Set n} {T : Setoid {n} {s} B} {_+B_ : B → B → B} {C : Set o} {U : Setoid {o} {t} C} {_+C_ : C → C → C} {G : Group S _+A_} {H : Group T _+B_} {I : Group U _+C_} {f : A → B} {g : B → C} (fHom : GroupHom G H f) (gHom : GroupHom H I g) → GroupHom G I (g ∘ f)
GroupHom.wellDefined (groupHomsCompose {G = G} {H} {I} {f} {g} fHom gHom) {x} {y} pr = GroupHom.wellDefined gHom (GroupHom.wellDefined fHom pr)
GroupHom.groupHom (groupHomsCompose {S = S} {_+A_ = _·A_} {T = T} {U = U} {_+C_ = _·C_} {G = G} {H} {I} {f} {g} fHom gHom) {x} {y} = answer
groupHomsCompose : {o t : _} {C : Set o} {U : Setoid {o} {t} C} {_+C_ : C → C → C} {I : Group U _+C_} {g : B → C} (gHom : GroupHom H I g) → GroupHom G I (g ∘ f)
GroupHom.wellDefined (groupHomsCompose {I} {f} gHom) {x} {y} pr = GroupHom.wellDefined gHom (GroupHom.wellDefined hom pr)
GroupHom.groupHom (groupHomsCompose {U = U} {_+C_ = _·C_} {I} {g} gHom) {x} {y} = answer
where
open Group I
answer : (Setoid._∼_ U) ((g ∘ f) (x ·A y)) ((g ∘ f) x ·C (g ∘ f) y)
answer = (Equivalence.transitive (Setoid.eq U)) (GroupHom.wellDefined gHom (GroupHom.groupHom fHom {x} {y}) ) (GroupHom.groupHom gHom {f x} {f y})
answer : (Setoid._∼_ U) ((g ∘ f) (x +A y)) ((g ∘ f) x ·C (g ∘ f) y)
answer = (Equivalence.transitive (Setoid.eq U)) (GroupHom.wellDefined gHom (GroupHom.groupHom hom {x} {y}) ) (GroupHom.groupHom gHom {f x} {f y})
homRespectsInverse : {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) → {x : A} → Setoid._∼_ T (underF (Group.inverse G x)) (Group.inverse H (underF x))
homRespectsInverse {T = T} {_·A_ = _·A_} {_·B_ = _·B_} {G = G} {H = H} {underF = f} fHom {x} = rightInversesAreUnique H (f x) (f (Group.inverse G x)) (transitive (symmetric (GroupHom.groupHom fHom)) (transitive (GroupHom.wellDefined fHom (Group.invLeft G)) (imageOfIdentityIsIdentity fHom)))
homRespectsInverse : {x : A} → Setoid._∼_ T (f (Group.inverse G x)) (Group.inverse H (f x))
homRespectsInverse {x} = rightInversesAreUnique H (f x) (f (Group.inverse G x)) (transitive (symmetric (GroupHom.groupHom hom)) (transitive (GroupHom.wellDefined hom (Group.invLeft G)) imageOfIdentityIsIdentity))
where
open Setoid T
open Equivalence eq
identityHom : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+A_ : A → A → A} (G : Group S _+A_) → GroupHom G G id
GroupHom.groupHom (identityHom {S = S} G) = Equivalence.reflexive (Setoid.eq S)
GroupHom.wellDefined (identityHom G) = id
zeroImpliesInverseZero : {x : A} → Setoid._∼_ T (f x) (Group.0G H) → Setoid._∼_ T (f (Group.inverse G x)) (Group.0G H)
zeroImpliesInverseZero {x} fx=0 = transitive homRespectsInverse (transitive (inverseWellDefined H fx=0) (invIdent H))
where
open Setoid T
open Equivalence eq