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Tidy up groups (#64)

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Patrick Stevens
2019-11-03 17:12:48 +00:00
committed by GitHub
parent e4daab7153
commit d95f510cdd
42 changed files with 1438 additions and 1038 deletions
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25
Groups/Homomorphisms/Definition.agda Normal file
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{-# 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
module Groups.Homomorphisms.Definition where
record GroupHom {m n o p : _} {A : Set m} {S : Setoid {m} {o} A} {_·A_ : A → A → A} {B : Set n} {T : Setoid {n} {p} B} {_·B_ : B → B → B} (G : Group S _·A_) (H : Group T _·B_) (f : A → B) : Set (m ⊔ n ⊔ o ⊔ p) where
open Group H
open Setoid T
field
groupHom : {x y : A} → f (x ·A y) ∼ (f x) ·B (f y)
wellDefined : {x y : A} → Setoid._∼_ S x y → f x ∼ f y
record InjectiveGroupHom {m n o p : _} {A : Set m} {S : Setoid {m} {o} A} {_·A_ : A → A → A} {B : Set n} {T : Setoid {n} {p} B} {_·B_ : B → B → B} {G : Group S _·A_} {H : Group T _·B_} {underf : A → B} (f : GroupHom G H underf) : Set (m ⊔ n ⊔ o ⊔ p) where
open Setoid S renaming (_∼_ to _∼A_)
open Setoid T renaming (_∼_ to _∼B_)
field
injective : SetoidInjection S T underf

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Groups/Homomorphisms/Lemmas.agda Normal file
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{-# 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.Lemmas 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
where
open Group H
open Setoid T
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)
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
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})
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)))
where
open Setoid T
open Equivalence eq