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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/Isomorphisms/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 Groups.Homomorphisms.Definition
open import Sets.EquivalenceRelations
module Groups.Isomorphisms.Definition where
record GroupIso {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 Setoid S renaming (_∼_ to _∼G_)
open Setoid T renaming (_∼_ to _∼H_)
field
groupHom : GroupHom G H f
bij : SetoidBijection S T f
record GroupsIsomorphic {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_) : Set (m ⊔ n ⊔ o ⊔ p) where
field
isomorphism : A → B
proof : GroupIso G H isomorphism

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Groups/Isomorphisms/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.Isomorphisms.Definition
open import Groups.Homomorphisms.Definition
open import Groups.Homomorphisms.Lemmas
module Groups.Isomorphisms.Lemmas where
groupIsosCompose : {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 : GroupIso G H f) (gHom : GroupIso H I g) → GroupIso G I (g ∘ f)
GroupIso.groupHom (groupIsosCompose fHom gHom) = groupHomsCompose (GroupIso.groupHom fHom) (GroupIso.groupHom gHom)
GroupIso.bij (groupIsosCompose {A = A} {S = S} {T = T} {C = C} {U = U} {f = f} {g = g} fHom gHom) = record { inj = record { injective = λ pr → (SetoidInjection.injective (SetoidBijection.inj (GroupIso.bij fHom))) (SetoidInjection.injective (SetoidBijection.inj (GroupIso.bij gHom)) pr) ; wellDefined = +WellDefined } ; surj = record { surjective = surj ; wellDefined = +WellDefined } }
where
open Setoid S renaming (_∼_ to _∼A_)
open Setoid U renaming (_∼_ to _∼C_)
+WellDefined : {x y : A} → (x ∼A y) → (g (f x) ∼C g (f y))
+WellDefined = GroupHom.wellDefined (groupHomsCompose (GroupIso.groupHom fHom) (GroupIso.groupHom gHom))
surj : {x : C} → Sg A (λ a → (g (f a) ∼C x))
surj {x} with SetoidSurjection.surjective (SetoidBijection.surj (GroupIso.bij gHom)) {x}
surj {x} | a , prA with SetoidSurjection.surjective (SetoidBijection.surj (GroupIso.bij fHom)) {a}
... | b , prB = b , transitive (GroupHom.wellDefined (GroupIso.groupHom gHom) prB) prA
where
open Equivalence (Setoid.eq U)
--groupIsoInvertible : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d}} {_+A_ : A → A → A} {_+B_ : B → B → B} {G : Group S _+A_} {H : Group T _+B_} {f : A → B} → (iso : GroupIso G H f) → GroupIso H G (Invertible.inverse (bijectionImpliesInvertible (GroupIso.bijective iso)))
--GroupHom.groupHom (GroupIso.groupHom (groupIsoInvertible {G = G} {H} {f} iso)) {x} {y} = {!!}
-- where
-- open Group G renaming (_·_ to _+G_)
-- open Group H renaming (_·_ to _+H_)
--GroupHom.wellDefined (GroupIso.groupHom (groupIsoInvertible {G = G} {H} {f} iso)) {x} {y} x∼y = {!GroupHom.wellDefined x∼y!}
--GroupIso.bijective (groupIsoInvertible {G = G} {H} {f} iso) = invertibleImpliesBijection (inverseIsInvertible (bijectionImpliesInvertible (GroupIso.bijective iso)))