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author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Sun, 29 Jan 2023 11:40:29 +0900 |
parents | src/Fundamental.agda@b89af4300407 |
children | 515de7624a0c |
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-- fundamental homomorphism theorem -- open import Level hiding ( suc ; zero ) module FundamentalHomorphism (c : Level) where open import Algebra open import Algebra.Structures open import Algebra.Definitions open import Algebra.Core open import Algebra.Bundles open import Data.Product open import Relation.Binary.PropositionalEquality open import Gutil0 import Gutil import Function.Definitions as FunctionDefinitions import Algebra.Morphism.Definitions as MorphismDefinitions open import Algebra.Morphism.Structures open import Tactic.MonoidSolver using (solve; solve-macro) -- -- Given two groups G and H and a group homomorphism f : G → H, -- let K be a normal subgroup in G and φ the natural surjective homomorphism G → G/K -- (where G/K is the quotient group of G by K). -- If K is a subset of ker(f) then there exists a unique homomorphism h: G/K → H such that f = h∘φ. -- https://en.wikipedia.org/wiki/Fundamental_theorem_on_homomorphisms -- -- f -- G --→ H -- | / -- φ | / h -- ↓ / -- G/K -- import Relation.Binary.Reasoning.Setoid as EqReasoning _<_∙_> : (m : Group c c ) → Group.Carrier m → Group.Carrier m → Group.Carrier m m < x ∙ y > = Group._∙_ m x y _<_≈_> : (m : Group c c ) → (f g : Group.Carrier m ) → Set c m < x ≈ y > = Group._≈_ m x y infixr 9 _<_∙_> -- -- Coset : N : NormalSubGroup → { a ∙ n | G ∋ a , N ∋ n } -- open GroupMorphisms import Axiom.Extensionality.Propositional postulate f-extensionality : { n m : Level} → Axiom.Extensionality.Propositional.Extensionality n m open import Tactic.MonoidSolver using (solve; solve-macro) record NormalSubGroup (A : Group c c ) : Set c where open Group A field ⟦_⟧ : Group.Carrier A → Group.Carrier A normal : IsGroupHomomorphism (GR A) (GR A) ⟦_⟧ comm : {a b : Carrier } → b ∙ ⟦ a ⟧ ≈ ⟦ a ⟧ ∙ b -- factor : (a b : Carrier) → Carrier is-factor : (a b : Carrier) → b ∙ ⟦ factor a b ⟧ ≈ a -- Set of a ∙ ∃ n ∈ N -- record an {A : Group c c } (N : NormalSubGroup A ) (n x : Group.Carrier A ) : Set c where open Group A open NormalSubGroup N field a : Group.Carrier A aN=x : a ∙ ⟦ n ⟧ ≈ x record aN {A : Group c c } (N : NormalSubGroup A ) : Set c where field n : Group.Carrier A is-an : (x : Group.Carrier A) → an N n x qid : {A : Group c c } (N : NormalSubGroup A ) → aN N qid {A} N = record { n = ε ; is-an = λ x → record { a = x ; aN=x = ? } } where open Group A open NormalSubGroup N _/_ : (A : Group c c ) (N : NormalSubGroup A ) → Group c c _/_ A N = record { Carrier = aN N ; _≈_ = λ f g → ⟦ n f ⟧ ≈ ⟦ n g ⟧ ; _∙_ = qadd ; ε = qid N ; _⁻¹ = ? ; isGroup = record { isMonoid = record { isSemigroup = record { isMagma = record { isEquivalence = record {refl = grefl ; trans = gtrans ; sym = gsym } ; ∙-cong = λ {x} {y} {u} {v} x=y u=v → ? } ; assoc = ? } ; identity = ? , (λ q → ? ) } ; inverse = ( (λ x → ? ) , (λ x → ? )) ; ⁻¹-cong = λ i=j → ? } } where open Group A open aN open an open NormalSubGroup N open IsGroupHomomorphism normal open EqReasoning (Algebra.Group.setoid A) open Gutil A qadd : (f g : aN N) → aN N qadd f g = record { n = n f ∙ n g ; is-an = λ x → record { a = x ⁻¹ ∙ ( a (is-an f x) ∙ a (is-an g x)) ; aN=x = qadd0 } } where qadd0 : {x : Carrier} → x ⁻¹ ∙ (a (is-an f x) ∙ a (is-an g x)) ∙ ⟦ n f ∙ n g ⟧ ≈ x qadd0 {x} = begin x ⁻¹ ∙ (a (is-an f x) ∙ a (is-an g x)) ∙ ⟦ n f ∙ n g ⟧ ≈⟨ ? ⟩ x ⁻¹ ∙ (a (is-an f x) ∙ a (is-an g x) ∙ ⟦ n f ∙ n g ⟧) ≈⟨ ? ⟩ x ⁻¹ ∙ (a (is-an f x) ∙ a (is-an g x) ∙ ( ⟦ n f ⟧ ∙ ⟦ n g ⟧ )) ≈⟨ ? ⟩ x ⁻¹ ∙ (a (is-an f x) ∙ ( a (is-an g x) ∙ ⟦ n f ⟧) ∙ ⟦ n g ⟧) ≈⟨ ? ⟩ x ⁻¹ ∙ (a (is-an f x) ∙ ( ⟦ n f ⟧ ∙ a (is-an g x) ) ∙ ⟦ n g ⟧) ≈⟨ ? ⟩ x ⁻¹ ∙ ((a (is-an f x) ∙ ⟦ n f ⟧ ) ∙ ( a (is-an g x) ∙ ⟦ n g ⟧)) ≈⟨ ? ⟩ x ⁻¹ ∙ ((a (is-an f x) ∙ ⟦ n f ⟧ ) ∙ ( a (is-an g x) ∙ ⟦ n g ⟧)) ≈⟨ ? ⟩ x ⁻¹ ∙ (x ∙ x) ≈⟨ ? ⟩ x ∎ -- K ⊂ ker(f) K⊆ker : (G H : Group c c) (K : NormalSubGroup G) (f : Group.Carrier G → Group.Carrier H ) → Set c K⊆ker G H K f = (x : Group.Carrier G ) → f ( ⟦ x ⟧ ) ≈ ε where open Group H open NormalSubGroup K open import Function.Surjection open import Function.Equality module GK (G : Group c c) (K : NormalSubGroup G) where open Group G open aN open an open NormalSubGroup K open IsGroupHomomorphism normal open EqReasoning (Algebra.Group.setoid G) open Gutil G φ : Group.Carrier G → Group.Carrier (G / K ) φ g = record { n = factor ε g ; is-an = λ x → record { a = x ∙ ( ⟦ factor ε g ⟧ ⁻¹) ; aN=x = ? } } φ-homo : IsGroupHomomorphism (GR G) (GR (G / K)) φ φ-homo = record {⁻¹-homo = λ g → ? ; isMonoidHomomorphism = record { ε-homo = ? ; isMagmaHomomorphism = record { homo = ? ; isRelHomomorphism = record { cong = ? } }}} φe : (Algebra.Group.setoid G) Function.Equality.⟶ (Algebra.Group.setoid (G / K)) φe = record { _⟨$⟩_ = φ ; cong = ? } where φ-cong : {f g : Carrier } → f ≈ g → ⟦ n (φ f ) ⟧ ≈ ⟦ n (φ g ) ⟧ φ-cong = ? -- inverse ofφ -- f = λ x → record { a = af ; n = fn ; aN=x = x ≈ af ∙ ⟦ fn ⟧ ) = (af)K , fn ≡ factor x af , af ≡ a (f x) -- (inv-φ f)K ≡ (af)K -- φ (inv-φ f) x → f (h0 x) -- f x → φ (inv-φ f) (h1 x) inv-φ : Group.Carrier (G / K ) → Group.Carrier G inv-φ f = ⟦ n f ⟧ ⁻¹ cong-i : {f g : Group.Carrier (G / K ) } → ⟦ n f ⟧ ≈ ⟦ n g ⟧ → inv-φ f ≈ inv-φ g cong-i = ? is-inverse : (f : aN K ) → ⟦ n (φ (inv-φ f) ) ⟧ ≈ ⟦ n f ⟧ is-inverse f = begin ⟦ n (φ (inv-φ f) ) ⟧ ≈⟨ grefl ⟩ ⟦ n (φ (⟦ n f ⟧ ⁻¹) ) ⟧ ≈⟨ grefl ⟩ ⟦ factor ε (⟦ n f ⟧ ⁻¹) ⟧ ≈⟨ ? ⟩ ( ⟦ n f ⟧ ∙ ⟦ n f ⟧ ⁻¹) ∙ ⟦ factor ε (⟦ n f ⟧ ⁻¹) ⟧ ≈⟨ ? ⟩ ⟦ n f ⟧ ∙ ( ⟦ n f ⟧ ⁻¹ ∙ ⟦ factor ε (⟦ n f ⟧ ⁻¹) ⟧ ) ≈⟨ ∙-cong grefl (is-factor ε _ ) ⟩ ⟦ n f ⟧ ∙ ε ≈⟨ ? ⟩ ⟦ n f ⟧ ∎ φ-surjective : Surjective φe φ-surjective = record { from = record { _⟨$⟩_ = inv-φ ; cong = λ {f} {g} → cong-i {f} {g} } ; right-inverse-of = is-inverse } gk00 : {x : Carrier } → ⟦ factor ε x ⟧ ⁻¹ ≈ x gk00 {x} = begin ⟦ factor ε x ⟧ ⁻¹ ≈⟨ ? ⟩ ( x ⁻¹ ∙ ( x ∙ ⟦ factor ε x ⟧) ) ⁻¹ ≈⟨ ? ⟩ ( x ⁻¹ ∙ ( x ∙ ⟦ factor ε x ⟧) ) ⁻¹ ≈⟨ ⁻¹-cong (∙-cong grefl (is-factor ε x)) ⟩ ( x ⁻¹ ∙ ε ) ⁻¹ ≈⟨ ? ⟩ ( x ⁻¹ ) ⁻¹ ≈⟨ ? ⟩ x ∎ gk01 : (x : Group.Carrier (G / K ) ) → (G / K) < φ ( inv-φ x ) ≈ x > gk01 x = begin ⟦ factor ε ( inv-φ x) ⟧ ≈⟨ ? ⟩ ( inv-φ x ) ⁻¹ ∙ ( inv-φ x ∙ ⟦ factor ε ( inv-φ x) ⟧) ≈⟨ ∙-cong grefl (is-factor ε _ ) ⟩ ( inv-φ x ) ⁻¹ ∙ ε ≈⟨ ? ⟩ ( ⟦ n x ⟧ ⁻¹) ⁻¹ ≈⟨ ? ⟩ ⟦ n x ⟧ ∎ record FundamentalHomomorphism (G H : Group c c ) (f : Group.Carrier G → Group.Carrier H ) (homo : IsGroupHomomorphism (GR G) (GR H) f ) (K : NormalSubGroup G ) (kf : K⊆ker G H K f) : Set c where open Group H open GK G K field h : Group.Carrier (G / K ) → Group.Carrier H h-homo : IsGroupHomomorphism (GR (G / K) ) (GR H) h is-solution : (x : Group.Carrier G) → f x ≈ h ( φ x ) unique : (h1 : Group.Carrier (G / K ) → Group.Carrier H) → (homo : IsGroupHomomorphism (GR (G / K)) (GR H) h1 ) → ( (x : Group.Carrier G) → f x ≈ h1 ( φ x ) ) → ( ( x : Group.Carrier (G / K)) → h x ≈ h1 x ) FundamentalHomomorphismTheorm : (G H : Group c c) (f : Group.Carrier G → Group.Carrier H ) (homo : IsGroupHomomorphism (GR G) (GR H) f ) (K : NormalSubGroup G ) → (kf : K⊆ker G H K f) → FundamentalHomomorphism G H f homo K kf FundamentalHomomorphismTheorm G H f homo K kf = record { h = h ; h-homo = h-homo ; is-solution = is-solution ; unique = unique } where open GK G K open Group H open Gutil H open NormalSubGroup K open IsGroupHomomorphism homo open aN h : Group.Carrier (G / K ) → Group.Carrier H h r = f ( inv-φ r ) h03 : (x y : Group.Carrier (G / K ) ) → h ( (G / K) < x ∙ y > ) ≈ h x ∙ h y h03 x y = {!!} h-homo : IsGroupHomomorphism (GR (G / K ) ) (GR H) h h-homo = record { isMonoidHomomorphism = record { isMagmaHomomorphism = record { isRelHomomorphism = record { cong = λ {x} {y} eq → {!!} } ; homo = h03 } ; ε-homo = {!!} } ; ⁻¹-homo = {!!} } open EqReasoning (Algebra.Group.setoid H) is-solution : (x : Group.Carrier G) → f x ≈ h ( φ x ) is-solution x = begin f x ≈⟨ gsym ( ⟦⟧-cong (gk00 )) ⟩ f ( Group._⁻¹ G ⟦ factor (Group.ε G) x ⟧ ) ≈⟨ grefl ⟩ h ( φ x ) ∎ unique : (h1 : Group.Carrier (G / K ) → Group.Carrier H) → (h1-homo : IsGroupHomomorphism (GR (G / K)) (GR H) h1 ) → ( (x : Group.Carrier G) → f x ≈ h1 ( φ x ) ) → ( ( x : Group.Carrier (G / K)) → h x ≈ h1 x ) unique h1 h1-homo h1-is-solution x = begin h x ≈⟨ grefl ⟩ f ( inv-φ x ) ≈⟨ h1-is-solution _ ⟩ h1 ( φ ( inv-φ x ) ) ≈⟨ IsGroupHomomorphism.⟦⟧-cong h1-homo (gk01 x) ⟩ h1 x ∎