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Free-group lemmas (#106)

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Patrick Stevens
2020-04-05 11:09:12 +01:00
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parent 61b5f8acc5
commit 2f07a8e972
18 changed files with 327 additions and 188 deletions
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{-# OPTIONS --safe --warning=error #-}
open import Numbers.Naturals.Semiring
open import Groups.FreeProduct.Definition
open import Groups.FreeProduct.Setoid
open import Groups.FreeProduct.Group
open import Groups.Definition
open import Groups.Homomorphisms.Definition
open import Groups.Isomorphisms.Definition
open import LogicalFormulae
open import Numbers.Integers.Addition
open import Numbers.Integers.Definition
open import Groups.FreeGroup.Definition
open import Groups.FreeGroup.Word
open import Groups.FreeGroup.Group
open import Groups.FreeGroup.UniversalProperty
open import Setoids.Setoids
module Groups.FreeProduct.Lemmas {i : _} {I : Set i} (decidableIndex : (x y : I) → ((x ≡ y) || ((x ≡ y) → False))) where
private
f : ReducedWord decidableIndex → ReducedSequence decidableIndex (λ _ → ℤDecideEquality) (λ _ → ℤGroup)
f = universalPropertyFunction decidableIndex (FreeProductGroup decidableIndex (λ _ → ℤDecideEquality) (λ _ → ℤGroup)) λ i → nonempty i (ofEmpty i (nonneg 1) λ ())
freeProductIso : GroupHom (freeGroup decidableIndex) (FreeProductGroup decidableIndex (λ _ → ℤDecideEquality) (λ _ → ℤGroup)) f
freeProductIso = universalPropertyHom decidableIndex (FreeProductGroup decidableIndex (λ _ → ℤDecideEquality) (λ _ → ℤGroup)) (λ i → nonempty i (ofEmpty i (nonneg 1) λ ()))
freeProductInj : (x y : ReducedWord decidableIndex) → (decidableIndex =RP λ _ → ℤDecideEquality) (λ _ → ℤGroup) (f x) (f y) → x ≡ y
freeProductInj empty empty pr = refl
freeProductInj empty (prependLetter (ofLetter x₁) y x) pr = exFalso {!!}
freeProductInj empty (prependLetter (ofInv x₁) y x) pr = {!!}
freeProductInj (prependLetter letter x x₁) y pr = {!!}
freeProductZ : GroupsIsomorphic (freeGroup decidableIndex) (FreeProductGroup decidableIndex (λ _ → ℤDecideEquality) (λ _ → ℤGroup))
GroupsIsomorphic.isomorphism (freeProductZ) = universalPropertyFunction decidableIndex (FreeProductGroup decidableIndex (λ _ → ℤDecideEquality) (λ _ → ℤGroup)) λ i → nonempty i (ofEmpty i (nonneg 1) λ ())
GroupIso.groupHom (GroupsIsomorphic.proof (freeProductZ)) = freeProductIso
SetoidInjection.wellDefined (SetoidBijection.inj (GroupIso.bij (GroupsIsomorphic.proof (freeProductZ)))) = GroupHom.wellDefined (freeProductIso)
SetoidInjection.injective (SetoidBijection.inj (GroupIso.bij (GroupsIsomorphic.proof (freeProductZ)))) {x} {y} = freeProductInj x y
SetoidSurjection.wellDefined (SetoidBijection.surj (GroupIso.bij (GroupsIsomorphic.proof (freeProductZ)))) = GroupHom.wellDefined freeProductIso
SetoidSurjection.surjective (SetoidBijection.surj (GroupIso.bij (GroupsIsomorphic.proof (freeProductZ)))) {empty} = empty , record {}
SetoidSurjection.surjective (SetoidBijection.surj (GroupIso.bij (GroupsIsomorphic.proof (freeProductZ)))) {nonempty i (ofEmpty .i (nonneg zero) nonZero)} = exFalso (nonZero refl)
SetoidSurjection.surjective (SetoidBijection.surj (GroupIso.bij (GroupsIsomorphic.proof (freeProductZ)))) {nonempty i (ofEmpty .i (nonneg (succ x)) nonZero)} = prependLetter (ofLetter i) empty (wordEmpty refl) , {!!}
SetoidSurjection.surjective (SetoidBijection.surj (GroupIso.bij (GroupsIsomorphic.proof (freeProductZ)))) {nonempty i (ofEmpty .i (negSucc x) nonZero)} = {!!}
SetoidSurjection.surjective (SetoidBijection.surj (GroupIso.bij (GroupsIsomorphic.proof (freeProductZ)))) {nonempty i (prependLetter .i g nonZero x x₁)} = {!!}
freeProductNonAbelian : {!!}
freeProductNonAbelian = {!!}