| import OAI.Algebra.Finitistic.Encoding.QuadraticPresentation |
| import OAI.Algebra.Finitistic.Selection.AuxiliaryGroupAlgebraPresentation |
|
|
| namespace OAI |
| noncomputable section |
| namespace LittleFinitistic.QuadraticPresentation |
|
|
| open GroupAlgebraPresentation in |
| /-- The Section 3 conversion applied to the exact Section 2 complex group algebra. -/ |
| theorem auxiliaryGroupAlgebra_hasFiniteQuadraticPresentation : |
| HasFiniteQuadraticPresentation ℂ (MonoidAlgebra ℂ Presentation.AuxiliaryGroup) := |
| finite_presentation_to_quadratic ℂ auxiliaryGroupAlgebra_hasFinitePresentation |
|
|
| abbrev AuxiliaryOldLetter := GroupAlgebraPresentation.Letter FinitePresentation.SeedGenerator |
| abbrev AuxiliaryOldEquation := GroupAlgebraPresentation.EquationIndex |
| FinitePresentation.SeedGenerator FinitePresentation.finiteRelators |
|
|
| def auxiliaryOriginalRelation (i : AuxiliaryOldEquation) : FreeAlgebra ℂ AuxiliaryOldLetter := |
| GroupAlgebraPresentation.equation ℂ FinitePresentation.SeedGenerator |
| FinitePresentation.finiteRelators i - 1 |
|
|
| def auxiliaryWordBound : ℕ := |
| Classical.choose (exists_word_bound ℂ AuxiliaryOldLetter auxiliaryOriginalRelation) |
|
|
| theorem auxiliaryWordBound_spec (i : AuxiliaryOldEquation) (w : FreeMonoid AuxiliaryOldLetter) |
| (hw : w ∈ (coefficients ℂ AuxiliaryOldLetter (auxiliaryOriginalRelation i)).support) : |
| w.toList.length ≤ auxiliaryWordBound := |
| Classical.choose_spec (exists_word_bound ℂ AuxiliaryOldLetter auxiliaryOriginalRelation) i w hw |
|
|
| abbrev AuxiliaryLetter := Letter AuxiliaryOldLetter auxiliaryWordBound |
| abbrev AuxiliaryEquation := EquationIndex AuxiliaryOldLetter auxiliaryWordBound (κ := AuxiliaryOldEquation) |
|
|
| def auxiliaryRelation : AuxiliaryEquation → FreeAlgebra ℂ AuxiliaryLetter := |
| relation ℂ AuxiliaryOldLetter auxiliaryOriginalRelation auxiliaryWordBound |
| |
| instance : Finite AuxiliaryLetter := inferInstanceAs (Finite (Letter AuxiliaryOldLetter auxiliaryWordBound)) |
| instance : Finite AuxiliaryEquation := inferInstanceAs |
| (Finite (EquationIndex AuxiliaryOldLetter auxiliaryWordBound (κ := AuxiliaryOldEquation))) |
|
|
| theorem auxiliaryRelation_degree (i : AuxiliaryEquation) : |
| auxiliaryRelation i ∈ degreeLE ℂ AuxiliaryLetter 2 := |
| relation_degree ℂ AuxiliaryOldLetter auxiliaryOriginalRelation auxiliaryWordBound i |
|
|
| /-- Concrete quadratic quotient and actual algebra equivalence, retaining the |
| proved stable group's generators and relators and eliminating its word auxiliaries. -/ |
| def auxiliaryQuadraticEquiv : Original ℂ AuxiliaryLetter auxiliaryRelation ≃ₐ[ℂ] |
| MonoidAlgebra ℂ Presentation.AuxiliaryGroup := |
| (quadraticEquiv ℂ AuxiliaryOldLetter auxiliaryOriginalRelation auxiliaryWordBound |
| auxiliaryWordBound_spec).trans GroupAlgebraPresentation.auxiliaryGroupAlgebraZeroEquiv |
|
|
| end LittleFinitistic.QuadraticPresentation |
|
|
| end |
|
|
| end OAI |
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|
