| import OAI.Algebra.Finitistic.Selected.SelectedChainDiagram |
| import OAI.Algebra.Finitistic.Rectification.RectificationLemma |
| import OAI.Algebra.Finitistic.Localization.UnderlyingRepresentatives |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory CategoryTheory.Limits HomologicalComplex |
| namespace LittleFinitistic.SelectedRectification |
| open SelectedFixedDiagram ExternalTensor.Section4 RelationBasisCoefficients |
|
|
| abbrev D0 := UnderlyingRepresentatives.complex P fixed.D0 |
| abbrev D1 := UnderlyingRepresentatives.complex P fixed.D1 |
| abbrev D2 := UnderlyingRepresentatives.complex P fixed.D2 |
| abbrev F := (UnderlyingRepresentatives.forget P).mapHomologicalComplex (.up ℤ) |
| def f (a : Fin (encoding.generators+1)) : D0 ⟶ D1 := F.map (SelectedChainDiagram.fixedChain.f a) |
| def g (b : Fin (encoding.generators+1)) : D1 ⟶ D2 := F.map (SelectedChainDiagram.fixedChain.g b) |
| def relations (r : RelationIndex P) : |
| Homotopy (relationMap (ConcreteQuadraticDiagram.quadratic P) D0 D1 D2 f g r) 0 := |
| UnderlyingRepresentatives.relationHomotopy P (encoding.generators+1) (encoding.generators+1) |
| (Module.finrank ℂ (QuadraticEncoding.relationSpace P)) fixed.D0 fixed.D1 fixed.D2 |
| SelectedChainDiagram.fixedChain.f SelectedChainDiagram.fixedChain.g (basisCoefficient P) |
| SelectedChainDiagram.fixedChain.homotopy r |
|
|
| /-- One explicit bimodule complex chosen before any evaluation module. -/ |
| def bimodule : BiComplex ℂ (QuadraticEncoding.B P) := |
| InducedRectification.total (ConcreteQuadraticDiagram.quadratic P) D0 D1 D2 f g relations |
|
|
| /-- The single fixed selecting complex satisfies the full rectification lemma, |
| including all original vertex and arrow maps, independently of Y. -/ |
| theorem verified : ∃ lo hi : ℤ, RectificationLemma.Verified P D0 D1 D2 f g relations lo hi := by |
| obtain ⟨l₀,h₀,b₀⟩ := UnderlyingRepresentatives.bounded P fixed.D0 |
| obtain ⟨l₁,h₁,b₁⟩ := UnderlyingRepresentatives.bounded P fixed.D1 |
| obtain ⟨l₂,h₂,b₂⟩ := UnderlyingRepresentatives.bounded P fixed.D2 |
| let lo := min l₀ (min l₁ l₂) |
| let hi := max h₀ (max h₁ h₂) |
| refine ⟨lo,hi,RectificationLemma.rectification P D0 D1 D2 f g relations lo hi |
| (fun n hn => b₀ n ?_) (fun n hn => b₁ n ?_) (fun n hn => b₂ n ?_) |
| (UnderlyingRepresentatives.projective P fixed.D0) |
| (UnderlyingRepresentatives.projective P fixed.D1) |
| (UnderlyingRepresentatives.projective P fixed.D2) |
| (UnderlyingRepresentatives.finite P fixed.D0) |
| (UnderlyingRepresentatives.finite P fixed.D1) |
| (UnderlyingRepresentatives.finite P fixed.D2)⟩ |
| all_goals dsimp [lo,hi] at hn; omega |
| end LittleFinitistic.SelectedRectification |
|
|
| end |
|
|
| end OAI |
|
|
