| import OAI.Algebra.Finitistic.Selected.SelectedRectification |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory CategoryTheory.Limits |
| namespace LittleFinitistic.SelectedWindow |
| open SelectedFixedDiagram SelectedRectification |
|
|
| / |
| structure Window where |
| lo : ℤ |
| hi : ℤ |
| bound0 : ∀ n, n < lo ∨ hi < n → IsZero (D0.X n) |
| bound1 : ∀ n, n < lo ∨ hi < n → IsZero (D1.X n) |
| bound2 : ∀ n, n < lo ∨ hi < n → IsZero (D2.X n) |
| |
| instance : Nonempty Window := 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 |
| refine ⟨⟨min l₀ (min l₁ l₂), max h₀ (max h₁ h₂), ?_, ?_, ?_⟩⟩ |
| · intro n hn; apply b₀ n; omega |
| · intro n hn; apply b₁ n; omega |
| · intro n hn; apply b₂ n; omega |
|
|
| def window : Window := Classical.choice inferInstance |
| abbrev lo := window.lo |
| abbrev hi := window.hi |
|
|
| / |
| theorem bounded (n : ℤ) (hn : n < lo - 2 ∨ hi < n) : IsZero (bimodule.X n) := |
| InducedRectification.total_bounded (ConcreteQuadraticDiagram.quadratic P) |
| D0 D1 D2 f g relations lo hi window.bound0 window.bound1 window.bound2 n hn |
|
|
| / |
| theorem verified : RectificationLemma.Verified P D0 D1 D2 f g relations lo hi := |
| RectificationLemma.rectification P D0 D1 D2 f g relations lo hi |
| window.bound0 window.bound1 window.bound2 |
| (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) |
| end LittleFinitistic.SelectedWindow |
|
|
| end |
|
|
| end OAI |
| |
