| import OAI.Algebra.Finitistic.Localization.NativeSingleEvaluation |
| import OAI.Algebra.Finitistic.Selected.SelectedWindow |
| import OAI.Algebra.Finitistic.Selected.SelectedIteration |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory CategoryTheory.Limits |
| attribute [local instance] HasDerivedCategory.standard HasExt.standard |
| namespace LittleFinitistic.SelectedDerivedFunctor |
| open SelectedFixedDiagram QuadraticEncoding FiniteSelectionFamily |
|
|
| / |
| def model : BimoduleTotal.BoundedProjective ℂ (B P) SelectedRectification.bimodule := |
| BimoduleTotal.nativeProjectiveModel SelectedRectification.bimodule |
| (SelectedWindow.lo-2) SelectedWindow.hi SelectedWindow.bounded SelectedWindow.verified.projective |
|
|
| abbrev Phi := model.derived |
|
|
| / |
| actual fixed selecting bimodule and its actual derived tensor endofunctor. -/ |
| def comparison (Y : S.{0}) : Phi.obj (derived Y) ≅ |
| SelectedTensorSplitting.object (representation Y).Evaluation := |
| model.nativeSingleEvaluation (representation Y).Evaluation |
|
|
| / |
| def iteration (j : ℕ) (Y : S.{0}) : |
| (Phi.obj^[j]) (derived Y) ≅ |
| ⨁ fun q : Fin (j+1) => ⨁ fun _ : Fin (Nat.choose j q.val) => |
| (derived ((H^[j]) Y))⟦(3 : ℤ)*(q.val : ℤ)⟧ := |
| SelectedIteration.iteration Phi comparison j Y |
|
|
| theorem isZero_iff (j : ℕ) (Y : S.{0}) : |
| IsZero ((Phi.obj^[j]) (derived Y)) ↔ Subsingleton (selectionIterate Y.val j) := |
| SelectedIteration.isZero_iff Phi comparison j Y |
|
|
| / |
| theorem extinction (m : ℕ) [NeZero m] : |
| (∀ j < m, ¬ IsZero ((Phi.obj^[j]) (derived (test m)))) ∧ |
| IsZero ((Phi.obj^[m]) (derived (test m))) := |
| SelectedIteration.extinction Phi comparison m |
| end LittleFinitistic.SelectedDerivedFunctor |
|
|
| end |
|
|
| end OAI |
|
|
