openAI-math-dataset / lean /OAI /Algebra /Finitistic /Localization /CompletedShiftedCornerEvaluation.lean
| import OAI.Algebra.Finitistic.Localization.CompletedCornerEvaluation | |
| import OAI.Algebra.Finitistic.Localization.CompletedShiftEvaluation | |
| namespace OAI | |
| noncomputable section | |
| open CategoryTheory CategoryTheory.Idempotents | |
| attribute [local instance] HasDerivedCategory.standard | |
| namespace LittleFinitistic.QuadraticEncoding | |
| universe u v | |
| variable {k : Type u} [Field k] {d t : ℕ} {P : Relations k d t} | |
| variable {Y : Type v} [AddCommGroup Y] [Module k Y] [FiniteDimensional k Y] | |
| variable (X : Representation P Y) (e a : PresentedAlgebra P) (he : e*e=e) | |
| /-- The same corner action evaluates correctly on every formal shift, including shift 3. -/ | |
| theorem completed_shifted_corner_square (n : ℤ) : | |
| (completedDerivedEvaluation X).map | |
| (((functorExtension₂ (BoundedQ P) (BoundedQ P)).obj (shiftFunctor (BoundedQ P) n)).map | |
| (quotientCornerEnd e a he)) ≫ (completedShiftedSummandImageIso X e he n).hom = | |
| (completedShiftedSummandImageIso X e he n).hom ≫ (evaluatedCornerEnd X e a)⟦n⟧' := by | |
| have hn := (completedDerivedShiftIso X n).hom.naturality (quotientCornerEnd e a he) | |
| have hm := congrArg (shiftFunctor (FiniteVectorDerived.{u,v} (k := k)) n).map | |
| (completed_corner_square X e a he) | |
| simp only [Functor.map_comp] at hm | |
| change _ ≫ ((completedDerivedShiftIso X n).hom.app (quotientSummand e he) ≫ | |
| (shiftFunctor _ n).map (completedSummandImageIso X e he).hom) = | |
| ((completedDerivedShiftIso X n).hom.app (quotientSummand e he) ≫ | |
| (shiftFunctor _ n).map (completedSummandImageIso X e he).hom) ≫ _ | |
| rw [← Category.assoc] | |
| change (((functorExtension₂ (BoundedQ P) (BoundedQ P)).obj (shiftFunctor (BoundedQ P) n) ⋙ | |
| completedDerivedEvaluation X).map (quotientCornerEnd e a he) ≫ _) ≫ _ = _ | |
| rw [hn, Category.assoc] | |
| change _ ≫ ((shiftFunctor _ n).map ((completedDerivedEvaluation X).map (quotientCornerEnd e a he)) ≫ _) = _ | |
| rw [hm, ← Category.assoc] | |
| end LittleFinitistic.QuadraticEncoding | |
| end | |
| end OAI | |