openAI-math-dataset / lean /OAI /Algebra /Finitistic /Localization /CompletedOddDoubleEvaluation.lean
| import OAI.Algebra.Finitistic.Localization.CompletedShiftedCornerEvaluation | |
| import OAI.Algebra.Finitistic.Localization.CompletedCornerEvaluation | |
| import Mathlib.CategoryTheory.Idempotents.Biproducts | |
| import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Biproducts | |
| namespace OAI | |
| noncomputable section | |
| open CategoryTheory CategoryTheory.Limits CategoryTheory.Idempotents | |
| attribute [local instance] HasDerivedCategory.standard | |
| namespace LittleFinitistic.QuadraticEncoding | |
| private theorem mapBiprod_square {C D : Type*} [Category C] [Category D] | |
| [Preadditive C] [Preadditive D] [HasBinaryBiproducts C] [HasBinaryBiproducts D] | |
| (F : C ⥤ D) [F.Additive] [PreservesBinaryBiproducts F] | |
| {A B : C} {X Y : D} (a : A ⟶ A) (b : B ⟶ B) (x : X ⟶ X) (y : Y ⟶ Y) | |
| (e : F.obj A ≅ X) (f : F.obj B ≅ Y) | |
| (ha : F.map a ≫ e.hom = e.hom ≫ x) (hb : F.map b ≫ f.hom = f.hom ≫ y) : | |
| F.map (biprod.map a b) ≫ (F.mapBiprod A B ≪≫ biprod.mapIso e f).hom = | |
| (F.mapBiprod A B ≪≫ biprod.mapIso e f).hom ≫ biprod.map x y := by | |
| apply biprod.hom_ext | |
| · simp only [Iso.trans_hom, Functor.mapBiprod_hom, biprod.mapIso_hom, | |
| Category.assoc, biprod.map_fst, biprod.map_fst_assoc, biprod.lift_fst_assoc] | |
| rw [← Category.assoc, ← Functor.map_comp, biprod.map_fst, Functor.map_comp, | |
| Category.assoc, ha] | |
| · simp only [Iso.trans_hom, Functor.mapBiprod_hom, biprod.mapIso_hom, | |
| Category.assoc, biprod.map_snd, biprod.map_snd_assoc, biprod.lift_snd_assoc] | |
| rw [← Category.assoc, ← Functor.map_comp, biprod.map_snd, Functor.map_comp, | |
| Category.assoc, hb] | |
| 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) | |
| local instance : HasFiniteBiproducts (Karoubi (BoundedQ P)) := Karoubi.karoubi_hasFiniteBiproducts | |
| local instance : HasBinaryBiproducts (Karoubi (BoundedQ P)) := hasBinaryBiproducts_of_finite_biproducts _ | |
| local instance : PreservesBinaryBiproducts (completedDerivedEvaluation X) := | |
| preservesBinaryBiproducts_of_preservesBiproducts _ | |
| /-- Evaluation of the formal odd double used in Section 3. -/ | |
| def completedOddDoubleImageIso (r : PresentedAlgebra P) (hr : r*r=r) (n : ℤ) : | |
| (completedDerivedEvaluation X).obj | |
| (quotientSummand r hr ⊞ | |
| ((functorExtension₂ (BoundedQ P) (BoundedQ P)).obj (shiftFunctor (BoundedQ P) n)).obj | |
| (quotientSummand r hr)) ≅ | |
| evaluatedImageDerived X r ⊞ (evaluatedImageDerived X r)⟦n⟧ := | |
| (completedDerivedEvaluation X).mapBiprod _ _ ≪≫ | |
| biprod.mapIso (completedSummandImageIso X r hr) (completedShiftedSummandImageIso X r hr n) | |
| /-- The two summands of the evaluated odd double carry exactly the original | |
| corner action and its shift. This is the action used at each diagram vertex. -/ | |
| theorem completed_oddDouble_corner_square (e a : PresentedAlgebra P) (he : e*e=e) (n : ℤ) : | |
| (completedDerivedEvaluation X).map | |
| (biprod.map (quotientCornerEnd e a he) | |
| (((functorExtension₂ (BoundedQ P) (BoundedQ P)).obj (shiftFunctor (BoundedQ P) n)).map | |
| (quotientCornerEnd e a he))) ≫ (completedOddDoubleImageIso X e he n).hom = | |
| (completedOddDoubleImageIso X e he n).hom ≫ | |
| biprod.map (evaluatedCornerEnd X e a) ((evaluatedCornerEnd X e a)⟦n⟧') := by | |
| exact mapBiprod_square (completedDerivedEvaluation X) _ _ _ _ | |
| (completedSummandImageIso X e he) (completedShiftedSummandImageIso X e he n) | |
| (completed_corner_square X e a he) (completed_shifted_corner_square X e a he n) | |
| end LittleFinitistic.QuadraticEncoding | |
| end | |
| end OAI | |