openAI-math-dataset / lean /OAI /Algebra /Finitistic /Rectification /CanonicalArrowCompatibility.lean
| import OAI.Algebra.Finitistic.Localization.NativeArrowBridge | |
| import OAI.Algebra.Finitistic.Rectification.VertexHomotopyEquivalence | |
| namespace OAI | |
| noncomputable section | |
| open CategoryTheory HomologicalComplex | |
| open scoped TensorProduct | |
| namespace LittleFinitistic.CanonicalArrowCompatibility | |
| open ExternalTensor.Section4 ExternalTensor.Section4.ArrowVertex | |
| open QuadraticEncoding | |
| universe u | |
| variable {k : Type u} [Field k] {d t : ℕ} (P : Relations k d t) | |
| lemma diagonal_original (i : Fin 3) (D : RightComplex (B P)) : | |
| (VertexColumnComplex.diagonalComplexIso (composition P) i D).hom = | |
| diagonalComplex k (B P) (ThreeVertexAlgebra.vertex (composition P) i) | |
| ((ThreeVertexAlgebra.directedSystem (composition P)).idem i) D := by | |
| apply NativeBridge.diagonal_eq_of_original | |
| intro n x | |
| exact VertexColumnModels.diagonal_symm_val (composition P) (D.X n) i x | |
| variable (D₀ D₁ D₂ : RightComplex (B P)) | |
| variable (f : Fin (d+1) → (D₀ ⟶ D₁)) (g : Fin (d+1) → (D₁ ⟶ D₂)) | |
| variable (H : ∀ r, Homotopy (relationMap (ConcreteQuadraticDiagram.quadratic P) D₀ D₁ D₂ f g r) 0) | |
| lemma arrow01_target (a : Fin (d+1)) : | |
| ThreeVertexAlgebra.vertex (composition P) 1 * arrow01 P a = arrow01 P a := by | |
| change ThreeVertexAlgebra.mul (composition P) (0,1,0,0,0,0) (0,0,0,basis a,0,0) = _ | |
| simp [ThreeVertexAlgebra.mul, arrow01] | |
| lemma arrow12_target (b : Fin (d+1)) : | |
| ThreeVertexAlgebra.vertex (composition P) 2 * arrow12 P b = arrow12 P b := by | |
| change ThreeVertexAlgebra.mul (composition P) (0,0,1,0,0,0) (0,0,0,0,basis b,0) = _ | |
| simp [ThreeVertexAlgebra.mul, arrow12] | |
| /-- Left multiplication by the original first arrow on the actual vertex complexes. -/ | |
| def arrow01Map (a : Fin (d+1)) := | |
| arrowComplex k (B P) (ThreeVertexAlgebra.vertex (composition P) 0) | |
| (ThreeVertexAlgebra.vertex (composition P) 1) (arrow01 P a) (arrow01_target P a) | |
| (InducedRectification.total (ConcreteQuadraticDiagram.quadratic P) D₀ D₁ D₂ f g H) | |
| /-- Left multiplication by the original second arrow on the actual vertex complexes. -/ | |
| def arrow12Map (b : Fin (d+1)) := | |
| arrowComplex k (B P) (ThreeVertexAlgebra.vertex (composition P) 1) | |
| (ThreeVertexAlgebra.vertex (composition P) 2) (arrow12 P b) (arrow12_target P b) | |
| (InducedRectification.total (ConcreteQuadraticDiagram.quadratic P) D₀ D₁ D₂ f g H) | |
| /-- The homotopy uses exactly the original first and second vertex equivalences. -/ | |
| def compatibility01 (a : Fin (d+1)) : Homotopy | |
| (FirstVertex.iota P D₀ D₁ D₂ f g H ≫ arrow01Map P D₀ D₁ D₂ f g H a) | |
| (f a ≫ SecondVertex.iota P D₀ D₁ D₂ f g H) := by | |
| rw [FirstVertex.iota, SecondVertex.iota_original, | |
| diagonal_original, diagonal_original] | |
| exact NativeBridge.nativeCompatibility01 (ConcreteQuadraticDiagram.quadratic P) | |
| D₀ D₁ D₂ f g H ((ThreeVertexAlgebra.directedSystem (composition P)).idem 0) | |
| ((ThreeVertexAlgebra.directedSystem (composition P)).idem 1) a (arrow01_target P a) | |
| /-- The homotopy uses exactly the original second and third vertex equivalences. -/ | |
| def compatibility12 (b : Fin (d+1)) : Homotopy | |
| (SecondVertex.iota P D₀ D₁ D₂ f g H ≫ arrow12Map P D₀ D₁ D₂ f g H b) | |
| (g b ≫ ThirdVertex.iota P D₀ D₁ D₂ f g H) := by | |
| rw [SecondVertex.iota_original, ThirdVertex.iota_original, | |
| diagonal_original, diagonal_original] | |
| exact NativeBridge.nativeCompatibility12 (ConcreteQuadraticDiagram.quadratic P) | |
| D₀ D₁ D₂ f g H ((ThreeVertexAlgebra.directedSystem (composition P)).idem 1) | |
| ((ThreeVertexAlgebra.directedSystem (composition P)).idem 2) b (arrow12_target P b) | |
| end LittleFinitistic.CanonicalArrowCompatibility | |
| end | |
| end OAI | |