| import Mathlib |
| import OAI.Combinatorics.Chromatic.Shuffle.TensorColumnSeriesFiltration |
| import OAI.Combinatorics.Chromatic.GradedAlgebra.ClearedSeparation |
|
|
| namespace OAI |
|
|
| section |
| namespace ElementaryPositivity.RawShuffle |
| open scoped TensorProduct |
| open ElementaryPositivity.LaurentAtInfinity ElementaryPositivity.SlopeArithmetic |
| open SeparationInfinity |
| variable {I : Type*} [Fintype I] [DecidableEq I] |
| noncomputable local instance clearBTensorS (d e : I → ℕ) : CommRing (S d⊗[ℚ]S e) := inferInstance |
| noncomputable local instance clearBTensorSA (d e : I → ℕ) : Algebra ℚ (S d⊗[ℚ]S e) := inferInstance |
| noncomputable local instance clearBTensorB (a : I → I → ℕ) (μ : (I → ℕ) → ℝ) (d e : I → ℕ) : |
| CommRing (B a μ d⊗[ℚ]B a μ e) := inferInstance |
| noncomputable local instance clearBTensorBA (a : I → I → ℕ) (μ : (I → ℕ) → ℝ) (d e : I → ℕ) : |
| Algebra ℚ (B a μ d⊗[ℚ]B a μ e) := inferInstance |
|
|
| noncomputable def clearedSeparationB (a : I → I → ℕ) (μ : (I → ℕ) → ℝ) (d e : I → ℕ) : |
| B a μ d⊗[ℚ]B a μ e →ₗ[ℚ] LaurentSeries (B a μ d⊗[ℚ]B a μ e) := |
| (mulPolynomial (mixedInverseKernelUnit a a μ d e * |
| (mixedInverseKernelUnit a (fun _ _ : I=>0) μ d e)⁻¹).val).comp |
| (relativeTaylorB a μ d e).toLinearMap |
|
|
| lemma mapRing_ratio_val {R T : Type*} [CommRing R] [CommRing T] |
| (f : R →+* T) (u v : (LaurentSeries R)ˣ) : |
| mapRing f (u*v⁻¹).val= |
| (Units.map (mapRing f).toMonoidHom u*(Units.map (mapRing f).toMonoidHom v)⁻¹).val := by |
| rw [Units.val_mul,map_mul] |
| rfl |
|
|
| lemma clearedSeparationB_mk (a : I → I → ℕ) (μ : (I → ℕ) → ℝ) (d e : I → ℕ) |
| (x : S d⊗[ℚ]S e) : |
| clearedSeparationB a μ d e (quotientTensor a μ d e x)= |
| mapRing (quotientTensor a μ d e).toRingHom |
| ((inverseKernelUnit a d e*(inverseKernelUnit (fun _ _ : I=>0) d e)⁻¹).val * |
| polynomial (relativeTaylor d e x)) := by |
| rw [mapRing_mul_polynomial,mapRing_ratio_val,relativeTaylor_quotient] |
| rfl |
|
|
| variable {A : I → Type*} [∀ i,Fintype (A i)] [∀ i,DecidableEq (A i)] |
| lemma unitalSeparationSeries_shuffle_grid (a : I → I → ℕ) (c η : I → ℝ) (hc : ∀ i,0<c i) |
| {d e α β : I → ℕ} (h : d+e=α+β) |
| (hs : α=0 ∨ β=0 ∨ slope c η α=slope c η β) |
| (f : S d) (g : S e) {s : ElementaryPositivity.PackConvolution.Pack (A:=A)} |
| (R : Realization (α+β) s) : |
| unitalSeparationSeries a c η hc hs |
| (quotientAlg a (slope c η) (α+β) (castS h (shufflePolynomial a f g)))= |
| clearedSeparationB a (slope c η) α β |
| (quotientTensor a (slope c η) α β |
| (gridTensor a f g (cutRealizationEquiv R (firstCut α β)).val |
| (leftRealization R (firstCut α β)) (rightRealization R (firstCut α β)))) := by |
| rw [unitalSeparationSeries_mk,separation_shuffle_grid a h f g R,clearedSeparationB_mk] |
|
|
| lemma clearedSeparationB_zero_of_quotient (a : I → I → ℕ) (μ : (I → ℕ) → ℝ) (d e : I → ℕ) |
| (x : S d⊗[ℚ]S e) (hx : quotientTensor a μ d e x=0) : |
| clearedSeparationB a μ d e (quotientTensor a μ d e x)=0 := by |
| rw [hx] |
| exact (clearedSeparationB a μ d e).map_zero |
|
|
| end ElementaryPositivity.RawShuffle |
|
|
| end |
| section |
| namespace ElementaryPositivity.RawShuffle |
| open scoped TensorProduct |
| open ElementaryPositivity.SlopeArithmetic ElementaryPositivity.LinearDetection |
| variable {I : Type*} [Fintype I] [DecidableEq I] |
|
|
| noncomputable local instance gridTensorBCommRing (a : I → I → ℕ) (μ : (I → ℕ) → ℝ) (d e : I → ℕ) : |
| CommRing (B a μ d ⊗[ℚ] B a μ e) := inferInstance |
| noncomputable local instance gridTensorBAlgebra (a : I → I → ℕ) (μ : (I → ℕ) → ℝ) (d e : I → ℕ) : |
| Algebra ℚ (B a μ d ⊗[ℚ] B a μ e) := inferInstance |
|
|
| lemma quotient_cellTransfer_filtered (a : I → I → ℕ) (c η : I → ℝ) (hc : ∀ i,0<c i) |
| (θ : ℝ) (d₁ e₁ d₂ e₂ : I → ℕ) |
| (hd : d₁=0 ∨ d₂=0 ∨ slope c η d₁=slope c η d₂) |
| (he : e₁=0 ∨ e₂=0 ∨ slope c η e₁=slope c η e₂) |
| (h₁ : d₁=0 ∨ e₁=0 ∨ slope c η d₁=slope c η e₁) |
| (h₂ : d₂=0 ∨ e₂=0 ∨ slope c η d₂=slope c η e₂) |
| (ih₁ : ∀ u v x y,x∈unitalSourceFiltration a c η hc θ d₁ u → |
| y∈unitalSourceFiltration a c η hc θ e₁ v → |
| shuffleBUnit a c η hc d₁ e₁ h₁ x y∈unitalSourceFiltration a c η hc θ (d₁+e₁) (u+v)) |
| (ih₂ : ∀ u v x y,x∈unitalSourceFiltration a c η hc θ d₂ u → |
| y∈unitalSourceFiltration a c η hc θ e₂ v → |
| shuffleBUnit a c η hc d₂ e₂ h₂ x y∈unitalSourceFiltration a c η hc θ (d₂+e₂) (u+v)) |
| (U V : ℤ) (f : S (d₁+d₂)) (g : S (e₁+e₂)) |
| (hf : quotientAlg a (slope c η) (d₁+d₂) f∈unitalSourceFiltration a c η hc θ (d₁+d₂) U) |
| (hg : quotientAlg a (slope c η) (e₁+e₂) g∈unitalSourceFiltration a c η hc θ (e₁+e₂) V) : |
| quotientTensor a (slope c η) (d₁+e₁) (d₂+e₂) |
| (cellTransfer a d₁ e₁ d₂ e₂ (fourGridPolynomial a f g d₁ e₁ d₂ e₂))∈ |
| unitalSourceTensorFiltration a c η hc θ (d₁+e₁) (d₂+e₂) (U+V) := by |
| let F := unitalSourceFiltration a c η hc θ |
| have hm (d) := unitalSourceFiltration_mul_mem a c η hc θ d |
| have htf := restrictionBUnit_unitalSourceTensorFiltration a c η hc θ d₁ d₂ hd |
| (firstCut d₁ d₂) U _ hf |
| have htg := restrictionBUnit_unitalSourceTensorFiltration a c η hc θ e₁ e₂ he |
| (firstCut e₁ e₂) V _ hg |
| have hx := interchange_mem_filtration (F d₁) (F d₂) (F e₁) (F e₂) U V _ _ htf htg |
| have hmul := additiveTensorFiltration_mul_mem |
| (additiveTensorFiltration (F d₁) (F e₁)) (additiveTensorFiltration (F d₂) (F e₂)) |
| (additiveTensorFiltration_mul_mem (F d₁) (F e₁) (hm d₁) (hm e₁)) |
| (additiveTensorFiltration_mul_mem (F d₂) (F e₂) (hm d₂) (hm e₂)) (U+V) _ |
| (fourQuotient a (slope c η) d₁ e₁ d₂ e₂ (fourCrossTensor a d₁ e₁ d₂ e₂)) hx |
| erw [←fourQuotient_grid_factor a c η hc d₁ e₁ d₂ e₂ hd he |
| (firstCut d₁ d₂) (firstCut e₁ e₂) f g] at hmul |
| change quotientTensor a (slope c η) (d₁+e₁) (d₂+e₂) |
| (twoTargetTransfer a d₁ e₁ d₂ e₂ _)∈_ |
| rw [quotient_twoTargetTransfer_unit a c η hc d₁ e₁ d₂ e₂ h₁ h₂] |
| exact tensor_map_mem_filtration _ _ (F (d₁+e₁)) (F (d₂+e₂)) |
| (shuffleTensorBUnit a c η hc d₁ e₁ h₁) (shuffleTensorBUnit a c η hc d₂ e₂ h₂) |
| (lift_mem_filtration _ _ _ (unitalSourceFiltration_antitone a c η hc θ (d₁+e₁)) _ ih₁) |
| (lift_mem_filtration _ _ _ (unitalSourceFiltration_antitone a c η hc θ (d₂+e₂)) _ ih₂) |
| (U+V) _ hmul |
|
|
| end ElementaryPositivity.RawShuffle |
|
|
| end |
|
|
| end OAI |
|
|
