| import Mathlib | |
| import OAI.Combinatorics.Chromatic.Walls.CenterLineSpecialization | |
| import OAI.Combinatorics.Chromatic.Shuffle.SymbolDetection | |
| import OAI.Combinatorics.Chromatic.Shuffle.CellSeparation | |
| namespace OAI | |
| section | |
| namespace ElementaryPositivity.RawShuffle | |
| open MvPolynomial HahnSeries | |
| open ElementaryPositivity.LaurentAtInfinity SeparationInfinity | |
| open scoped TensorProduct | |
| variable {I : Type*} [Fintype I] [DecidableEq I] | |
| attribute [local instance] cancelTensorRing cancelTensorAlg cancelFourRing cancelFourAlg | |
| attribute [local instance] cancelTensorSelf cancelTensorNonUnital cancelFourNonUnital cancelSeriesRing | |
| attribute [local instance] cellSepTensorSemiring cellSepTensorNonAssoc cellSepFourSemiring cellSepFourNonAssoc cellSepFourModule | |
| lemma mappedInverseKernel_ratio {R : Type*} [CommRing R] [Algebra ℚ R] | |
| (a b : I → I → ℕ) (d e : I → ℕ) (f : S d⊗[ℚ]S e →ₐ[ℚ]R) : | |
| mapRing f.toRingHom ((inverseKernelUnit a d e)*(inverseKernelUnit b d e)⁻¹).val= | |
| (mappedInverseKernel a d e f).val * ((mappedInverseKernel b d e f)⁻¹).val := by | |
| rw [Units.val_mul,map_mul] | |
| rfl | |
| lemma cellTransfer_normalized (a : I → I → ℕ) (d₁ e₁ d₂ e₂ : I → ℕ) | |
| (f : S (d₁+d₂)) (g : S (e₁+e₂)) : | |
| ((inverseKernelUnit a (d₁+e₁) (d₂+e₂))* | |
| (inverseKernelUnit (fun _ _ : I=>0) (d₁+e₁) (d₂+e₂))⁻¹).val * | |
| polynomial (relativeTaylor (d₁+e₁) (d₂+e₂) | |
| (cellTransfer a d₁ e₁ d₂ e₂ (fourGridPolynomial a f g d₁ e₁ d₂ e₂)))= | |
| ((-1 : ℚ)^eulerForm a d₂ e₁) • | |
| mapLinear (twoTargetTransfer a d₁ e₁ d₂ e₂) (fourInputSeries a d₁ e₁ d₂ e₂ f g) := by | |
| rw [cellTransfer,LinearMap.comp_apply,←twoTarget_polynomial_relativeTaylor, | |
| ←twoTarget_Laurent_projection a d₁ e₁ d₂ e₂ (firstCut d₁ e₁) (firstCut d₂ e₂), | |
| mappedInverseKernel_ratio,fourGrid_normalized] | |
| exact (mapLinear (twoTargetTransfer a d₁ e₁ d₂ e₂)).map_smul _ _ | |
| end ElementaryPositivity.RawShuffle | |
| end | |
| section | |
| namespace ElementaryPositivity.RawShuffle | |
| open scoped TensorProduct | |
| open ElementaryPositivity.LaurentAtInfinity | |
| open SplitTree | |
| variable {I : Type*} [Fintype I] [DecidableEq I] | |
| omit [DecidableEq I] in | |
| lemma eulerForm_transpose (a : I → I → ℕ) (d e : I → ℕ) : | |
| eulerForm (fun i j=>a j i) d e=eulerForm a e d := by | |
| unfold eulerForm | |
| congr 1 | |
| · exact Finset.sum_congr rfl (fun i hi=>mul_comm _ _) | |
| · rw [Finset.sum_comm] | |
| apply Finset.sum_congr rfl | |
| intro i hi | |
| apply Finset.sum_congr rfl | |
| intro j hj | |
| ring | |
| namespace SeparationInfinity | |
| noncomputable def mixedInverseKernelUnit (a b : I → I → ℕ) | |
| (μ : (I → ℕ) → ℝ) (d e : I → ℕ) : | |
| (LaurentSeries (B a μ d⊗[ℚ] B a μ e))ˣ := | |
| Units.map (mapRing (R:=S d⊗[ℚ]S e) (S:=B a μ d⊗[ℚ]B a μ e) | |
| (Algebra.TensorProduct.map (quotientAlg a μ d) (quotientAlg a μ e)).toRingHom).toMonoidHom | |
| (inverseKernelUnit b d e) | |
| lemma pointed_mixedInverseKernel (a b : I → I → ℕ) (c η : I → ℝ) | |
| (hc : ∀ i,0<c i) (θ : ℝ) (L R : SplitTree I) | |
| (hL : L.OnSlope c η θ) (hR : R.OnSlope c η θ) | |
| (v : L.Centers → ℚ) (w : R.Centers → ℚ) : | |
| mapRing (pointedTensorRestriction a c η hc θ L R hL hR v w).toRingHom | |
| (mixedInverseKernelUnit a b (SlopeArithmetic.slope c η) L.dim R.dim).val = | |
| mapRing (algebraMap ℚ (tensor (quotientFamily a (SlopeArithmetic.slope c η)) (.node L R))) | |
| (mapRing (crossPointEval L R v w) (rawInverseKernelUnit b L.dim R.dim).val) := by | |
| apply HahnSeries.ext | |
| funext k | |
| change pointedTensorRestriction a c η hc θ L R hL hR v w | |
| (Algebra.TensorProduct.map (quotientAlg a (SlopeArithmetic.slope c η) L.dim) | |
| (quotientAlg a (SlopeArithmetic.slope c η) R.dim) ((inverseKernelUnit b L.dim R.dim).val.coeff k))=_ | |
| rw [pointedTensorRestriction_quotient] | |
| have h:=congrArg (fun f : LaurentSeries (MvPolynomial (CellVars L.dim R.dim) ℚ)=>f.coeff k) | |
| (realize_inverseKernelUnit b L.dim R.dim) | |
| exact congrArg (fun f=>algebraMap ℚ _ (crossPointEval L R v w f)) h | |
| lemma refined_mixedKernel_degreeZero (a b : I → I → ℕ) (c η : I → ℝ) | |
| (hc : ∀ i,0<c i) (θ : ℝ) (L R : SplitTree I) | |
| (hL : L.OnSlope c η θ) (hR : R.OnSlope c η θ) | |
| (v : L.Centers → ℚ) (w : R.Centers → ℚ) : | |
| mapRing (degreeZeroTensor a (SlopeArithmetic.slope c η) (.node L R)).toRingHom | |
| (mapRing (refinedAtB a c η hc θ L R hL hR v w).toRingHom | |
| (mixedInverseKernelUnit a b (SlopeArithmetic.slope c η) L.dim R.dim).val)= | |
| mapRing (algebraMap ℚ (tensor (quotientFamily a (SlopeArithmetic.slope c η)) (.node L R))) | |
| (scalarCrossKernel b L R v w).val := by | |
| trans mapRing (pointedTensorRestriction a c η hc θ L R hL hR v w).toRingHom | |
| (mixedInverseKernelUnit a b (SlopeArithmetic.slope c η) L.dim R.dim).val | |
| · apply HahnSeries.ext | |
| funext k | |
| exact degreeZero_refinedAtB a c η hc θ L R hL hR v w _ | |
| · rw [pointed_mixedInverseKernel] | |
| congr 1 | |
| exact congrArg Units.val (crossPointEval_inverseKernel_scalar b L R v w) | |
| noncomputable def braidingRatioUnit (a : I → I → ℕ) (μ : (I → ℕ) → ℝ) | |
| (d e : I → ℕ) : (LaurentSeries (B a μ d⊗[ℚ]B a μ e))ˣ := | |
| mixedInverseKernelUnit a a μ d e * | |
| (mixedInverseKernelUnit a (fun i j=>a j i) μ d e)⁻¹ | |
| end SeparationInfinity | |
| namespace SplitTree | |
| omit [DecidableEq I] in | |
| lemma scalarCrossKernel_transpose (a : I → I → ℕ) (c η : I → ℝ) | |
| (θ : ℝ) (hχ : SlopeEulerSymmetric a c η θ) (L R : SplitTree I) | |
| (hL : L.OnSlope c η θ) (hR : R.OnSlope c η θ) | |
| (v : L.Centers → ℚ) (w : R.Centers → ℚ) : | |
| scalarCrossKernel (fun i j=>a j i) L R v w=scalarCrossKernel a L R v w := by | |
| unfold scalarCrossKernel | |
| apply Finset.prod_congr rfl | |
| intro z hz | |
| apply Finset.prod_congr rfl | |
| intro t ht | |
| rw [eulerForm_transpose] | |
| congr 1 | |
| obtain ⟨hz,hzs⟩:=L.leafDimension_onSlope c η θ hL z | |
| obtain ⟨ht,hts⟩:=R.leafDimension_onSlope c η θ hR t | |
| exact (hχ _ _ hz ht hzs hts).symm | |
| end SplitTree | |
| namespace SeparationInfinity | |
| lemma refined_braidingRatio_degreeZero (a : I → I → ℕ) (c η : I → ℝ) | |
| (hc : ∀ i,0<c i) (θ : ℝ) (hχ : SlopeEulerSymmetric a c η θ) | |
| (L R : SplitTree I) (hL : L.OnSlope c η θ) (hR : R.OnSlope c η θ) | |
| (v : L.Centers → ℚ) (w : R.Centers → ℚ) : | |
| mapRing (degreeZeroTensor a (SlopeArithmetic.slope c η) (.node L R)).toRingHom | |
| (mapRing (refinedAtB a c η hc θ L R hL hR v w).toRingHom | |
| (braidingRatioUnit a (SlopeArithmetic.slope c η) L.dim R.dim).val)=1 := by | |
| let H := (mapRing (degreeZeroTensor a (SlopeArithmetic.slope c η) (.node L R)).toRingHom).comp | |
| (mapRing (refinedAtB a c η hc θ L R hL hR v w).toRingHom) | |
| let Φ := Units.map H.toMonoidHom | |
| have he : Φ (mixedInverseKernelUnit a a (SlopeArithmetic.slope c η) L.dim R.dim)= | |
| Φ (mixedInverseKernelUnit a (fun i j=>a j i) (SlopeArithmetic.slope c η) L.dim R.dim) := by | |
| apply Units.ext | |
| change H _=H _ | |
| dsimp only [H,RingHom.comp_apply] | |
| rw [refined_mixedKernel_degreeZero,refined_mixedKernel_degreeZero, | |
| scalarCrossKernel_transpose a c η θ hχ L R hL hR] | |
| have hh : Φ (braidingRatioUnit a (SlopeArithmetic.slope c η) L.dim R.dim)=1 := by | |
| rw [braidingRatioUnit,map_mul,map_inv,he,mul_inv_cancel] | |
| exact congrArg Units.val hh | |
| end SeparationInfinity | |
| end ElementaryPositivity.RawShuffle | |
| end | |
| end OAI | |