| import Mathlib |
| import OAI.Combinatorics.Chromatic.Shuffle.CanonicalCard |
| import OAI.Combinatorics.Chromatic.Shuffle.LaurentKernel |
| import OAI.Combinatorics.Chromatic.GradedAlgebra.LaurentRectangularKernel |
| import OAI.Combinatorics.Chromatic.GradedAlgebra.LaurentScalarNaturality |
|
|
| namespace OAI |
|
|
| section |
| namespace ElementaryPositivity.RawShuffle |
| open MvPolynomial HahnSeries |
| open ElementaryPositivity.LaurentAtInfinity ElementaryPositivity.RectangularKernel |
| open scoped TensorProduct |
| variable {I : Type*} [Fintype I] [DecidableEq I] |
|
|
| noncomputable local instance clearingTensorRing (d e : I → ℕ) : CommRing (S d⊗[ℚ]S e) := inferInstance |
| noncomputable local instance clearingTensorAlg (d e : I → ℕ) : Algebra ℚ (S d⊗[ℚ]S e) := inferInstance |
|
|
| lemma rawInverseKernel_zero (d e : I → ℕ) : |
| (SeparationInfinity.rawInverseKernelUnit (fun _ _ : I=>0) d e).val= |
| ∏ i,∏ x : Fin (d i),∏ y : Fin (e i), |
| (affineUnit (X (Sum.inr ⟨i,y⟩)-X (Sum.inl ⟨i,x⟩) : MvPolynomial (CellVars d e) ℚ)).val := by |
| have h : SeparationInfinity.rawInverseKernelUnit (fun _ _ : I=>0) d e= |
| rectangular (fun i j : I=>if i=j then 1 else 0) |
| (fun i j x y=>affineUnit (X (Sum.inr ⟨j,y⟩)-X (Sum.inl ⟨i,x⟩) : MvPolynomial (CellVars d e) ℚ)) := by |
| simp only [SeparationInfinity.rawInverseKernelUnit,SeparationInfinity.rawFactor, |
| SeparationInfinity.inverseExponent,Nat.cast_zero,sub_zero,rectangular] |
| rw [h,rectangular_diagonal] |
| simp only [Units.coe_prod] |
|
|
| lemma polynomial_crossDifference (d e : I → ℕ) : |
| polynomial (relativeRaw d e (crossDifference d e))= |
| crossSign d e • (SeparationInfinity.rawInverseKernelUnit (fun _ _ : I=>0) d e).val := by |
| rw [relativeRaw_crossDifference,rawInverseKernel_zero] |
| rw [rat_smul_series] |
| simp only [map_prod,crossSign,←Finset.prod_mul_distrib] |
| apply Finset.prod_congr rfl |
| intro i hi |
| apply Finset.prod_congr rfl |
| intro x hx |
| apply Finset.prod_congr rfl |
| intro y hy |
| simp only [map_add,polynomial_X,polynomial_C,affineUnit_val,map_neg,map_one,neg_one_mul] |
| simp only [map_sub] |
| ring |
|
|
| lemma polynomial_crossTensor (d e : I → ℕ) : |
| polynomial (relativeTaylor d e (crossTensor d e))= |
| crossSign d e • (SeparationInfinity.inverseKernelUnit (fun _ _ : I=>0) d e).val := by |
| apply SeparationInfinity.realize_injective d e |
| rw [polynomial_map,map_relativeTaylor,tensorValue_crossTensor] |
| rw [mapRing_rat_smul, |
| SeparationInfinity.realize_inverseKernelUnit,polynomial_crossDifference] |
|
|
| omit [DecidableEq I] in |
| lemma crossSign_sq (d e : I → ℕ) : crossSign d e * crossSign d e=1 := by |
| rw [crossSign,←Finset.prod_mul_distrib] |
| simp only [←Finset.prod_mul_distrib,neg_mul_neg,one_mul,Finset.prod_const_one] |
|
|
| end ElementaryPositivity.RawShuffle |
|
|
| end |
|
|
| end OAI |
|
|
