| import Mathlib |
| import OAI.Combinatorics.Chromatic.Shuffle.HNExpansion |
|
|
| namespace OAI |
|
|
| section |
| namespace ElementaryPositivity.RawShuffle |
| open MvPolynomial |
| open ElementaryPositivity.ShufflePolynomiality ElementaryPositivity.PackConvolution |
| variable {I : Type*} [Fintype I] [DecidableEq I] |
| variable {A : I → Type*} [∀ i,Fintype (A i)] [∀ i,DecidableEq (A i)] |
|
|
| omit [DecidableEq I] [∀ i,Fintype (A i)] [∀ i,DecidableEq (A i)] in |
| lemma sameDen_reverse_eq (s t : Pack (A:=A)) : |
| sameDen t s = (-1 : MvPolynomial (Σi,A i) ℚ) ^ (∑ i,(s i).card*(t i).card) * sameDen s t := by |
| unfold sameDen |
| calc |
| _ = ∏ i,∏ x∈s i,∏ y∈t i, |
| (-1 : MvPolynomial (Σi,A i) ℚ)*diagonal ⟨i,x⟩ ⟨i,y⟩ := by |
| apply Finset.prod_congr rfl |
| intro i hi |
| rw [Finset.prod_comm] |
| apply Finset.prod_congr rfl |
| intro x hx |
| apply Finset.prod_congr rfl |
| intro y hy |
| unfold diagonal |
| ring |
| _ = _ := by |
| simp only [Finset.prod_mul_distrib,Finset.prod_const,←pow_mul] |
| rw [Finset.prod_pow_eq_pow_sum] |
| congr 2 |
| apply Finset.sum_congr rfl |
| intro i hi |
| exact Nat.mul_comm _ _ |
|
|
| omit [∀ i,Fintype (A i)] in |
| lemma crossKernelPolynomial_explicit (a : I → I → ℕ) {s : Pack (A:=A)} |
| (p : PackConvolution.Cut s) (q : PackConvolution.Cut (left p)) |
| (r : PackConvolution.Cut (right p)) : |
| crossKernelPolynomial a p q r = |
| (-1 : MvPolynomial (Σi,A i) ℚ) ^ (∑ i,(right q i).card*(left r i).card) * |
| sameDen (left q) (left r) * sameDen (right q) (right r) * |
| arrowNum a (left q) (right r) * arrowNum a (left r) (right q) := by |
| apply (IsFractionRing.injective (MvPolynomial (Σi,A i) ℚ) |
| (FractionRing (MvPolynomial (Σi,A i) ℚ))) |
| rw [←crossKernelPolynomial_spec,kernel_polynomial_ratio,kernel_polynomial_ratio] |
| have hq : (fun i=>left q i ∪ right q i)=left p := funext fun i=>(q i).property.2 |
| have hr : (fun i=>left r i ∪ right r i)=right p := funext fun i=>(r i).property.2 |
| have hd : sameDen (left p) (right p)= |
| sameDen (left q) (left r)*sameDen (left q) (right r)* |
| ((-1 : MvPolynomial (Σi,A i) ℚ) ^ (∑ i,(right q i).card*(left r i).card)* |
| sameDen (left r) (right q))*sameDen (right q) (right r) := by |
| conv_lhs => rw [←hq,←hr] |
| rw [sameDen_union_left (left q) (right q) (fun i=>left r i ∪ right r i) (fun i=>(q i).property.1), |
| sameDen_union_right (left q) (left r) (right r) (fun i=>(r i).property.1), |
| sameDen_union_right (right q) (left r) (right r) (fun i=>(r i).property.1)] |
| rw [sameDen_reverse_eq (left r) (right q)] |
| have he : (∑ i,(left r i).card*(right q i).card)=∑ i,(right q i).card*(left r i).card := by |
| apply Finset.sum_congr rfl |
| intro i hi |
| exact Nat.mul_comm _ _ |
| rw [he] |
| ring |
| have hne := crossDen_ne_zero p q r |
| have h₁ : algebraMap (MvPolynomial (Σi,A i) ℚ) (FractionRing (MvPolynomial (Σi,A i) ℚ)) |
| (sameDen (left q) (right r))≠0 := by |
| simpa only [map_zero] using |
| (IsFractionRing.injective (MvPolynomial (Σi,A i) ℚ) (FractionRing (MvPolynomial (Σi,A i) ℚ))).ne |
| (mul_ne_zero_iff.mp hne).1 |
| have h₂ : algebraMap (MvPolynomial (Σi,A i) ℚ) (FractionRing (MvPolynomial (Σi,A i) ℚ)) |
| (sameDen (left r) (right q))≠0 := by |
| simpa only [map_zero] using |
| (IsFractionRing.injective (MvPolynomial (Σi,A i) ℚ) (FractionRing (MvPolynomial (Σi,A i) ℚ))).ne |
| (mul_ne_zero_iff.mp hne).2 |
| rw [hd] |
| simp only [map_mul,map_pow,map_neg,map_one] |
| field_simp |
|
|
| end ElementaryPositivity.RawShuffle |
|
|
| end |
| section |
| namespace ElementaryPositivity.RawShuffle |
| open MvPolynomial |
| open ElementaryPositivity.ShufflePolynomiality ElementaryPositivity.PackConvolution |
| variable {I : Type*} [Fintype I] [DecidableEq I] |
|
|
| noncomputable def fourForwardDifference (d₁ e₁ d₂ e₂ : I → ℕ) |
| (i j : I) (x : Fin (d₁ i)⊕Fin (e₁ i)) (y : Fin (d₂ j)⊕Fin (e₂ j)) : |
| MvPolynomial (CellVars d₁ e₁⊕CellVars d₂ e₂) ℚ := |
| X (Sum.inr ((Equiv.sigmaSumDistrib _ _) ⟨j,y⟩)) - |
| X (Sum.inl ((Equiv.sigmaSumDistrib _ _) ⟨i,x⟩)) |
|
|
| omit [DecidableEq I] in |
| lemma fourSameDen11 (d₁ e₁ d₂ e₂ : I → ℕ) : |
| rename (fourPackEquiv d₁ e₁ d₂ e₂) (sameDen (left (canonicalLeftCut d₁ e₁ d₂ e₂)) (left (canonicalRightCut d₁ e₁ d₂ e₂)))= |
| ∏ i,∏ x:Fin (d₁ i),∏ y:Fin (d₂ i),fourForwardDifference d₁ e₁ d₂ e₂ i i (.inl x) (.inl y) := by |
| simp only [sameDen,map_prod,canonicalLeftCut,canonicalRightCut,embedPackCut, |
| sumPackCut,left,right,Finset.prod_map,ShufflePolynomiality.diagonal,map_sub,rename_X] |
| rfl |
|
|
| omit [DecidableEq I] in |
| lemma fourSameDen22 (d₁ e₁ d₂ e₂ : I → ℕ) : |
| rename (fourPackEquiv d₁ e₁ d₂ e₂) (sameDen (right (canonicalLeftCut d₁ e₁ d₂ e₂)) (right (canonicalRightCut d₁ e₁ d₂ e₂)))= |
| ∏ i,∏ x:Fin (e₁ i),∏ y:Fin (e₂ i),fourForwardDifference d₁ e₁ d₂ e₂ i i (.inr x) (.inr y) := by |
| simp only [sameDen,map_prod,canonicalLeftCut,canonicalRightCut,embedPackCut, |
| sumPackCut,left,right,Finset.prod_map,ShufflePolynomiality.diagonal,map_sub,rename_X] |
| rfl |
|
|
| omit [DecidableEq I] in |
| lemma fourArrow12 (a : I → I → ℕ) (d₁ e₁ d₂ e₂ : I → ℕ) : |
| rename (fourPackEquiv d₁ e₁ d₂ e₂) (arrowNum a (left (canonicalLeftCut d₁ e₁ d₂ e₂)) (right (canonicalRightCut d₁ e₁ d₂ e₂)))= |
| ∏ i,∏ j,∏ x:Fin (d₁ i),∏ y:Fin (e₂ j),fourForwardDifference d₁ e₁ d₂ e₂ i j (.inl x) (.inr y)^a i j := by |
| simp only [arrowNum,map_prod,map_pow,canonicalLeftCut,canonicalRightCut,embedPackCut, |
| sumPackCut,left,right,Finset.prod_map,ShufflePolynomiality.diagonal,map_sub,rename_X] |
| rfl |
|
|
| omit [Fintype I] [DecidableEq I] in |
| lemma fourReverseDifference (d₁ e₁ d₂ e₂ : I → ℕ) (i j : I) |
| (x : Fin (d₂ i)) (y : Fin (e₁ j)) : |
| rename (fourPackEquiv d₁ e₁ d₂ e₂) |
| (ShufflePolynomiality.diagonal ⟨i,Sum.inr (Sum.inl x)⟩ ⟨j,Sum.inl (Sum.inr y)⟩)= |
| -fourForwardDifference d₁ e₁ d₂ e₂ j i (.inr y) (.inl x) := by |
| simp only [ShufflePolynomiality.diagonal,map_sub,rename_X] |
| unfold fourForwardDifference |
| rw [neg_sub] |
| rfl |
|
|
| omit [DecidableEq I] in |
| lemma fourArrow21 (a : I → I → ℕ) (d₁ e₁ d₂ e₂ : I → ℕ) : |
| rename (fourPackEquiv d₁ e₁ d₂ e₂) (arrowNum a (left (canonicalRightCut d₁ e₁ d₂ e₂)) (right (canonicalLeftCut d₁ e₁ d₂ e₂)))= |
| ∏ i,∏ j,∏ x:Fin (d₂ i),∏ y:Fin (e₁ j),(-fourForwardDifference d₁ e₁ d₂ e₂ j i (.inr y) (.inl x))^a i j := by |
| simp only [arrowNum,map_prod,map_pow,canonicalLeftCut,canonicalRightCut,embedPackCut, |
| sumPackCut,left,right,Finset.prod_map] |
| apply Finset.prod_congr rfl |
| intro i hi |
| apply Finset.prod_congr rfl |
| intro j hj |
| apply Finset.prod_congr rfl |
| intro x hx |
| apply Finset.prod_congr rfl |
| intro y hy |
| congr 1 |
| exact fourReverseDifference d₁ e₁ d₂ e₂ i j x y |
|
|
| lemma fourCrossPolynomial_explicit (a : I → I → ℕ) (d₁ e₁ d₂ e₂ : I → ℕ) : |
| fourCrossPolynomial a d₁ e₁ d₂ e₂ = |
| (-1 : MvPolynomial (CellVars d₁ e₁⊕CellVars d₂ e₂) ℚ) ^ (∑ i,e₁ i*d₂ i) * |
| (∏ i,∏ x:Fin (d₁ i),∏ y:Fin (d₂ i),fourForwardDifference d₁ e₁ d₂ e₂ i i (.inl x) (.inl y)) * |
| (∏ i,∏ x:Fin (e₁ i),∏ y:Fin (e₂ i),fourForwardDifference d₁ e₁ d₂ e₂ i i (.inr x) (.inr y)) * |
| (∏ i,∏ j,∏ x:Fin (d₁ i),∏ y:Fin (e₂ j),fourForwardDifference d₁ e₁ d₂ e₂ i j (.inl x) (.inr y)^a i j) * |
| (∏ i,∏ j,∏ x:Fin (d₂ i),∏ y:Fin (e₁ j),(-fourForwardDifference d₁ e₁ d₂ e₂ j i (.inr y) (.inl x))^a i j) := by |
| rw [fourCrossPolynomial,crossKernelPolynomial_explicit] |
| simp only [map_mul,map_pow,map_neg,map_one,fourSameDen11,fourSameDen22,fourArrow12,fourArrow21] |
| congr 5 |
| simp only [canonicalLeftCut,canonicalRightCut,embedPackCut,sumPackCut,left,right, |
| Finset.card_map,Finset.card_univ,Fintype.card_fin] |
|
|
| end ElementaryPositivity.RawShuffle |
|
|
| end |
|
|
| end OAI |
|
|
