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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
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