| import Mathlib | |
| import OAI.Combinatorics.Chromatic.Shuffle.TensorSymbolSwap | |
| import OAI.Combinatorics.Chromatic.Shuffle.NormalizedTripleSymbol | |
| namespace OAI | |
| section | |
| namespace ElementaryPositivity.RawShuffle | |
| open ElementaryPositivity.LinearDetection ElementaryPositivity.LinearFiltration | |
| open SeparationInfinity | |
| open scoped TensorProduct | |
| variable {I : Type*} [Fintype I] [DecidableEq I] | |
| lemma tensorSeparationLeft_filtration (a : I → I → ℕ) (c η : I → ℝ) | |
| (hc : ∀ i,0<c i) (θ : ℝ) (d e f : I → ℕ) | |
| (hs : SlopeArithmetic.slope c η d=SlopeArithmetic.slope c η e) | |
| (W : ℤ) (x : B a (SlopeArithmetic.slope c η) (d+e)⊗[ℚ]B a (SlopeArithmetic.slope c η) f) | |
| (hx : x∈sourceTensorFiltration a c η hc θ (d+e) f W) : | |
| TensorProduct.map (separationCoefficient a c η hc hs 0) LinearMap.id x∈ | |
| sourceTripleFiltration a c η hc θ d e f W := | |
| tensor_map_mem_filtration (sourceFiltration a c η hc θ (d+e)) (sourceFiltration a c η hc θ f) | |
| (sourceTensorFiltration a c η hc θ d e) (sourceFiltration a c η hc θ f) | |
| (separationCoefficient a c η hc hs 0) LinearMap.id | |
| (fun w=>separationCoefficient_filtration a c η hc θ d e hs w 0) (fun _ _ h=>h) W x hx | |
| noncomputable def tensorSeparationLeft (a : I → I → ℕ) (c η : I → ℝ) | |
| (hc : ∀ i,0<c i) (θ : ℝ) (d e f : I → ℕ) | |
| (hs : SlopeArithmetic.slope c η d=SlopeArithmetic.slope c η e) (W : ℤ) : | |
| sourceTensorFiltration a c η hc θ (d+e) f W →ₗ[ℚ] | |
| sourceTripleFiltration a c η hc θ d e f W := | |
| let C : B a (SlopeArithmetic.slope c η) (d+e) →ₗ[ℚ] | |
| B a (SlopeArithmetic.slope c η) d⊗[ℚ]B a (SlopeArithmetic.slope c η) e := | |
| separationCoefficient a c η hc hs 0 | |
| let F : (B a (SlopeArithmetic.slope c η) (d+e)⊗[ℚ]B a (SlopeArithmetic.slope c η) f) →ₗ[ℚ] | |
| ((B a (SlopeArithmetic.slope c η) d⊗[ℚ]B a (SlopeArithmetic.slope c η) e)⊗[ℚ] | |
| B a (SlopeArithmetic.slope c η) f) := TensorProduct.map C LinearMap.id | |
| (F.comp (sourceTensorFiltration a c η hc θ (d+e) f W).subtype).codRestrict | |
| (sourceTripleFiltration a c η hc θ d e f W) | |
| (fun x=>tensorSeparationLeft_filtration a c η hc θ d e f hs W x.val x.property) | |
| namespace SplitTree | |
| open MvPolynomial ElementaryPositivity.CenterCalculus | |
| lemma normalizedTripleSymbol_tmul_eq (a : I → I → ℕ) (c η : I → ℝ) | |
| (hc : ∀ i,0<c i) (θ : ℝ) (L M N : SplitTree I) | |
| (hL : L.OnSlope c η θ) (hM : M.OnSlope c η θ) (hN : N.OnSlope c η θ) | |
| (hχL : L.PairSymmetric a) (hχM : M.PairSymmetric a) (hχN : N.PairSymmetric a) | |
| (W U V : ℤ) (hw : W=U+V) | |
| (x : sourceTensorFiltration a c η hc θ L.dim M.dim U) | |
| (y : sourceFiltration a c η hc θ N.dim V) : | |
| normalizedTripleSymbol a c η hc θ L M N hL hM hN hχL hχM hχN W | |
| ⟨x.val⊗ₜ[ℚ]y.val,Submodule.subset_span ⟨U,V,x.val,y.val,hw.le,x.property,y.property,rfl⟩⟩= | |
| box (normalizedTensorSymbol a c η hc θ L M hL hM hχL hχM U x) | |
| (normalizedSymbol a c η hc θ N hN hχN V y) := by | |
| subst W | |
| exact normalizedTripleSymbol_tmul a c η hc θ L M N hL hM hN hχL hχM hχN U V x y | |
| theorem tensorSeparationLeft_normalized_symbol (a : I → I → ℕ) (c η : I → ℝ) | |
| (hc : ∀ i,0<c i) (θ : ℝ) (hχ : SlopeEulerSymmetric a c η θ) | |
| (L M N : SplitTree I) (hL : L.OnSlope c η θ) (hM : M.OnSlope c η θ) | |
| (hN : N.OnSlope c η θ) (W : ℤ) | |
| (x : sourceTensorFiltration a c η hc θ (L.dim+M.dim) N.dim W) : | |
| normalizedTripleSymbol a c η hc θ L M N hL hM hN | |
| (pairSymmetric_of_slope a c η θ hχ L hL) (pairSymmetric_of_slope a c η θ hχ M hM) | |
| (pairSymmetric_of_slope a c η θ hχ N hN) W | |
| (tensorSeparationLeft a c η hc θ L.dim M.dim N.dim | |
| ((slope_dim c η hc hL).trans (slope_dim c η hc hM).symm) W x)= | |
| normalizedTensorSymbol a c η hc θ (.node L M) N ⟨hL,hM⟩ hN | |
| (pairSymmetric_of_slope a c η θ hχ (.node L M) ⟨hL,hM⟩) | |
| (pairSymmetric_of_slope a c η θ hχ N hN) W x := by | |
| let hLM := (slope_dim c η hc hL).trans (slope_dim c η hc hM).symm | |
| let χL:=pairSymmetric_of_slope a c η θ hχ L hL | |
| let χM:=pairSymmetric_of_slope a c η θ hχ M hM | |
| let χN:=pairSymmetric_of_slope a c η θ hχ N hN | |
| let χLM:=pairSymmetric_of_slope a c η θ hχ (.node L M) ⟨hL,hM⟩ | |
| let F:=(normalizedTripleSymbol a c η hc θ L M N hL hM hN χL χM χN W).comp | |
| (tensorSeparationLeft a c η hc θ L.dim M.dim N.dim hLM W) | |
| let G:=normalizedTensorSymbol a c η hc θ (.node L M) N ⟨hL,hM⟩ hN χLM χN W | |
| have he : F=G := by | |
| apply linearMap_span_ext | |
| rintro _ ⟨u,v,p,q,huv,hp,hq,rfl⟩ | |
| have hq' := sourceFiltration_antitone a c η hc θ N.dim (show W-u≤v by omega) hq | |
| let p0:=separationCoefficientRestricted a c η hc θ L.dim M.dim hLM u 0 ⟨p,hp⟩ | |
| have hp0:=normalizedTripleSymbol_tmul_eq a c η hc θ L M N hL hM hN χL χM χN | |
| W u (W-u) (by omega) p0 ⟨q,hq'⟩ | |
| change normalizedTripleSymbol a c η hc θ L M N hL hM hN χL χM χN W | |
| ⟨p0.val⊗ₜ[ℚ]q,_⟩=G ⟨p⊗ₜ[ℚ]q,_⟩ | |
| rw [hp0] | |
| have hp1:=coproduct_normalized_symbol a c η hc θ hχ L M hL hM u ⟨p,hp⟩ | |
| have hp2:=normalizedTensorSymbol_tmul_eq a c η hc θ (.node L M) N ⟨hL,hM⟩ hN χLM χN | |
| W u (W-u) (by omega) ⟨p,hp⟩ ⟨q,hq'⟩ | |
| change box (normalizedTensorSymbol a c η hc θ L M hL hM χL χM u p0) | |
| (normalizedSymbol a c η hc θ N hN χN (W-u) ⟨q,hq'⟩)=_ | |
| change normalizedTensorSymbol a c η hc θ L M hL hM χL χM u p0= | |
| normalizedSymbol a c η hc θ (.node L M) ⟨hL,hM⟩ χLM u ⟨p,hp⟩ at hp1 | |
| rw [hp1] | |
| exact hp2.symm | |
| exact LinearMap.congr_fun he x | |
| end SplitTree | |
| end ElementaryPositivity.RawShuffle | |
| end | |
| end OAI | |