| import Mathlib |
| import OAI.Combinatorics.Chromatic.Shuffle.PrimitiveOperators |
| import OAI.Combinatorics.Chromatic.Shuffle.OriginalDegree |
| import OAI.Combinatorics.Chromatic.Shuffle.GlobalTensorDetection |
| import OAI.Combinatorics.Chromatic.Walls.UnitalGradeComparison |
|
|
| namespace OAI |
|
|
| section |
| namespace ElementaryPositivity.RawShuffle |
| open scoped TensorProduct DirectSum |
| open DimensionSplit ElementaryPositivity.SlopeArithmetic |
| variable {I : Type*} [Fintype I] [DecidableEq I] |
| attribute [local instance] Classical.propDecidable |
| variable (a : I → I → ℕ) (c η : I → ℝ) (hc : ∀ i,0<c i) (θ : ℝ) |
| [Fact (SlopeEulerSymmetric a c η θ)] |
|
|
| lemma globalPrimitive_iff_local (d : slopeDimensions c η hc θ) (hd : d.val≠0) (W : ℤ) |
| (x : UnitalSourceGrade a c η hc θ d.val W) : |
| DirectSum.lof ℚ _ (unitalComponent a c η hc θ) (d,W) x ∈ globalPrimitives a c η hc θ ↔ |
| unitalGradeNonzeroEquiv a c η hc θ d.val hd W x ∈ |
| primitiveSpace a c η hc θ Fact.out d.val W := by |
| rw [globalPrimitive_iff_split_zero a c η hc θ (d,W) hd x, |
| primitiveSpace_iff_coproducts_zero] |
| constructor |
| · intro hx L R hL hR hde |
| have hls : OnSlopeOrZero c η θ L.dim := Or.inr (SplitTree.slope_dim c η hc hL) |
| have hrs : OnSlopeOrZero c η θ R.dim := Or.inr (SplitTree.slope_dim c η hc hR) |
| let s : SlopeSplit c η hc θ d.val := ⟨ofPair L.dim R.dim hde,by |
| constructor |
| · exact hls |
| · simpa only [right_ofPair] using hrs⟩ |
| have hz := hx s (SplitTree.dim_ne_zero hL) (by simpa only [s,right_ofPair] using SplitTree.dim_ne_zero hR) |
| rw [globalSplitCoproduct_dimensions a c η hc θ (d,W) s L.dim R.dim rfl |
| (right_ofPair _ _ hde) hls hrs hde x] at hz |
| have hz' := (globalTensorGradeInclusion_injective a c η hc θ ⟨L.dim,hls⟩ ⟨R.dim,hrs⟩ W) |
| (hz.trans (map_zero _).symm) |
| have hcomp := unitalGradeCoproduct_nonzero a c η hc θ L.dim R.dim |
| (SplitTree.dim_ne_zero hL) (SplitTree.dim_ne_zero hR) (hls.compatible hrs) |
| ((SplitTree.slope_dim c η hc hL).trans (SplitTree.slope_dim c η hc hR).symm) W |
| (unitalGradeCast a c η hc θ hde.symm rfl x) |
| rw [hz',map_zero,unitalGradeNonzeroEquiv_cast a c η hc θ hde.symm hd] at hcomp |
| exact hcomp.symm |
| · intro hx s hl hr |
| have hL : (SplitTree.leaf (left s.val)).OnSlope c η θ := ⟨hl,s.property.1.resolve_left hl⟩ |
| have hR : (SplitTree.leaf (right s.val)).OnSlope c η θ := ⟨hr,s.property.2.resolve_left hr⟩ |
| have hz := hx (.leaf (left s.val)) (.leaf (right s.val)) hL hR (left_add_right s.val) |
| have hcomp := unitalGradeCoproduct_nonzero a c η hc θ (left s.val) (right s.val) |
| hl hr (s.property.1.compatible s.property.2) (hL.2.trans hR.2.symm) W |
| (unitalGradeCast a c η hc θ (left_add_right s.val).symm rfl x) |
| rw [unitalGradeNonzeroEquiv_cast a c η hc θ (left_add_right s.val).symm hd,hz] at hcomp |
| have hz' := (unitalTensorGradeNonzeroEquiv a c η hc θ (left s.val) (right s.val) hl hr W).injective |
| (hcomp.trans (map_zero _).symm) |
| change globalTensorGradeInclusion a c η hc θ _ _ _ |
| (unitalGradeCoproduct a c η hc θ _ _ _ _ (unitalGradeCast a c η hc θ _ _ x))=0 |
| rw [hz',map_zero] |
|
|
| end ElementaryPositivity.RawShuffle |
|
|
| end |
| section |
| namespace ElementaryPositivity.RawShuffle |
| open MvPolynomial CommonTranslation |
| variable {I : Type*} [Fintype I] [DecidableEq I] |
|
|
| omit [DecidableEq I] in |
| lemma derivativeS_component (d : I → ℕ) (ℓ : ℤ) (f : S d) : |
| derivativeS d (componentS d ℓ f)=componentS d (ℓ-1) (derivativeS d f) := by |
| classical |
| rw [←sum_componentS d f] |
| simp only [map_sum] |
| apply Finset.sum_congr rfl |
| intro m hm |
| have hh : (derivativeS d (componentS d m f)).val.IsWeightedHomogeneous (fun _=>(1:ℤ)) (m-1) := by |
| rw [derivativeS_val] |
| exact D_homogeneous _ _ (componentS_homogeneous d m f) |
| rw [componentS_componentS,componentS_of_homogeneous d _ hh] |
| by_cases h : ℓ=m |
| · subst m |
| simp |
| · simp only [ite_eq_right h,ite_eq_right (by omega : ℓ-1≠m-1),map_zero] |
|
|
| lemma derivativeB_component (a : I → I → ℕ) (μ : (I → ℕ) → ℝ) (d : I → ℕ) (ℓ : ℤ) (f : B a μ d) : |
| derivativeB a μ d (componentB a μ d ℓ f)=componentB a μ d (ℓ-1) (derivativeB a μ d f) := by |
| induction f using Submodule.Quotient.induction_on with |
| | H f => |
| change derivativeB a μ d (quotientAlg a μ d (componentS d ℓ f))= |
| componentB a μ d (ℓ-1) (derivativeB a μ d (quotientAlg a μ d f)) |
| rw [derivativeB_mk,derivativeB_mk] |
| change quotientAlg a μ d (derivativeS d (componentS d ℓ f))= |
| quotientAlg a μ d (componentS d (ℓ-1) (derivativeS d f)) |
| rw [derivativeS_component] |
|
|
| omit [DecidableEq I] in |
| lemma meanS_homogeneous (d : I → ℕ) : |
| (meanS d).val.IsWeightedHomogeneous (fun _=>(1:ℤ)) 1 := by |
| classical |
| apply (weightedHomogeneousSubmodule ℚ (fun _ : (Σ i,Fin (d i))=>(1:ℤ)) 1).smul_mem |
| exact IsWeightedHomogeneous.sum _ _ 1 (fun v _=>isWeightedHomogeneous_X ℚ (fun _=>(1:ℤ)) v) |
|
|
| omit [DecidableEq I] in |
| lemma componentS_mean (d : I → ℕ) (ℓ : ℤ) (f : S d) : |
| componentS d (ℓ+1) (meanS d*f)=meanS d*componentS d ℓ f := by |
| apply Subtype.ext |
| change weightedHomogeneousComponent (fun _=>(1:ℤ)) (ℓ+1) ((meanS d).val*f.val)=_ |
| rw [add_comm ℓ 1,Homogeneity.component_mul _ _ (meanS_homogeneous d) ℓ] |
| rfl |
|
|
| lemma meanMultiplyB_component (a : I → I → ℕ) (μ : (I → ℕ) → ℝ) (d : I → ℕ) (ℓ : ℤ) (f : B a μ d) : |
| componentB a μ d (ℓ+1) (meanMultiplyB a μ d f)=meanMultiplyB a μ d (componentB a μ d ℓ f) := by |
| induction f using Submodule.Quotient.induction_on with |
| | H f => |
| change componentB a μ d (ℓ+1) (quotientAlg a μ d (meanS d)*quotientAlg a μ d f)= |
| quotientAlg a μ d (meanS d)*quotientAlg a μ d (componentS d ℓ f) |
| rw [←map_mul,←map_mul] |
| change quotientAlg a μ d (componentS d (ℓ+1) (meanS d*f))=_ |
| rw [componentS_mean] |
|
|
| variable (a : I → I → ℕ) (c η : I → ℝ) (hc : ∀ i,0<c i) (θ : ℝ) |
| lemma gradeDerivative_component (d : I → ℕ) (W ℓ : ℤ) (f : SourceAssociatedGrade a c η hc θ d W) : |
| gradeDerivative a c η hc θ d W (associatedComponent a c η hc θ d W ℓ f)= |
| associatedComponent a c η hc θ d W (ℓ-1) (gradeDerivative a c η hc θ d W f) := by |
| induction f using Submodule.Quotient.induction_on with |
| | H f => |
| apply congrArg (LinearFiltration.mk (sourceFiltration a c η hc θ d W) (sourceFiltration a c η hc θ d (W+1))) |
| apply Subtype.ext |
| exact derivativeB_component a (SlopeArithmetic.slope c η) d ℓ f.val |
|
|
| lemma gradeMeanMultiply_component (d : I → ℕ) (W ℓ : ℤ) (f : SourceAssociatedGrade a c η hc θ d W) : |
| associatedComponent a c η hc θ d W (ℓ+1) (gradeMeanMultiply a c η hc θ d W f)= |
| gradeMeanMultiply a c η hc θ d W (associatedComponent a c η hc θ d W ℓ f) := by |
| induction f using Submodule.Quotient.induction_on with |
| | H f => |
| apply congrArg (LinearFiltration.mk (sourceFiltration a c η hc θ d W) (sourceFiltration a c η hc θ d (W+1))) |
| apply Subtype.ext |
| exact meanMultiplyB_component a (SlopeArithmetic.slope c η) d ℓ f.val |
|
|
| end ElementaryPositivity.RawShuffle |
|
|
| end |
| section |
| namespace ElementaryPositivity.RawShuffle |
| variable {I : Type*} [Fintype I] [DecidableEq I] |
| attribute [local instance] Classical.propDecidable |
| variable (a : I → I → ℕ) (c η : I → ℝ) (hc : ∀ i,0<c i) (θ : ℝ) |
| (hχ : SlopeEulerSymmetric a c η θ) (d : I → ℕ) (W : ℤ) |
|
|
| noncomputable local instance : AddCommGroup (primitiveSpace a c η hc θ hχ d W) := Submodule.addCommGroup _ |
| noncomputable local instance : Module ℚ (primitiveSpace a c η hc θ hχ d W) := Submodule.module _ |
| noncomputable local instance : AddCommGroup (LinearMap.ker (primitiveDerivative a c η hc θ hχ d W)) := Submodule.addCommGroup _ |
| noncomputable local instance : Module ℚ (LinearMap.ker (primitiveDerivative a c η hc θ hχ d W)) := Submodule.module _ |
|
|
| noncomputable def primitiveComponent (ℓ : ℤ) : |
| Module.End ℚ (primitiveSpace a c η hc θ hχ d W) := |
| LinearFiltration.restrict _ _ (associatedComponent a c η hc θ d W ℓ) |
| (associatedComponent_primitive a c η hc θ hχ d W ℓ) |
|
|
| lemma primitiveDerivative_component (ℓ : ℤ) (x : primitiveSpace a c η hc θ hχ d W) : |
| primitiveDerivative a c η hc θ hχ d W (primitiveComponent a c η hc θ hχ d W ℓ x)= |
| primitiveComponent a c η hc θ hχ d W (ℓ-1) (primitiveDerivative a c η hc θ hχ d W x) := |
| Subtype.ext (gradeDerivative_component a c η hc θ d W ℓ x.val) |
|
|
| lemma primitiveMeanMultiply_component (ℓ : ℤ) (x : primitiveSpace a c η hc θ hχ d W) : |
| primitiveComponent a c η hc θ hχ d W (ℓ+1) (primitiveMeanMultiply a c η hc θ hχ d W x)= |
| primitiveMeanMultiply a c η hc θ hχ d W (primitiveComponent a c η hc θ hχ d W ℓ x) := |
| Subtype.ext (gradeMeanMultiply_component a c η hc θ d W ℓ x.val) |
|
|
| lemma primitiveComponent_ker (ℓ : ℤ) (x : primitiveSpace a c η hc θ hχ d W) |
| (hx : x∈LinearMap.ker (primitiveDerivative a c η hc θ hχ d W)) : |
| primitiveComponent a c η hc θ hχ d W ℓ x∈LinearMap.ker (primitiveDerivative a c η hc θ hχ d W) := by |
| rw [LinearMap.mem_ker] at hx ⊢ |
| rw [primitiveDerivative_component,hx,map_zero] |
|
|
| noncomputable def stringBaseComponent (ℓ : ℤ) : |
| Module.End ℚ (LinearMap.ker (primitiveDerivative a c η hc θ hχ d W)) := |
| LinearFiltration.restrict _ _ (primitiveComponent a c η hc θ hχ d W ℓ) |
| (primitiveComponent_ker a c η hc θ hχ d W ℓ) |
|
|
| lemma stringBaseComponent_twice (ℓ m : ℤ) |
| (x : LinearMap.ker (primitiveDerivative a c η hc θ hχ d W)) : |
| stringBaseComponent a c η hc θ hχ d W ℓ (stringBaseComponent a c η hc θ hχ d W m x)= |
| if ℓ=m then stringBaseComponent a c η hc θ hχ d W m x else 0 := by |
| split_ifs with h |
| · apply Subtype.ext; apply Subtype.ext |
| exact (associatedComponent_twice a c η hc θ d W ℓ m x.val.val).trans (ite_eq_left h) |
| · apply Subtype.ext; apply Subtype.ext |
| exact (associatedComponent_twice a c η hc θ d W ℓ m x.val.val).trans (ite_eq_right h) |
| |
| noncomputable def homogeneousStringBases : Set (LinearMap.ker (primitiveDerivative a c η hc θ hχ d W)) := |
| {x | ∃ ℓ,stringBaseComponent a c η hc θ hχ d W ℓ x=x} |
|
|
| lemma homogeneousStringBases_span : |
| ⊤ ≤ Submodule.span ℚ (homogeneousStringBases a c η hc θ hχ d W) := by |
| intro x hx |
| obtain ⟨s,hs,he⟩:=associatedComponent_finite_decomposition a c η hc θ d W x.val.val |
| have he' : (∑ ℓ∈s,stringBaseComponent a c η hc θ hχ d W ℓ x)=x := by |
| apply Subtype.ext; apply Subtype.ext |
| simp only [Submodule.coe_sum] |
| exact he |
| rw [←he'] |
| apply Submodule.sum_mem |
| intro ℓ hℓ |
| apply Submodule.subset_span |
| exact ⟨ℓ,(stringBaseComponent_twice a c η hc θ hχ d W ℓ ℓ x).trans (ite_eq_left rfl)⟩ |
| |
| noncomputable def StringBaseIndex : Type _ := |
| (linearIndepOn_empty ℚ (id : LinearMap.ker (primitiveDerivative a c η hc θ hχ d W) → _)).extend |
| (Set.empty_subset (homogeneousStringBases a c η hc θ hχ d W)) |
|
|
| noncomputable def homogeneousStringBaseBasis : |
| Module.Basis (StringBaseIndex a c η hc θ hχ d W) ℚ |
| (LinearMap.ker (primitiveDerivative a c η hc θ hχ d W)) := |
| Module.Basis.ofSpan (homogeneousStringBases_span a c η hc θ hχ d W) |
|
|
| lemma homogeneousStringBaseBasis_exists_degree (i : StringBaseIndex a c η hc θ hχ d W) : |
| ∃ ℓ,stringBaseComponent a c η hc θ hχ d W ℓ (homogeneousStringBaseBasis a c η hc θ hχ d W i)= |
| homogeneousStringBaseBasis a c η hc θ hχ d W i := |
| Module.Basis.ofSpan_subset (homogeneousStringBases_span a c η hc θ hχ d W) (Set.mem_range_self i) |
|
|
| noncomputable def stringBaseDegree (i : StringBaseIndex a c η hc θ hχ d W) : ℤ := |
| (homogeneousStringBaseBasis_exists_degree a c η hc θ hχ d W i).choose |
|
|
| lemma stringBaseDegree_spec (i : StringBaseIndex a c η hc θ hχ d W) : |
| primitiveComponent a c η hc θ hχ d W (stringBaseDegree a c η hc θ hχ d W i) |
| (homogeneousStringBaseBasis a c η hc θ hχ d W i).val= |
| (homogeneousStringBaseBasis a c η hc θ hχ d W i).val := |
| congrArg Subtype.val (homogeneousStringBaseBasis_exists_degree a c η hc θ hχ d W i).choose_spec |
|
|
| lemma stringBaseDegree_nonneg (i : StringBaseIndex a c η hc θ hχ d W) : |
| 0 ≤ stringBaseDegree a c η hc θ hχ d W i := by |
| by_contra h |
| have hs:=congrArg Subtype.val (stringBaseDegree_spec a c η hc θ hχ d W i) |
| change associatedComponent a c η hc θ d W _ _=_ at hs |
| rw [associatedComponent_negative a c η hc θ d W _ (lt_of_not_ge h),LinearMap.zero_apply] at hs |
| exact (homogeneousStringBaseBasis a c η hc θ hχ d W).ne_zero i |
| (Subtype.ext (Subtype.ext hs.symm)) |
|
|
| lemma primitiveMeanMultiply_pow_component (ℓ : ℤ) (n : ℕ) (x : primitiveSpace a c η hc θ hχ d W) |
| (hx : primitiveComponent a c η hc θ hχ d W ℓ x=x) : |
| primitiveComponent a c η hc θ hχ d W (ℓ+n) ((primitiveMeanMultiply a c η hc θ hχ d W^n) x)= |
| ((primitiveMeanMultiply a c η hc θ hχ d W^n) x) := by |
| induction n with |
| | zero => simpa only [Nat.cast_zero,add_zero,pow_zero,Module.End.one_apply] using hx |
| | succ n ih => |
| rw [Nat.cast_succ,←add_assoc,pow_succ',Module.End.mul_apply,primitiveMeanMultiply_component,ih] |
|
|
| noncomputable def homogeneousPrimitiveStringBasis (hd : d≠0) : |
| Module.Basis (Σ _ : ℕ,StringBaseIndex a c η hc θ hχ d W) ℚ |
| (primitiveSpace a c η hc θ hχ d W) := |
| (Finsupp.basis (fun _ : ℕ=>homogeneousStringBaseBasis a c η hc θ hχ d W)).map |
| (primitiveStringDecomposition a c η hc θ hχ d hd W) |
|
|
| lemma homogeneousPrimitiveStringBasis_apply (hd : d≠0) |
| (i : Σ _ : ℕ,StringBaseIndex a c η hc θ hχ d W) : |
| homogeneousPrimitiveStringBasis a c η hc θ hχ d W hd i= |
| (primitiveMeanMultiply a c η hc θ hχ d W^i.1) (homogeneousStringBaseBasis a c η hc θ hχ d W i.2).val := by |
| rw [homogeneousPrimitiveStringBasis,Module.Basis.map_apply,Finsupp.coe_basis] |
| exact PrimitiveStrings.evaluate_single _ _ _ _ |
|
|
| lemma homogeneousPrimitiveStringBasis_degree (hd : d≠0) |
| (i : Σ _ : ℕ,StringBaseIndex a c η hc θ hχ d W) : |
| primitiveComponent a c η hc θ hχ d W (stringBaseDegree a c η hc θ hχ d W i.2+i.1) |
| (homogeneousPrimitiveStringBasis a c η hc θ hχ d W hd i)= |
| homogeneousPrimitiveStringBasis a c η hc θ hχ d W hd i := by |
| rw [homogeneousPrimitiveStringBasis_apply] |
| exact primitiveMeanMultiply_pow_component a c η hc θ hχ d W _ _ _ |
| (stringBaseDegree_spec a c η hc θ hχ d W i.2) |
|
|
| end ElementaryPositivity.RawShuffle |
|
|
| end |
|
|
| end OAI |
|
|
