| import OAI.Analysis.CircleDomains.Modulus.BarrierModulus |
| import OAI.Analysis.CircleDomains.Probability.CountableLawExtraction |
|
|
| namespace OAI |
|
|
| /-! Failure of a small compatible barrier yields a probability law, |
| using ADMS duality, Bogachev lifting, and the resulting modulus bound. -/ |
|
|
| open Set Function Filter MeasureTheory |
| open scoped ENNReal NNReal Topology |
|
|
| noncomputable section |
| namespace CircleDomainRigidity |
|
|
| attribute [local instance] sphereMetric sphereBorel |
| local instance : BorelSpace Sphere := ⟨rfl⟩ |
| theorem exists_positive_barrier_duality_parameter (C : ℝ) {β : ℝ≥0∞} (hβ : 0 < β) : |
| ∃r : ℝ≥0, 0 < r ∧ (r : ℝ≥0∞) < (β / (2 * barrierConstructionConstant C)) ^ 2 := by |
| have hden : 2 * barrierConstructionConstant C ≠ ∞ := |
| ENNReal.mul_ne_top (by norm_num) (barrierConstructionConstant_ne_top C) |
| have hpos : 0 < (β / (2 * barrierConstructionConstant C)) ^ 2 := |
| pos_iff_ne_zero.mpr (pow_ne_zero 2 (ENNReal.div_pos hβ.ne' hden).ne') |
| obtain ⟨r, hr, hrb⟩ := ENNReal.lt_iff_exists_nnreal_btwn.mp hpos |
| exact ⟨r, ENNReal.coe_pos.mp hr, hrb⟩ |
|
|
| variable [MeasurableSpace C(PathTime, Sphere)] [BorelSpace C(PathTime, Sphere)] |
|
|
| theorem code_law_of_no_small_compatible_barrier |
| (hcoarea : WeightedPlanarScalarCoarea) (hLusin : FiniteTorusAnalyticMeasurability) (hsard : SmoothPlanarSard) |
| (hsch : PlanarSchoenflies) |
| (d : PathCodeData Sphere Sphere) (hU : IsOpen d.quotientDomain) |
| (hG : IsOpen d.physicalDomain) (hdisj : Disjoint d.quotientDomain d.dust) |
| (π : Sphere → Sphere) (hπ : Continuous π) (hclosed : IsClosedMap π) (hsurj : Surjective π) |
| (hr : ∀x y, π x = π y ↔ x = y ∨ |
| (x ∈ d.physicalDomainᶜ ∧ y ∈ connectedComponentIn d.physicalDomainᶜ x)) |
| (he : ∀x : d.physicalDomain, (d.domainEquiv x).val = π x.val) |
| (hB : IsClosed d.dust) (hTD : IsTotallyDisconnected d.dust) |
| (hBU : d.dustᶜ ⊆ d.quotientDomain) |
| (hdense : Dense d.quotientDomain) (hinfty : π OnePoint.infty ∉ d.dust) |
| (hbad : ∀x ∈ d.physicalDomainᶜ, π x ∈ d.dust) |
| (ν : Measure Sphere) [IsFiniteMeasure ν] [NullSingletonClass ν] |
| (hinf : OnePoint.infty ∈ d.physicalDomain) |
| (EStar : Set Sphere) (hE : IsClosed d.anchor) (hEU : d.anchor ⊆ d.quotientDomain) |
| (hEStar : IsCompact EStar) (hEStarE : EStar ⊆ interior d.anchor) |
| (hinfE : π OnePoint.infty ∈ interior d.anchor) |
| (p : Sphere) (hpDust : p ∈ d.dust) (hp : p ≠ π OnePoint.infty) |
| {g₀ : ℂ → ℝ≥0∞} (D : ℕ → WeightedCollar d.physicalDomain π d.dust p g₀) |
| (hnest : ∀n, closure (D (n+1)).region ⊆ (D n).region) |
| (hshrink : ∀V ∈ 𝓝 p, ∃k : ℕ, ∀n ≥ k, closure (D n).region ⊆ V) |
| (hED : Disjoint d.anchor (closure (D 0).region)) |
| (f : d.quotientDomain → Sphere) (hf : Continuous f) |
| (W₁ W₂ : Set Sphere) (hW₁ : IsClosed W₁) (hW₂ : IsOpen W₂) (hW₁₂ : W₁ ⊆ W₂) |
| (hfiber : ∀x, IsConnected (π ⁻¹' {x})) |
| (harcs : ∀C : Set Sphere, IsCompact C → LocallyConnectedSpace C → |
| HasSimpleArcsInConnectedOpens C) |
| (previous : ℕ → PhysicalRealCoordinate d π) |
| (hlevels : ∀i, d.levels i = Prod.swap ⁻¹' |
| circleBandGraph d.quotientDomain (previous i).quotient) |
| (hweights : ∀i, (∑'x, d.levelWeights i x ^ 2) ≠ ∞) |
| (hpoint : ∀i x, volume {t | (t, x) ∈ d.levels i} ≤ d.levelWeights i x) |
| (flags : Set Sphere) (hflags : flags.Countable) |
| (C : ℝ) (hC : 0 ≤ C) |
| (hderiv : ∀z : ℂ, |
| ‖fderiv ℝ (fun z : ℂ => (roundSphereEquiv (z : Sphere) : RoundAmbient)) z‖ ≤ |
| C / (1 + ‖z‖ ^ 2)) |
| (htarget : d.target = p) |
| (hcontours : d.gateContours = fun j => frontier (D j).region) |
| (hgates : d.gateSets = fun _ => Subtype.val '' (f ⁻¹' closure W₂)) |
| {A : Type*} [TopologicalSpace A] [Countable A] [MeasurableSpace A] [BorelSpace A] |
| (a : A → Sphere) (ha : Injective a) (haB : range a ⊆ d.dust) |
| (c : A → ℝ≥0∞) (hc : (∑'s, c s ^ 2) < ∞) |
| (ω : TupleType → ℝ≥0∞) (hω : ∀i, ω i ≠ 0) |
| [PolishSpace A] [StandardBorelSpace (ConstrainedCostCode d a c ν)] |
| (β : ℝ≥0∞) |
| (hno : ∀R : CompatibleRealBarrier d π previous flags EStar (fun j => (D j).region) f W₁, |
| β ≤ R.controlledSize) |
| (hωsum : (∑'i, ω i) < ∞) |
| (hdual : MeasureImageDuality (ConstrainedCostCode d a c ν) (PathCostSpace Sphere A)) |
| (hlift : MeasurePlanLifting (ConstrainedCostCode d a c ν) (PathCostSpace Sphere A)) |
| (r : ℝ≥0) (hrpos : 0 < r) |
| (hrbound : (r : ℝ≥0∞) < (β / (2 * barrierConstructionConstant C)) ^ 2) : |
| ∃(P : ProbabilityMeasure (BasePathCode d)) (q : PathCostSpace Sphere A → ℝ≥0∞), |
| Measurable q ∧ |
| meanCodeCost d a c P = (pathCostReference sphericalArea c (tupleReference ω)).withDensity q ∧ |
| (∫⁻z, q z ^ 2 ∂pathCostReference sphericalArea c (tupleReference ω)) ≤ (r : ℝ≥0∞)⁻¹ ∧ |
| (∀ᵐ Γ ∂(P : Measure (BasePathCode d)), pathCostMeasure d a c Γ univ < ∞) ∧ |
| (∀ᵐ Γ ∂(P : Measure (BasePathCode d)), ν (residualImage d.dust Γ.val) = 0) := by |
| let : IsFiniteMeasure (tupleReference ω) := finite_tupleReference hωsum |
| have hmod := hrbound.trans_le (barrier_modulus_lower_bound hcoarea hLusin hsard hsch |
| d hU hG hdisj π hπ hclosed hsurj hr he hB hTD hBU hdense hinfty hbad ν hinf |
| EStar hE hEU hEStar hEStarE hinfE p hpDust hp D hnest hshrink hED f hf W₁ W₂ |
| hW₁ hW₂ hW₁₂ hfiber harcs previous hlevels hweights hpoint flags hflags C hC hderiv |
| htarget hcontours hgates a ha haB c hc ω hω β hno) |
| exact positive_modulus_code_law d hU hB a c ν sphericalArea (tupleReference ω) |
| hc hdual hlift r hrpos hmod |
|
|
| theorem prescribed_countable_law_of_no_small_barrier |
| (hcoarea : WeightedPlanarScalarCoarea) (hLusin : FiniteTorusAnalyticMeasurability) (hsard : SmoothPlanarSard) |
| (hsch : PlanarSchoenflies) |
| (d : PathCodeData Sphere Sphere) (hU : IsOpen d.quotientDomain) |
| (hG : IsOpen d.physicalDomain) (hdisj : Disjoint d.quotientDomain d.dust) |
| (π : Sphere → Sphere) (hπ : Continuous π) (hclosed : IsClosedMap π) (hsurj : Surjective π) |
| (hr : ∀x y, π x = π y ↔ x = y ∨ |
| (x ∈ d.physicalDomainᶜ ∧ y ∈ connectedComponentIn d.physicalDomainᶜ x)) |
| (he : ∀x : d.physicalDomain, (d.domainEquiv x).val = π x.val) |
| (hB : IsClosed d.dust) (hTD : IsTotallyDisconnected d.dust) |
| (hBU : d.dustᶜ ⊆ d.quotientDomain) |
| (hdense : Dense d.quotientDomain) (hinfty : π OnePoint.infty ∉ d.dust) |
| (hbad : ∀x ∈ d.physicalDomainᶜ, π x ∈ d.dust) |
| (ν : Measure Sphere) [IsFiniteMeasure ν] [NullSingletonClass ν] |
| (hinf : OnePoint.infty ∈ d.physicalDomain) |
| (EStar : Set Sphere) (hE : IsClosed d.anchor) (hEU : d.anchor ⊆ d.quotientDomain) |
| (hEStar : IsCompact EStar) (hEStarE : EStar ⊆ interior d.anchor) |
| (hinfE : π OnePoint.infty ∈ interior d.anchor) |
| (p : Sphere) (hpDust : p ∈ d.dust) (hp : p ≠ π OnePoint.infty) |
| {g₀ : ℂ → ℝ≥0∞} (D : ℕ → WeightedCollar d.physicalDomain π d.dust p g₀) |
| (hnest : ∀n, closure (D (n+1)).region ⊆ (D n).region) |
| (hshrink : ∀V ∈ 𝓝 p, ∃k : ℕ, ∀n ≥ k, closure (D n).region ⊆ V) |
| (hED : Disjoint d.anchor (closure (D 0).region)) |
| (f : d.quotientDomain → Sphere) (hf : Continuous f) |
| (W₁ W₂ : Set Sphere) (hW₁ : IsClosed W₁) (hW₂ : IsOpen W₂) (hW₁₂ : W₁ ⊆ W₂) |
| (hfiber : ∀x, IsConnected (π ⁻¹' {x})) |
| (harcs : ∀C : Set Sphere, IsCompact C → LocallyConnectedSpace C → |
| HasSimpleArcsInConnectedOpens C) |
| (previous : ℕ → PhysicalRealCoordinate d π) |
| (hlevels : ∀i, d.levels i = Prod.swap ⁻¹' |
| circleBandGraph d.quotientDomain (previous i).quotient) |
| (hweights : ∀i, (∑'x, d.levelWeights i x ^ 2) ≠ ∞) |
| (hpoint : ∀i x, volume {t | (t, x) ∈ d.levels i} ≤ d.levelWeights i x) |
| (flags : Set Sphere) (hflags : flags.Countable) |
| (C : ℝ) (hC : 0 ≤ C) |
| (hderiv : ∀z : ℂ, |
| ‖fderiv ℝ (fun z : ℂ => (roundSphereEquiv (z : Sphere) : RoundAmbient)) z‖ ≤ |
| C / (1 + ‖z‖ ^ 2)) |
| (htarget : d.target = p) |
| (hcontours : d.gateContours = fun j => frontier (D j).region) |
| (hgates : d.gateSets = fun _ => Subtype.val '' (f ⁻¹' closure W₂)) |
| {A : Type*} [TopologicalSpace A] [Countable A] [MeasurableSpace A] [BorelSpace A] |
| (a : A → Sphere) (ha : Injective a) (haB : range a ⊆ d.dust) |
| (c : A → ℝ≥0∞) (hc : (∑'s, c s ^ 2) < ∞) |
| (ω : TupleType → ℝ≥0∞) (hω : ∀i, ω i ≠ 0) |
| [PolishSpace A] [StandardBorelSpace (ConstrainedCostCode d a c ν)] |
| (β : ℝ≥0∞) |
| (hno : ∀R : CompatibleRealBarrier d π previous flags EStar (fun j => (D j).region) f W₁, |
| β ≤ R.controlledSize) |
| (hωsum : (∑'i, ω i) < ∞) |
| (hdual : MeasureImageDuality (ConstrainedCostCode d a c ν) (PathCostSpace Sphere A)) |
| (hlift : MeasurePlanLifting (ConstrainedCostCode d a c ν) (PathCostSpace Sphere A)) |
| (r : ℝ≥0) (hrpos : 0 < r) |
| (hrbound : (r : ℝ≥0∞) < (β / (2 * barrierConstructionConstant C)) ^ 2) |
| (hc0 : ∀s, c s ≠ 0) (hct : ∀s, c s ≠ ∞) |
| (hν : ν d.dustᶜ = 0) |
| (hunique : ∀x ∈ d.dust \ flags, ∀i t u, |
| (t, x) ∈ d.levels i → (u, x) ∈ d.levels i → t = u) : |
| ∃(P : ProbabilityMeasure (BasePathCode d)) (g : Sphere → ℝ≥0∞) (q : LevelTuple → ℝ≥0∞), |
| Measurable g ∧ Measurable q ∧ |
| meanCodeOccupation d P = sphericalArea.withDensity g ∧ |
| (∫⁻y, g y ^ 2 ∂sphericalArea) ≤ (r : ℝ≥0∞)⁻¹ ∧ |
| countingSquareNorm (fun s => meanVisitCount (codePathLaw d P) (a s)) ≤ |
| (r : ℝ≥0∞)⁻¹ ^ (1 / 2 : ℝ) + 1 ∧ |
| meanCodeTuple d P = (tupleReference ω).withDensity q ∧ |
| (∫⁻t, q t ^ 2 ∂tupleReference ω) ≤ (r : ℝ≥0∞)⁻¹ ∧ |
| (∀k, slotLaw d P k ≪ tupleReference ω) ∧ |
| (∫⁻x, pathHitProbability (codePathLaw d P) x ∂ν) = 0 ∧ |
| (∀ᵐ Γ ∂(P : Measure (BasePathCode d)), ν (residualImage d.dust Γ.val) = 0) := by |
| have hmod := hrbound.trans_le (barrier_modulus_lower_bound hcoarea hLusin hsard hsch |
| d hU hG hdisj π hπ hclosed hsurj hr he hB hTD hBU hdense hinfty hbad ν hinf |
| EStar hE hEU hEStar hEStarE hinfE p hpDust hp D hnest hshrink hED f hf W₁ W₂ |
| hW₁ hW₂ hW₁₂ hfiber harcs previous hlevels hweights hpoint flags hflags C hC hderiv |
| htarget hcontours hgates a ha haB c hc ω hω β hno) |
| have hH (i : ℕ) : IsClosed (d.levels i) := by |
| rw [hlevels] |
| exact (isCompact_circleBandGraph (fun x _ => by |
| rw [abs_of_nonneg ((previous i).quotient_unit x).1] |
| exact ((previous i).quotient_unit x).2)).isClosed.preimage continuous_swap |
| exact prescribed_countable_law_of_positive_modulus d hU hB hH a ha c hc0 hct hc ν |
| flags hflags hν hunique sphericalArea ω hωsum hdual hlift r hrpos hmod |
|
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|
|
| end CircleDomainRigidity |
|
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| end |
|
|
| end OAI |
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|
