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9655094 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 | 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
end CircleDomainRigidity
end
end OAI
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