| import OAI.Analysis.CircleDomains.Modulus.BarrierTrialMetric |
| import OAI.Analysis.CircleDomains.Modulus.CodeModulusExtraction |
|
|
| namespace OAI |
|
|
| /-! A uniform modulus bound, derived from the constructed compatible |
| barrier and valid for the actual finite constrained coding-measure family. -/ |
|
|
| open Set Function Filter MeasureTheory |
| open scoped ENNReal Topology |
|
|
| noncomputable section |
| namespace CircleDomainRigidity |
|
|
| attribute [local instance] sphereMetric sphereBorel |
| local instance : BorelSpace Sphere := ⟨rfl⟩ |
| variable [MeasurableSpace C(PathTime, Sphere)] |
|
|
| theorem barrier_modulus_lower_bound |
| (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) [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) : |
| (β / (2 * barrierConstructionConstant C)) ^ 2 ≤ |
| measureFamilyModulus (ConstrainedCostCode d a c ν) (PathCostSpace Sphere A) |
| (pathCostReference sphericalArea c (tupleReference ω)) (constrainedCodeMeasure d a c ν) := by |
| apply le_modulus_of_admissible_norm_bound |
| intro h hh hadm |
| by_cases heh : (∫⁻z, h z ^ 2 ∂pathCostReference sphericalArea c (tupleReference ω)) < ∞ |
| · apply ENNReal.div_le_of_le_mul' |
| exact trial_metric_norm_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 |
| h hh heh ((costAdmissible_iff_constrained d a c ν h).mpr hadm) β hno |
| · have ht : (∫⁻z, h z ^ 2 ∂pathCostReference sphericalArea c (tupleReference ω)) = ∞ := |
| top_unique (not_lt.mp heh) |
| simp [squareIntegralNorm, ht] |
|
|
| end CircleDomainRigidity |
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| end |
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| end OAI |
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