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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 | 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
end
end OAI
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