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