import OAI.Analysis.CircleDomains.Topology.FiniteJordanModelInput
import OAI.Analysis.CircleDomains.Modulus.NormalizedUniformization

namespace OAI

/-! Circle-domain uniformization from the stated geometric, analytic,
and probability hypotheses for local and finite models. -/
open Set Function Filter MeasureTheory
open scoped Topology ENNReal NNReal
noncomputable section
namespace CircleDomainRigidity
open FiniteTransfer
attribute [local instance] sphereMetric sphereBorel
local instance : BorelSpace Sphere := ⟨rfl⟩
variable [MeasurableSpace C(PathTime, Sphere)] [BorelSpace C(PathTime, Sphere)]

theorem koebeCircleDomainConjecture_of_constructed_quotient
    (hcross : UnitDiskAlternatingContinua)
    (hplane : PlanarLipschitzFiberCoarea)
    (hcoarea : WeightedPlanarScalarCoarea) (hLusin : FiniteTorusAnalyticMeasurability)
    (hsard : SmoothPlanarSard) (hsch : PlanarSchoenflies) (hsep : UnitDiskCrosscutSeparation)
    (hcollar : SmoothRegularJordanCollars) (hselect : BorelPathRelationSelection)
    (hgreen : UnorientedFiniteCircleGreenFormula) (harea : ExteriorCoefficientAreaTheorem)
    (hcompact : SmoothCompactMetricSphereUniformization) (hfinite : FiniteJordanModelUniformization)
    (hmontel : MontelExhaustion) (hhurwitz : HurwitzUnivalentLimit) (hzero : HurwitzZeroFreeLimit)
    (harcs : ∀ C : Set Sphere, IsCompact C → LocallyConnectedSpace C →
      HasSimpleArcsInConnectedOpens C)
    (hstandard : ∀ e : PathCodeData Sphere Sphere, [StandardBorelSpace (BasePathCode e)] →
      StandardBorelProbabilityLaws (BasePathCode e))
    (hpath : PolishProbabilityLaws C(PathTime, Sphere))
    (hdual : ∀ (e : PathCodeData Sphere Sphere)
      (I : Set ℕ) [PolishSpace I] (b : I → Sphere) (c : I → ℝ≥0∞)
      (ν : Measure Sphere) [IsFiniteMeasure ν]
      [StandardBorelSpace (ConstrainedCostCode e b c ν)],
      MeasureImageDuality (ConstrainedCostCode e b c ν) (PathCostSpace Sphere I))
    (hlift : ∀ (e : PathCodeData Sphere Sphere)
      (I : Set ℕ) [PolishSpace I] (b : I → Sphere) (c : I → ℝ≥0∞)
      (ν : Measure Sphere) [IsFiniteMeasure ν]
      [StandardBorelSpace (ConstrainedCostCode e b c ν)],
      MeasurePlanLifting (ConstrainedCostCode e b c ν) (PathCostSpace Sphere I))
    : KoebeCircleDomainConjecture := by
  intro U hU hconn
  obtain ⟨x, hx⟩ := hconn.nonempty
  let N := pointToInfinity x
  have hG : IsOpen (N '' U) := N.isOpenMap U hU
  have hGconn : IsConnected (N '' U) := hconn.image _ N.continuous.continuousOn
  have hinf : OnePoint.infty ∈ N '' U := ⟨x, hx, pointToInfinity_apply_self x⟩
  have hN := isConformalEquivalence_pointToInfinity x U
  by_cases hwhole : N '' U = univ
  · exact ⟨N '' U, N, hwhole ▸ isCircleDomain_univ, hN⟩
  · obtain ⟨V, f, hV, hf⟩ := circle_uniformization_of_infty_mem_of_shrinking hcross hplane
      hcoarea hLusin hsard hsch hsep hcollar hselect hgreen harea hcompact hfinite
      hmontel hhurwitz hzero harcs hstandard hpath hdual hlift hG hGconn hinf hwhole
    exact ⟨V, f ∘ N, hV, hN.trans hf⟩


theorem koebeCircleDomainConjecture_of_constructed_contours
    (_hmoore : MooreSphereDecomposition)
    (hcross : UnitDiskAlternatingContinua)
    (hplane : PlanarLipschitzFiberCoarea)
    (hcoarea : WeightedPlanarScalarCoarea) (hLusin : FiniteTorusAnalyticMeasurability)
    (hsard : SmoothPlanarSard) (hsch : PlanarSchoenflies) (hsep : UnitDiskCrosscutSeparation)
    (hcollar : SmoothRegularJordanCollars) (hselect : BorelPathRelationSelection)
    (hgreen : UnorientedFiniteCircleGreenFormula) (harea : ExteriorCoefficientAreaTheorem)
    (hcompact : SmoothCompactMetricSphereUniformization) (hfinite : FiniteJordanModelUniformization)
    (hmontel : MontelExhaustion) (hhurwitz : HurwitzUnivalentLimit) (hzero : HurwitzZeroFreeLimit)
    (harcs : ∀ C : Set Sphere, IsCompact C → LocallyConnectedSpace C →
      HasSimpleArcsInConnectedOpens C)
    (hstandard : ∀ e : PathCodeData Sphere Sphere, [StandardBorelSpace (BasePathCode e)] →
      StandardBorelProbabilityLaws (BasePathCode e))
    (hpath : PolishProbabilityLaws C(PathTime, Sphere))
    (hdual : ∀ (e : PathCodeData Sphere Sphere)
      (I : Set ℕ) [PolishSpace I] (b : I → Sphere) (c : I → ℝ≥0∞)
      (ν : Measure Sphere) [IsFiniteMeasure ν]
      [StandardBorelSpace (ConstrainedCostCode e b c ν)],
      MeasureImageDuality (ConstrainedCostCode e b c ν) (PathCostSpace Sphere I))
    (hlift : ∀ (e : PathCodeData Sphere Sphere)
      (I : Set ℕ) [PolishSpace I] (b : I → Sphere) (c : I → ℝ≥0∞)
      (ν : Measure Sphere) [IsFiniteMeasure ν]
      [StandardBorelSpace (ConstrainedCostCode e b c ν)],
      MeasurePlanLifting (ConstrainedCostCode e b c ν) (PathCostSpace Sphere I))
    : KoebeCircleDomainConjecture := by
  exact koebeCircleDomainConjecture_of_constructed_quotient hcross hplane hcoarea hLusin hsard hsch hsep hcollar hselect hgreen harea hcompact hfinite hmontel hhurwitz hzero harcs hstandard hpath hdual hlift

theorem koebeCircleDomainConjecture_of_published_inputs
    (hmoore : MooreSphereDecomposition)
    (hcross : UnitDiskAlternatingContinua)
    (hplane : PlanarLipschitzFiberCoarea)
    (hcoarea : WeightedPlanarScalarCoarea) (hLusin : FiniteTorusAnalyticMeasurability)
    (hsard : SmoothPlanarSard) (hclass : CompactOneManifoldClassification)
    (hsch : PlanarSchoenflies) (hsep : UnitDiskCrosscutSeparation)
    (hcollar : SmoothRegularJordanCollars) (hselect : BorelPathRelationSelection)
    (hgreen : UnorientedFiniteCircleGreenFormula) (harea : ExteriorCoefficientAreaTheorem)
    (hcompact : SmoothCompactMetricSphereUniformization) (hfinite : FiniteCircleUniformization)
    (hmontel : MontelExhaustion) (hhurwitz : HurwitzUnivalentLimit) (hzero : HurwitzZeroFreeLimit)
    (harcs : ∀ C : Set Sphere, IsCompact C → LocallyConnectedSpace C →
      HasSimpleArcsInConnectedOpens C)
    (hstandard : ∀ e : PathCodeData Sphere Sphere, [StandardBorelSpace (BasePathCode e)] →
      StandardBorelProbabilityLaws (BasePathCode e))
    (hpath : PolishProbabilityLaws C(PathTime, Sphere))
    (hdual : ∀ (e : PathCodeData Sphere Sphere)
      (I : Set ℕ) [PolishSpace I] (b : I → Sphere) (c : I → ℝ≥0∞)
      (ν : Measure Sphere) [IsFiniteMeasure ν]
      [StandardBorelSpace (ConstrainedCostCode e b c ν)],
      MeasureImageDuality (ConstrainedCostCode e b c ν) (PathCostSpace Sphere I))
    (hlift : ∀ (e : PathCodeData Sphere Sphere)
      (I : Set ℕ) [PolishSpace I] (b : I → Sphere) (c : I → ℝ≥0∞)
      (ν : Measure Sphere) [IsFiniteMeasure ν]
      [StandardBorelSpace (ConstrainedCostCode e b c ν)],
      MeasurePlanLifting (ConstrainedCostCode e b c ν) (PathCostSpace Sphere I))
    : KoebeCircleDomainConjecture := by
  have _compatibilityInput : CompactOneManifoldClassification := hclass
  clear hclass _compatibilityInput
  exact koebeCircleDomainConjecture_of_constructed_contours hmoore hcross hplane hcoarea hLusin hsard hsch hsep hcollar hselect hgreen harea hcompact (finiteCircleUniformization_jordanModels hfinite) hmontel hhurwitz hzero harcs hstandard hpath hdual hlift

end CircleDomainRigidity

end

end OAI