nevanlinna/Nevanlinna/holomorphic_JensenFormula.lean

import Nevanlinna.analyticOn_zeroSet
import Nevanlinna.harmonicAt_examples
import Nevanlinna.harmonicAt_meanValue
import Nevanlinna.specialFunctions_CircleIntegral_affine


theorem jensen_case_R_eq_one
  (f : ℂ → ℂ)
  (h₁f : ∀ z ∈ Metric.closedBall (0 : ℂ) 1, HolomorphicAt f z)
  (h₂f : f 0 ≠ 0)
  (S : Finset ℕ)
  (a : S → ℂ)
  (ha : ∀ s, a s ∈ Metric.ball 0 1)
  (F : ℂ → ℂ)
  (h₁F : ∀ z ∈ Metric.closedBall (0 : ℂ) 1, HolomorphicAt F z)
  (h₂F : ∀ z ∈ Metric.closedBall (0 : ℂ) 1, F z ≠ 0)
  (h₃F : f = fun z ↦ (F z) * ∏ s : S, (z - a s)) :
  Real.log ‖f 0‖ = -∑ s, Real.log (‖a s‖⁻¹) + (2 * Real.pi)⁻¹ * ∫ (x : ℝ) in (0)..2 * Real.pi, Real.log ‖f (circleMap 0 1 x)‖ := by

  have h₁U : IsPreconnected (Metric.closedBall (0 : ℂ) 1) := by sorry
  have h₂U : IsCompact (Metric.closedBall (0 : ℂ) 1) := by sorry
  have h₁f : AnalyticOn ℂ f (Metric.closedBall (0 : ℂ) 1) := by sorry
  have h₂f : ∃ u ∈ (Metric.closedBall (0 : ℂ) 1), f u ≠ 0 := by sorry

  let α := AnalyticOnCompact.eliminateZeros h₁U h₂U h₁f h₂f
  obtain ⟨g, A, h'₁g, h₂g, h₃g⟩ := α
  have h₁g : ∀ z ∈ Metric.closedBall 0 1, HolomorphicAt F z := by sorry



  let logAbsF := fun w ↦ Real.log ‖F w‖

  have t₀ : ∀ z ∈ Metric.closedBall 0 1, HarmonicAt logAbsF z := by
    intro z hz
    apply logabs_of_holomorphicAt_is_harmonic
    apply h₁F z hz
    exact h₂F z hz

  have t₁ : (∫ (x : ℝ) in (0)..2 * Real.pi, logAbsF (circleMap 0 1 x)) = 2 * Real.pi * logAbsF 0 := by
    apply harmonic_meanValue₁ 1 Real.zero_lt_one t₀

  have t₂ : ∀ s, f (a s) = 0 := by
    intro s
    rw [h₃F]
    simp
    right
    apply Finset.prod_eq_zero_iff.2
    use s
    simp

  let logAbsf := fun w ↦ Real.log ‖f w‖
  have s₀ : ∀ z ∈ Metric.closedBall (0 : ℂ) 1, f z ≠ 0 → logAbsf z = logAbsF z + ∑ s, Real.log ‖z - a s‖ := by
    intro z h₁z h₂z
    dsimp [logAbsf]
    rw [h₃F]
    simp_rw [Complex.abs.map_mul]
    rw [Complex.abs_prod]
    rw [Real.log_mul]
    rw [Real.log_prod]
    rfl
    intro s hs
    simp
    by_contra ha'
    rw [ha'] at h₂z
    exact h₂z (t₂ s)
    -- Complex.abs (F z) ≠ 0
    simp
    exact h₂F z h₁z
    -- ∏ I : { x // x ∈ S }, Complex.abs (z - a I) ≠ 0
    by_contra h'
    obtain ⟨s, h's, h''⟩ := Finset.prod_eq_zero_iff.1 h'
    simp at h''
    rw [h''] at h₂z
    let A := t₂ s
    exact h₂z A

  have s₁ : ∀ z ∈ Metric.closedBall (0 : ℂ) 1, f z ≠ 0 → logAbsF z = logAbsf z - ∑ s, Real.log ‖z - a s‖ := by
    intro z h₁z h₂z
    rw [s₀ z h₁z]
    simp
    assumption

  have : 0 ∈ Metric.closedBall (0 : ℂ) 1 := by simp
  rw [s₁ 0 this h₂f] at t₁

  have h₀ {x : ℝ} : f (circleMap 0 1 x) ≠ 0 := by
    rw [h₃F]
    simp
    constructor
    · have : (circleMap 0 1 x) ∈ Metric.closedBall (0 : ℂ) 1 := by simp
      exact h₂F (circleMap 0 1 x) this
    · by_contra h'
      obtain ⟨s, _, h₂s⟩  := Finset.prod_eq_zero_iff.1 h'
      have : circleMap 0 1 x = a s := by
        rw [← sub_zero (circleMap 0 1 x)]
        nth_rw 2 [← h₂s]
        simp
      let A := ha s
      rw [← this] at A
      simp at A

  have {θ} : (circleMap 0 1 θ) ∈ Metric.closedBall (0 : ℂ) 1 := by simp
  simp_rw [s₁ (circleMap 0 1 _) this h₀] at t₁
  rw [intervalIntegral.integral_sub] at t₁
  rw [intervalIntegral.integral_finset_sum] at t₁

  simp_rw [int₀ (ha _)] at t₁
  simp at t₁
  rw [t₁]
  simp
  have {w : ℝ} : Real.pi⁻¹ * 2⁻¹ * (2 * Real.pi * w) = w := by
    ring_nf
    simp [mul_inv_cancel Real.pi_ne_zero]
  rw [this]
  simp
  rfl
  -- ∀ i ∈ Finset.univ, IntervalIntegrable (fun x => Real.log ‖circleMap 0 1 x - a i‖) MeasureTheory.volume 0 (2 * Real.pi)
  intro i _
  apply Continuous.intervalIntegrable
  apply continuous_iff_continuousAt.2
  intro x
  have : (fun x => Real.log ‖circleMap 0 1 x - a i‖) = Real.log ∘ Complex.abs ∘ (fun x ↦ circleMap 0 1 x - a i) :=
    rfl
  rw [this]
  apply ContinuousAt.comp
  apply Real.continuousAt_log
  simp
  by_contra ha'
  let A := ha i
  rw [← ha'] at A
  simp at A
  apply ContinuousAt.comp
  apply Complex.continuous_abs.continuousAt
  fun_prop
  -- IntervalIntegrable (fun x => logAbsf (circleMap 0 1 x)) MeasureTheory.volume 0 (2 * Real.pi)
  apply Continuous.intervalIntegrable
  apply continuous_iff_continuousAt.2
  intro x
  have : (fun x => logAbsf (circleMap 0 1 x)) = Real.log ∘ Complex.abs ∘ f ∘ (fun x ↦ circleMap 0 1 x) :=
    rfl
  rw [this]
  apply ContinuousAt.comp
  simp
  exact h₀
  apply ContinuousAt.comp
  apply Complex.continuous_abs.continuousAt
  apply ContinuousAt.comp
  apply ContDiffAt.continuousAt (f := f) (𝕜 := ℝ) (n := 1)
  apply HolomorphicAt.contDiffAt
  apply h₁f
  simp
  let A := continuous_circleMap 0 1
  apply A.continuousAt
  -- IntervalIntegrable (fun x => ∑ s : { x // x ∈ S }, Real.log ‖circleMap 0 1 x - a s‖) MeasureTheory.volume 0 (2 * Real.pi)
  apply Continuous.intervalIntegrable
  apply continuous_finset_sum
  intro i _
  apply continuous_iff_continuousAt.2
  intro x
  have : (fun x => Real.log ‖circleMap 0 1 x - a i‖) = Real.log ∘ Complex.abs ∘ (fun x ↦ circleMap 0 1 x - a i) :=
    rfl
  rw [this]
  apply ContinuousAt.comp
  apply Real.continuousAt_log
  simp
  by_contra ha'
  let A := ha i
  rw [← ha'] at A
  simp at A
  apply ContinuousAt.comp
  apply Complex.continuous_abs.continuousAt
  fun_prop