nevanlinna/Nevanlinna/holomorphic_JensenFormula2.lean

import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.Analysis.Analytic.IsolatedZeros
import Nevanlinna.analyticOn_zeroSet
import Nevanlinna.harmonicAt_examples
import Nevanlinna.harmonicAt_meanValue
import Nevanlinna.specialFunctions_CircleIntegral_affine

open Real



theorem jensen_case_R_eq_one
  (f : ℂ → ℂ)
  (h₁f : AnalyticOn ℂ f (Metric.closedBall (0 : ℂ) 1))
  (h₂f : f 0 ≠ 0) :
  log ‖f 0‖ = -∑ᶠ s, (h₁f.order s).toNat * log (‖s.1‖⁻¹) + (2 * π)⁻¹ * ∫ (x : ℝ) in (0)..(2 * π), log ‖f (circleMap 0 1 x)‖ := by

  have h₁U : IsPreconnected (Metric.closedBall (0 : ℂ) 1) :=
    (convex_closedBall (0 : ℂ) 1).isPreconnected

  have h₂U : IsCompact (Metric.closedBall (0 : ℂ) 1) :=
    isCompact_closedBall 0 1

  have h'₂f : ∃ u ∈ (Metric.closedBall (0 : ℂ) 1), f u ≠ 0 := by
    use 0; simp; exact h₂f

  obtain ⟨F, h₁F, h₂F, h₃F⟩ := AnalyticOnCompact.eliminateZeros₂ h₁U h₂U h₁f h'₂f

  have h'₁F : ∀ z ∈ Metric.closedBall (0 : ℂ) 1, HolomorphicAt F z := by
    intro z h₁z
    apply AnalyticAt.holomorphicAt
    exact h₁F z h₁z

  let G := fun z ↦ log ‖F z‖ + ∑ s ∈ (finiteZeros h₁U h₂U h₁f h'₂f).toFinset, (h₁f.order s).toNat * log ‖z - s‖

  have decompose_f : ∀ z ∈ Metric.closedBall (0 : ℂ) 1, f z ≠ 0 → log ‖f z‖ = G z := by
    intro z h₁z h₂z

    conv =>
      left
      arg 1
      rw [h₃F]
      rw [smul_eq_mul]
      rw [norm_mul]
      rw [norm_prod]
      left
      arg 2
      intro b
      rw [norm_pow]
    simp only [Complex.norm_eq_abs, Finset.univ_eq_attach]
    rw [Real.log_mul]
    rw [Real.log_prod]
    conv =>
      left
      left
      arg 2
      intro s
      rw [Real.log_pow]
    dsimp [G]
    abel

    -- ∀ x ∈ ⋯.toFinset, Complex.abs (z - ↑x) ^ (h'₁f.order x).toNat ≠ 0
    have : ∀ x ∈ (finiteZeros h₁U h₂U h₁f h'₂f).toFinset, Complex.abs (z - ↑x) ^ (h₁f.order x).toNat ≠ 0 := by
      intro s hs
      simp at hs
      simp
      intro h₂s
      rw [h₂s] at h₂z
      tauto
    exact this

    -- ∏ x ∈ ⋯.toFinset, Complex.abs (z - ↑x) ^ (h'₁f.order x).toNat ≠ 0
    have : ∀ x ∈ (finiteZeros h₁U h₂U h₁f h'₂f).toFinset, Complex.abs (z - ↑x) ^ (h₁f.order x).toNat ≠ 0 := by
      intro s hs
      simp at hs
      simp
      intro h₂s
      rw [h₂s] at h₂z
      tauto
    rw [Finset.prod_ne_zero_iff]
    exact this

    -- Complex.abs (F z) ≠ 0
    simp
    exact h₂F z h₁z


  have int_logAbs_f_eq_int_G : ∫ (x : ℝ) in (0)..2 * π, log ‖f (circleMap 0 1 x)‖ = ∫ (x : ℝ) in (0)..2 * π, G (circleMap 0 1 x) := by

    rw [intervalIntegral.integral_congr_ae]
    rw [MeasureTheory.ae_iff]
    apply Set.Countable.measure_zero
    simp

    have t₀ : {a | a ∈ Ι 0 (2 * π) ∧ ¬log ‖f (circleMap 0 1 a)‖ = G (circleMap 0 1 a)}
      ⊆ (circleMap 0 1)⁻¹' (Metric.closedBall 0 1 ∩ f⁻¹' {0}) := by
      intro a ha
      simp at ha
      simp
      by_contra C
      have : (circleMap 0 1 a) ∈ Metric.closedBall 0 1 :=
        circleMap_mem_closedBall 0 (zero_le_one' ℝ) a
      exact ha.2 (decompose_f (circleMap 0 1 a) this C)

    apply Set.Countable.mono t₀
    apply Set.Countable.preimage_circleMap
    apply Set.Finite.countable
    let A := finiteZeros h₁U h₂U h₁f h'₂f

    have : (Metric.closedBall 0 1 ∩ f ⁻¹' {0}) = (Metric.closedBall 0 1).restrict f ⁻¹' {0} := by
      ext z
      simp
      tauto
    rw [this]
    exact Set.Finite.image Subtype.val A
    exact Ne.symm (zero_ne_one' ℝ)


  have decompose_int_G : ∫ (x : ℝ) in (0)..2 * π, G (circleMap 0 1 x)
    = (∫ (x : ℝ) in (0)..2 * π, log (Complex.abs (F (circleMap 0 1 x))))
        + ∑ x ∈ (finiteZeros h₁U h₂U h₁f h'₂f).toFinset, (h₁f.order x).toNat * ∫ (x_1 : ℝ) in (0)..2 * π, log (Complex.abs (circleMap 0 1 x_1 - ↑x)) := by
    dsimp [G]
    rw [intervalIntegral.integral_add]
    rw [intervalIntegral.integral_finset_sum]
    simp_rw [intervalIntegral.integral_const_mul]

    --  ∀ i ∈ (finiteZeros h₁U h₂U h'₁f h'₂f).toFinset,
    --  IntervalIntegrable (fun x => (h'₁f.order i).toNat *
    --    log (Complex.abs (circleMap 0 1 x - ↑i))) MeasureTheory.volume 0 (2 * π)
    intro i _
    apply IntervalIntegrable.const_mul
    --simp at this
    by_cases h₂i : ‖i.1‖ = 1
    -- case pos
    exact int'₂ h₂i
    -- case neg
    apply Continuous.intervalIntegrable
    apply continuous_iff_continuousAt.2
    intro x
    have : (fun x => log (Complex.abs (circleMap 0 1 x - ↑i))) = log ∘ Complex.abs ∘ (fun x ↦ circleMap 0 1 x - ↑i) :=
      rfl
    rw [this]
    apply ContinuousAt.comp
    apply Real.continuousAt_log
    simp

    by_contra ha'
    conv at h₂i =>
      arg 1
      rw [← ha']
      rw [Complex.norm_eq_abs]
      rw [abs_circleMap_zero 1 x]
      simp
    tauto
    apply ContinuousAt.comp
    apply Complex.continuous_abs.continuousAt
    fun_prop
    -- IntervalIntegrable (fun x => log (Complex.abs (F (circleMap 0 1 x)))) MeasureTheory.volume 0 (2 * π)
    apply Continuous.intervalIntegrable
    apply continuous_iff_continuousAt.2
    intro x
    have : (fun x => log (Complex.abs (F (circleMap 0 1 x)))) = log ∘ Complex.abs ∘ F ∘ (fun x ↦ circleMap 0 1 x) :=
      rfl
    rw [this]
    apply ContinuousAt.comp
    apply Real.continuousAt_log
    simp [h₂F]
    -- ContinuousAt (⇑Complex.abs ∘ F ∘ fun x => circleMap 0 1 x) x
    apply ContinuousAt.comp
    apply Complex.continuous_abs.continuousAt
    apply ContinuousAt.comp
    apply DifferentiableAt.continuousAt (𝕜 := ℂ )
    apply HolomorphicAt.differentiableAt
    simp [h'₁F]
    -- ContinuousAt (fun x => circleMap 0 1 x) x
    apply Continuous.continuousAt
    apply continuous_circleMap

    have : (fun x => ∑ s ∈ (finiteZeros h₁U h₂U h₁f h'₂f).toFinset, (h₁f.order s).toNat * log (Complex.abs (circleMap 0 1 x - ↑s)))
      = ∑ s ∈ (finiteZeros h₁U h₂U h₁f h'₂f).toFinset, (fun x => (h₁f.order s).toNat * log (Complex.abs (circleMap 0 1 x - ↑s))) := by
      funext x
      simp
    rw [this]
    apply IntervalIntegrable.sum
    intro i _
    apply IntervalIntegrable.const_mul
    --have : i.1 ∈ Metric.closedBall (0 : ℂ) 1 := i.2
    --simp at this
    by_cases h₂i : ‖i.1‖ = 1
    -- case pos
    exact int'₂ h₂i
    -- case neg
    --have : i.1 ∈ Metric.ball (0 : ℂ) 1 := by sorry
    apply Continuous.intervalIntegrable
    apply continuous_iff_continuousAt.2
    intro x
    have : (fun x => log (Complex.abs (circleMap 0 1 x - ↑i))) = log ∘ Complex.abs ∘ (fun x ↦ circleMap 0 1 x - ↑i) :=
      rfl
    rw [this]
    apply ContinuousAt.comp
    apply Real.continuousAt_log
    simp

    by_contra ha'
    conv at h₂i =>
      arg 1
      rw [← ha']
      rw [Complex.norm_eq_abs]
      rw [abs_circleMap_zero 1 x]
      simp
    tauto
    apply ContinuousAt.comp
    apply Complex.continuous_abs.continuousAt
    fun_prop

  have t₁ : (∫ (x : ℝ) in (0)..2 * Real.pi, log ‖F (circleMap 0 1 x)‖) = 2 * Real.pi * log ‖F 0‖ := by
    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 h'₁F z hz
      exact h₂F z hz

    apply harmonic_meanValue₁ 1 Real.zero_lt_one t₀

  simp_rw [← Complex.norm_eq_abs] at decompose_int_G
  rw [t₁] at decompose_int_G

  conv at decompose_int_G =>
    right
    right
    arg 2
    intro x
    right
    rw [int₃ x.2]
  simp at decompose_int_G

  rw [int_logAbs_f_eq_int_G]
  rw [decompose_int_G]
  rw [h₃F]
  simp
  have {l : ℝ} : π⁻¹ * 2⁻¹ * (2 * π * l) = l := by
    calc π⁻¹ * 2⁻¹ * (2 * π * l)
    _ = π⁻¹ * (2⁻¹ * 2) * π * l := by ring
    _ = π⁻¹ * π * l := by ring
    _ = (π⁻¹ * π) * l := by ring
    _ = 1 * l := by
      rw [inv_mul_cancel₀]
      exact pi_ne_zero
    _ = l := by simp
  rw [this]
  rw [log_mul]
  rw [log_prod]
  simp

  rw [finsum_eq_sum_of_support_subset _ (s := (finiteZeros h₁U h₂U h₁f h'₂f).toFinset)]
  simp
  simp
  intro x ⟨h₁x, _⟩
  simp

  dsimp [AnalyticOn.order] at h₁x
  simp only [Function.mem_support, ne_eq, Nat.cast_eq_zero, not_or] at h₁x
  exact AnalyticAt.supp_order_toNat (AnalyticOn.order.proof_1 h₁f x) h₁x

  --
  intro x hx
  simp at hx
  simp
  intro h₁x
  nth_rw 1 [← h₁x] at h₂f
  tauto

  --
  rw [Finset.prod_ne_zero_iff]
  intro x hx
  simp at hx
  simp
  intro h₁x
  nth_rw 1 [← h₁x] at h₂f
  tauto

  --
  simp
  apply h₂F
  simp