nevanlinna/Nevanlinna/specialFunctions_CircleIntegral_affine.lean

import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.Analysis.SpecialFunctions.Log.NegMulLog
import Mathlib.MeasureTheory.Integral.CircleIntegral
import Mathlib.MeasureTheory.Measure.Restrict
import Nevanlinna.specialFunctions_Integral_log_sin

open scoped Interval Topology
open Real Filter MeasureTheory intervalIntegral




lemma int₁₁ : ∫ (x : ℝ) in (0)..π, log (4 * sin x ^ 2) = 0 := by

  have t₁ {x : ℝ} : x ∈ Set.Ioo 0 π → log (4 * sin x ^ 2) = log 4 + 2 * log (sin x) := by
    intro hx
    rw [log_mul, log_pow]
    rfl
    exact Ne.symm (NeZero.ne' 4)
    apply pow_ne_zero 2
    apply (fun a => Ne.symm (ne_of_lt a))
    exact sin_pos_of_mem_Ioo hx


  have t₂ : Set.EqOn (fun y ↦ log (4 * sin y ^ 2)) (fun y ↦ log 4 + 2 * log (sin y)) (Set.Ioo 0 π) := by
    intro x hx
    simp
    rw [t₁ hx]

  rw [intervalIntegral.integral_congr_volume pi_pos t₂]
  rw [intervalIntegral.integral_add]
  rw [intervalIntegral.integral_const_mul]
  simp
  rw [integral_log_sin₂]
  have : (4 : ℝ) = 2 * 2 := by norm_num
  rw [this, log_mul]
  ring
  norm_num
  norm_num
  -- IntervalIntegrable (fun x => log 4) volume 0 π
  simp
  -- IntervalIntegrable (fun x => 2 * log (sin x)) volume 0 π
  apply IntervalIntegrable.const_mul
  exact intervalIntegrable_log_sin


lemma int₁ :
  ∫ x in (0)..(2 * π), log ‖circleMap 0 1 x - 1‖ = 0 := by

  have {x : ℝ} : log ‖circleMap 0 1 x - 1‖ = log (4 * sin (x / 2) ^ 2) / 2 := by
    dsimp [Complex.abs]
    rw [log_sqrt (Complex.normSq_nonneg (circleMap 0 1 x - 1))]
    congr
    calc Complex.normSq (circleMap 0 1 x - 1)
    _ = (cos x - 1) * (cos x - 1) + sin x * sin x := by
      dsimp [circleMap]
      rw [Complex.normSq_apply]
      simp
    _ = sin x ^ 2 + cos x ^ 2 + 1 - 2 * cos x := by
      ring
    _ = 2 - 2 * cos x := by
      rw [sin_sq_add_cos_sq]
      norm_num
    _ = 2 - 2 * cos (2 * (x / 2)) := by
      rw [← mul_div_assoc]
      congr; norm_num
    _ = 4 - 4 * Real.cos (x / 2) ^ 2 := by
      rw [cos_two_mul]
      ring
    _ = 4 * sin (x / 2) ^ 2 := by
      nth_rw 1 [← mul_one 4, ← sin_sq_add_cos_sq (x / 2)]
      ring
  simp_rw [this]
  simp

  have : ∫ (x : ℝ) in (0)..2 * π, log (4 * sin (x / 2) ^ 2) =  2 * ∫ (x : ℝ) in (0)..π, log (4 * sin x ^ 2) := by
    have : 1 = 2 * (2 : ℝ)⁻¹ := by exact Eq.symm (mul_inv_cancel_of_invertible 2)
    nth_rw 1 [← one_mul (∫ (x : ℝ) in (0)..2 * π, log (4 * sin (x / 2) ^ 2))]
    rw [← mul_inv_cancel_of_invertible 2, mul_assoc]
    let f := fun y ↦ log (4 * sin y ^ 2)
    have {x : ℝ} : log (4 * sin (x / 2) ^ 2) = f (x / 2) := by simp
    conv =>
      left
      right
      right
      arg 1
      intro x
      rw [this]
    rw [intervalIntegral.inv_mul_integral_comp_div 2]
    simp
  rw [this]
  simp
  exact int₁₁