Splitting off files

Nevanlinna/specialFunctions_CircleIntegral_affine.lean

@@ -0,0 +1,331 @@
+import Mathlib.Analysis.SpecialFunctions.Integrals
+import Mathlib.Analysis.SpecialFunctions.Log.NegMulLog
+import Mathlib.MeasureTheory.Integral.CircleIntegral
+import Mathlib.MeasureTheory.Measure.Restrict
+
+open scoped Interval Topology
+open Real Filter MeasureTheory intervalIntegral
+
+
+
+lemma logsinBound : ∀ x ∈ (Set.Icc 0 1), ‖(log ∘ sin) x‖ ≤ ‖log ((π / 2)⁻¹ * x)‖ := by
+
+  intro x hx
+  by_cases h'x : x = 0
+  · rw [h'x]; simp
+
+  -- Now handle the case where x ≠ 0
+  have l₀ : log ((π / 2)⁻¹ * x) ≤ 0 := by
+    apply log_nonpos
+    apply mul_nonneg
+    apply le_of_lt
+    apply inv_pos.2
+    apply div_pos
+    exact pi_pos
+    exact zero_lt_two
+    apply (Set.mem_Icc.1 hx).1
+    simp
+    apply mul_le_one
+    rw [div_le_one pi_pos]
+    exact two_le_pi
+    exact (Set.mem_Icc.1 hx).1
+    exact (Set.mem_Icc.1 hx).2
+
+  have l₁ : 0 ≤ sin x := by
+    apply sin_nonneg_of_nonneg_of_le_pi (Set.mem_Icc.1 hx).1
+    trans (1 : ℝ)
+    exact (Set.mem_Icc.1 hx).2
+    trans π / 2
+    exact one_le_pi_div_two
+    norm_num [pi_nonneg]
+  have l₂ : log (sin x) ≤ 0 := log_nonpos l₁ (sin_le_one x)
+
+  simp only [norm_eq_abs, Function.comp_apply]
+  rw [abs_eq_neg_self.2 l₀]
+  rw [abs_eq_neg_self.2 l₂]
+  simp only [neg_le_neg_iff, ge_iff_le]
+
+  have l₃ : x ∈ (Set.Ioi 0) := by
+    simp
+    exact lt_of_le_of_ne (Set.mem_Icc.1 hx).1 ( fun a => h'x (id (Eq.symm a)) )
+
+  have l₅ : 0 < (π / 2)⁻¹ * x := by
+    apply mul_pos
+    apply inv_pos.2
+    apply div_pos pi_pos zero_lt_two
+    exact l₃
+
+  have : ∀ x ∈ (Set.Icc 0 (π / 2)), (π / 2)⁻¹ * x ≤ sin x := by
+    intro x hx
+
+    have i₀ : 0 ∈ Set.Icc 0 π :=
+      Set.left_mem_Icc.mpr pi_nonneg
+    have i₁ : π / 2 ∈ Set.Icc 0 π :=
+      Set.mem_Icc.mpr ⟨div_nonneg pi_nonneg zero_le_two, half_le_self pi_nonneg⟩
+
+    have i₂ : 0 ≤ 1 - (π / 2)⁻¹ * x := by
+      rw [sub_nonneg]
+      calc (π / 2)⁻¹ * x
+      _ ≤ (π / 2)⁻¹ * (π / 2) := by
+        apply mul_le_mul_of_nonneg_left
+        exact (Set.mem_Icc.1 hx).2
+        apply inv_nonneg.mpr (div_nonneg pi_nonneg zero_le_two)
+      _ = 1 := by
+        apply inv_mul_cancel
+        apply div_ne_zero_iff.mpr
+        constructor
+        · exact pi_ne_zero
+        · exact Ne.symm (NeZero.ne' 2)
+
+    have i₃ : 0 ≤ (π / 2)⁻¹ * x := by
+      apply mul_nonneg
+      apply inv_nonneg.2
+      apply div_nonneg
+      exact pi_nonneg
+      exact zero_le_two
+      exact (Set.mem_Icc.1 hx).1
+
+    have i₄ : 1 - (π / 2)⁻¹ * x + (π / 2)⁻¹ * x = 1 := by ring
+
+    let B := strictConcaveOn_sin_Icc.concaveOn.2 i₀ i₁ i₂ i₃ i₄
+    simp [Real.sin_pi_div_two] at B
+    rw [(by ring_nf; rw [mul_inv_cancel pi_ne_zero, one_mul] : 2 / π * x * (π / 2) = x)] at B
+    simpa
+
+  apply log_le_log l₅
+  apply this
+  apply Set.mem_Icc.mpr
+  constructor
+  · exact le_of_lt l₃
+  · trans 1
+    exact (Set.mem_Icc.1 hx).2
+    exact one_le_pi_div_two
+
+
+lemma intervalIntegrable_log_sin₁ : IntervalIntegrable (log ∘ sin) volume 0 1 := by
+
+  have int_log : IntervalIntegrable (fun x ↦ ‖log x‖) volume 0 1 := by
+    apply IntervalIntegrable.norm
+    rw [← neg_neg log]
+    apply IntervalIntegrable.neg
+    apply intervalIntegrable_deriv_of_nonneg (g := fun x ↦ -(x * log x - x))
+    · exact (continuous_mul_log.continuousOn.sub continuous_id.continuousOn).neg
+    · intro x hx
+      norm_num at hx
+      convert ((hasDerivAt_mul_log hx.left.ne.symm).sub (hasDerivAt_id x)).neg using 1
+      norm_num
+    · intro x hx
+      norm_num at hx
+      rw [Pi.neg_apply, Left.nonneg_neg_iff]
+      exact (log_nonpos_iff hx.left).mpr hx.right.le
+
+
+  have int_log : IntervalIntegrable (fun x ↦ ‖log ((π / 2)⁻¹ * x)‖) volume 0 1 := by
+
+    have A := IntervalIntegrable.comp_mul_right int_log (π / 2)⁻¹
+    simp only [norm_eq_abs] at A
+    conv =>
+      arg 1
+      intro x
+      rw [mul_comm]
+    simp only [norm_eq_abs]
+    apply IntervalIntegrable.mono A
+    simp
+    trans Set.Icc 0 (π / 2)
+    exact Set.Icc_subset_Icc (Preorder.le_refl 0) one_le_pi_div_two
+    exact Set.Icc_subset_uIcc
+    exact Preorder.le_refl volume
+
+  apply IntervalIntegrable.mono_fun' (g := fun x ↦ ‖log ((π / 2)⁻¹ * x)‖)
+  exact int_log
+
+  -- AEStronglyMeasurable (log ∘ sin) (volume.restrict (Ι 0 1))
+  apply ContinuousOn.aestronglyMeasurable
+  apply ContinuousOn.comp (t := Ι 0 1)
+  apply ContinuousOn.mono (s := {0}ᶜ)
+  exact continuousOn_log
+  intro x hx
+  by_contra contra
+  simp at contra
+  rw [contra, Set.left_mem_uIoc] at hx
+  linarith
+  exact continuousOn_sin
+
+  -- Set.MapsTo sin (Ι 0 1) (Ι 0 1)
+  rw [Set.uIoc_of_le (zero_le_one' ℝ)]
+  exact fun x hx ↦ ⟨sin_pos_of_pos_of_le_one hx.1 hx.2, sin_le_one x⟩
+
+  -- MeasurableSet (Ι 0 1)
+  exact measurableSet_uIoc
+
+  -- (fun x => ‖(log ∘ sin) x‖) ≤ᶠ[ae (volume.restrict (Ι 0 1))] ‖log‖
+  dsimp [EventuallyLE]
+  rw [MeasureTheory.ae_restrict_iff]
+  apply MeasureTheory.ae_of_all
+  intro x hx
+  have : x ∈ Set.Icc 0 1 := by
+    simp
+    simp at hx
+    constructor
+    · exact le_of_lt hx.1
+    · exact hx.2
+  let A := logsinBound x this
+  simp only [Function.comp_apply, norm_eq_abs] at A
+  exact A
+
+  apply measurableSet_le
+  apply Measurable.comp'
+  exact continuous_abs.measurable
+  exact Measurable.comp' measurable_log continuous_sin.measurable
+  -- Measurable fun a => |log ((π / 2)⁻¹ * a)|
+  apply Measurable.comp'
+  exact continuous_abs.measurable
+  apply Measurable.comp'
+  exact measurable_log
+  exact measurable_const_mul (π / 2)⁻¹
+
+lemma intervalIntegrable_log_sin₂ : IntervalIntegrable (log ∘ sin) volume 0 (π / 2) := by
+
+  apply IntervalIntegrable.trans (b := 1)
+  exact intervalIntegrable_log_sin₁
+
+  -- IntervalIntegrable (log ∘ sin) volume 1 (π / 2)
+  apply ContinuousOn.intervalIntegrable
+  apply ContinuousOn.comp continuousOn_log continuousOn_sin
+  intro x hx
+  rw [Set.uIcc_of_le, Set.mem_Icc] at hx
+  have : 0 < sin x := by
+    apply Real.sin_pos_of_pos_of_lt_pi
+    · calc 0
+      _ < 1 := Real.zero_lt_one
+      _ ≤ x := hx.1
+    · calc x
+      _ ≤ π / 2 := hx.2
+      _ < π := div_two_lt_of_pos pi_pos
+  by_contra h₁x
+  simp at h₁x
+  rw [h₁x] at this
+  simp at this
+  exact one_le_pi_div_two
+
+lemma intervalIntegrable_log_sin : IntervalIntegrable (log ∘ sin) volume 0 π := by
+  apply IntervalIntegrable.trans (b := π / 2)
+  exact intervalIntegrable_log_sin₂
+  let A := IntervalIntegrable.comp_sub_left intervalIntegrable_log_sin₂ π
+  simp at A
+  let B := IntervalIntegrable.symm A
+  have : π - π / 2 = π / 2 := by linarith
+  rwa [this] at B
+
+lemma intervalIntegrable_log_cos : IntervalIntegrable (log ∘ cos) volume 0 (π / 2) := by
+  let A := IntervalIntegrable.comp_sub_left intervalIntegrable_log_sin₂ (π / 2)
+  simp only [Function.comp_apply, sub_zero, sub_self] at A
+  simp_rw [sin_pi_div_two_sub] at A
+  have : (fun x => log (cos x)) = log ∘ cos := rfl
+  apply IntervalIntegrable.symm
+  rwa [← this]
+
+
+lemma integral_log_sin : ∫ (x : ℝ) in (0)..(π / 2), log (sin x) =  -log 2 * π/2 := by
+
+  have t₀ {x : ℝ} : sin (2 * x) = 2 * sin x * cos x := sin_two_mul x
+
+  have t₁ {x : ℝ} : log (sin (2 * x)) = log 2 + log (sin x) + log (cos x) := by
+    rw [sin_two_mul x, log_mul, log_mul]
+    exact Ne.symm (NeZero.ne' 2)
+    sorry
+    sorry
+    sorry
+
+  have t₂ {x : ℝ} : log (sin x) = log (sin (2 * x)) - log 2 - log (cos x) := by
+    rw [t₁]
+    ring
+
+  conv =>
+    left
+    arg 1
+    intro x
+    rw [t₂]
+
+  rw [intervalIntegral.integral_sub, intervalIntegral.integral_sub]
+  rw [intervalIntegral.integral_const]
+  rw [intervalIntegral.integral_comp_mul_left (c := 2) (f := fun x ↦ log (sin x))]
+  simp
+  have : 2 * (π / 2) = π := by linarith
+  rw [this]
+
+  have : ∫ (x : ℝ) in (0)..π, log (sin x) = 2 * ∫ (x : ℝ) in (0)..(π / 2), log (sin x) := by
+    sorry
+  rw [this]
+  have : ∫ (x : ℝ) in (0)..(π / 2), log (sin x) = ∫ (x : ℝ) in (0)..(π / 2), log (cos x) := by
+    sorry
+  rw [← this]
+  simp
+  linarith
+
+  exact Ne.symm (NeZero.ne' 2)
+  -- IntervalIntegrable (fun x => log (sin (2 * x))) volume 0 (π / 2)
+  sorry
+  -- IntervalIntegrable (fun x => log 2) volume 0 (π / 2)
+  simp
+  -- IntervalIntegrable (fun x => log (sin (2 * x)) - log 2) volume 0 (π / 2)
+  apply IntervalIntegrable.sub
+  -- -- IntervalIntegrable (fun x => log (sin (2 * x))) volume 0 (π / 2)
+  sorry
+  -- -- IntervalIntegrable (fun x => log 2) volume 0 (π / 2)
+  simp
+  -- -- IntervalIntegrable (fun x => log (cos x)) volume 0 (π / 2)
+  exact intervalIntegrable_log_cos
+
+
+lemma int₁₁ : ∫ (x : ℝ) in (0)..π, log (4 * sin x ^ 2) = 0 := by
+  sorry
+
+
+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

Nevanlinna/specialFunctions_Integral_log.lean

@@ -0,0 +1,388 @@
+import Mathlib.Analysis.SpecialFunctions.Integrals
+import Mathlib.Analysis.SpecialFunctions.Log.NegMulLog
+import Mathlib.MeasureTheory.Integral.CircleIntegral
+import Mathlib.MeasureTheory.Measure.Restrict
+
+open scoped Interval Topology
+open Real Filter MeasureTheory intervalIntegral
+
+-- The following theorem was suggested by Gareth Ma on Zulip
+
+
+lemma logsinBound : ∀ x ∈ (Set.Icc 0 1), ‖(log ∘ sin) x‖ ≤ ‖log ((π / 2)⁻¹ * x)‖ := by
+
+  intro x hx
+  by_cases h'x : x = 0
+  · rw [h'x]; simp
+
+  -- Now handle the case where x ≠ 0
+  have l₀ : log ((π / 2)⁻¹ * x) ≤ 0 := by
+    apply log_nonpos
+    apply mul_nonneg
+    apply le_of_lt
+    apply inv_pos.2
+    apply div_pos
+    exact pi_pos
+    exact zero_lt_two
+    apply (Set.mem_Icc.1 hx).1
+    simp
+    apply mul_le_one
+    rw [div_le_one pi_pos]
+    exact two_le_pi
+    exact (Set.mem_Icc.1 hx).1
+    exact (Set.mem_Icc.1 hx).2
+
+  have l₁ : 0 ≤ sin x := by
+    apply sin_nonneg_of_nonneg_of_le_pi (Set.mem_Icc.1 hx).1
+    trans (1 : ℝ)
+    exact (Set.mem_Icc.1 hx).2
+    trans π / 2
+    exact one_le_pi_div_two
+    norm_num [pi_nonneg]
+  have l₂ : log (sin x) ≤ 0 := log_nonpos l₁ (sin_le_one x)
+
+  simp only [norm_eq_abs, Function.comp_apply]
+  rw [abs_eq_neg_self.2 l₀]
+  rw [abs_eq_neg_self.2 l₂]
+  simp only [neg_le_neg_iff, ge_iff_le]
+
+  have l₃ : x ∈ (Set.Ioi 0) := by
+    simp
+    exact lt_of_le_of_ne (Set.mem_Icc.1 hx).1 ( fun a => h'x (id (Eq.symm a)) )
+
+  have l₅ : 0 < (π / 2)⁻¹ * x := by
+    apply mul_pos
+    apply inv_pos.2
+    apply div_pos pi_pos zero_lt_two
+    exact l₃
+
+  have : ∀ x ∈ (Set.Icc 0 (π / 2)), (π / 2)⁻¹ * x ≤ sin x := by
+    intro x hx
+
+    have i₀ : 0 ∈ Set.Icc 0 π :=
+      Set.left_mem_Icc.mpr pi_nonneg
+    have i₁ : π / 2 ∈ Set.Icc 0 π :=
+      Set.mem_Icc.mpr ⟨div_nonneg pi_nonneg zero_le_two, half_le_self pi_nonneg⟩
+
+    have i₂ : 0 ≤ 1 - (π / 2)⁻¹ * x := by
+      rw [sub_nonneg]
+      calc (π / 2)⁻¹ * x
+      _ ≤ (π / 2)⁻¹ * (π / 2) := by
+        apply mul_le_mul_of_nonneg_left
+        exact (Set.mem_Icc.1 hx).2
+        apply inv_nonneg.mpr (div_nonneg pi_nonneg zero_le_two)
+      _ = 1 := by
+        apply inv_mul_cancel
+        apply div_ne_zero_iff.mpr
+        constructor
+        · exact pi_ne_zero
+        · exact Ne.symm (NeZero.ne' 2)
+
+    have i₃ : 0 ≤ (π / 2)⁻¹ * x := by
+      apply mul_nonneg
+      apply inv_nonneg.2
+      apply div_nonneg
+      exact pi_nonneg
+      exact zero_le_two
+      exact (Set.mem_Icc.1 hx).1
+
+    have i₄ : 1 - (π / 2)⁻¹ * x + (π / 2)⁻¹ * x = 1 := by ring
+
+    let B := strictConcaveOn_sin_Icc.concaveOn.2 i₀ i₁ i₂ i₃ i₄
+    simp [Real.sin_pi_div_two] at B
+    rw [(by ring_nf; rw [mul_inv_cancel pi_ne_zero, one_mul] : 2 / π * x * (π / 2) = x)] at B
+    simpa
+
+  apply log_le_log l₅
+  apply this
+  apply Set.mem_Icc.mpr
+  constructor
+  · exact le_of_lt l₃
+  · trans 1
+    exact (Set.mem_Icc.1 hx).2
+    exact one_le_pi_div_two
+
+
+lemma intervalIntegrable_log_sin₁ : IntervalIntegrable (log ∘ sin) volume 0 1 := by
+
+  have int_log : IntervalIntegrable (fun x ↦ ‖log x‖) volume 0 1 := by
+    apply IntervalIntegrable.norm
+    rw [← neg_neg log]
+    apply IntervalIntegrable.neg
+    apply intervalIntegrable_deriv_of_nonneg (g := fun x ↦ -(x * log x - x))
+    · exact (continuous_mul_log.continuousOn.sub continuous_id.continuousOn).neg
+    · intro x hx
+      norm_num at hx
+      convert ((hasDerivAt_mul_log hx.left.ne.symm).sub (hasDerivAt_id x)).neg using 1
+      norm_num
+    · intro x hx
+      norm_num at hx
+      rw [Pi.neg_apply, Left.nonneg_neg_iff]
+      exact (log_nonpos_iff hx.left).mpr hx.right.le
+
+
+  have int_log : IntervalIntegrable (fun x ↦ ‖log ((π / 2)⁻¹ * x)‖) volume 0 1 := by
+
+    have A := IntervalIntegrable.comp_mul_right int_log (π / 2)⁻¹
+    simp only [norm_eq_abs] at A
+    conv =>
+      arg 1
+      intro x
+      rw [mul_comm]
+    simp only [norm_eq_abs]
+    apply IntervalIntegrable.mono A
+    simp
+    trans Set.Icc 0 (π / 2)
+    exact Set.Icc_subset_Icc (Preorder.le_refl 0) one_le_pi_div_two
+    exact Set.Icc_subset_uIcc
+    exact Preorder.le_refl volume
+
+  apply IntervalIntegrable.mono_fun' (g := fun x ↦ ‖log ((π / 2)⁻¹ * x)‖)
+  exact int_log
+
+  -- AEStronglyMeasurable (log ∘ sin) (volume.restrict (Ι 0 1))
+  apply ContinuousOn.aestronglyMeasurable
+  apply ContinuousOn.comp (t := Ι 0 1)
+  apply ContinuousOn.mono (s := {0}ᶜ)
+  exact continuousOn_log
+  intro x hx
+  by_contra contra
+  simp at contra
+  rw [contra, Set.left_mem_uIoc] at hx
+  linarith
+  exact continuousOn_sin
+
+  -- Set.MapsTo sin (Ι 0 1) (Ι 0 1)
+  rw [Set.uIoc_of_le (zero_le_one' ℝ)]
+  exact fun x hx ↦ ⟨sin_pos_of_pos_of_le_one hx.1 hx.2, sin_le_one x⟩
+
+  -- MeasurableSet (Ι 0 1)
+  exact measurableSet_uIoc
+
+  -- (fun x => ‖(log ∘ sin) x‖) ≤ᶠ[ae (volume.restrict (Ι 0 1))] ‖log‖
+  dsimp [EventuallyLE]
+  rw [MeasureTheory.ae_restrict_iff]
+  apply MeasureTheory.ae_of_all
+  intro x hx
+  have : x ∈ Set.Icc 0 1 := by
+    simp
+    simp at hx
+    constructor
+    · exact le_of_lt hx.1
+    · exact hx.2
+  let A := logsinBound x this
+  simp only [Function.comp_apply, norm_eq_abs] at A
+  exact A
+
+  apply measurableSet_le
+  apply Measurable.comp'
+  exact continuous_abs.measurable
+  exact Measurable.comp' measurable_log continuous_sin.measurable
+  -- Measurable fun a => |log ((π / 2)⁻¹ * a)|
+  apply Measurable.comp'
+  exact continuous_abs.measurable
+  apply Measurable.comp'
+  exact measurable_log
+  exact measurable_const_mul (π / 2)⁻¹
+
+lemma intervalIntegrable_log_sin₂ : IntervalIntegrable (log ∘ sin) volume 0 (π / 2) := by
+
+  apply IntervalIntegrable.trans (b := 1)
+  exact intervalIntegrable_log_sin₁
+
+  -- IntervalIntegrable (log ∘ sin) volume 1 (π / 2)
+  apply ContinuousOn.intervalIntegrable
+  apply ContinuousOn.comp continuousOn_log continuousOn_sin
+  intro x hx
+  rw [Set.uIcc_of_le, Set.mem_Icc] at hx
+  have : 0 < sin x := by
+    apply Real.sin_pos_of_pos_of_lt_pi
+    · calc 0
+      _ < 1 := Real.zero_lt_one
+      _ ≤ x := hx.1
+    · calc x
+      _ ≤ π / 2 := hx.2
+      _ < π := div_two_lt_of_pos pi_pos
+  by_contra h₁x
+  simp at h₁x
+  rw [h₁x] at this
+  simp at this
+  exact one_le_pi_div_two
+
+lemma intervalIntegrable_log_sin : IntervalIntegrable (log ∘ sin) volume 0 π := by
+  apply IntervalIntegrable.trans (b := π / 2)
+  exact intervalIntegrable_log_sin₂
+  let A := IntervalIntegrable.comp_sub_left intervalIntegrable_log_sin₂ π
+  simp at A
+  let B := IntervalIntegrable.symm A
+  have : π - π / 2 = π / 2 := by linarith
+  rwa [this] at B
+
+lemma intervalIntegrable_log_cos : IntervalIntegrable (log ∘ cos) volume 0 (π / 2) := by
+  let A := IntervalIntegrable.comp_sub_left intervalIntegrable_log_sin₂ (π / 2)
+  simp only [Function.comp_apply, sub_zero, sub_self] at A
+  simp_rw [sin_pi_div_two_sub] at A
+  have : (fun x => log (cos x)) = log ∘ cos := rfl
+  apply IntervalIntegrable.symm
+  rwa [← this]
+
+
+theorem logInt
+  {t : ℝ}
+  (ht : 0 < t) :
+  ∫ x in (0 : ℝ)..t, log x = t * log t - t := by
+  rw [← integral_add_adjacent_intervals (b := 1)]
+  trans (-1) + (t * log t - t + 1)
+  · congr
+    · -- ∫ x in 0..1, log x = -1, same proof as before
+      rw [integral_eq_sub_of_hasDerivAt_of_tendsto (f := fun x ↦ x * log x - x) (fa := 0) (fb := -1)]
+      · simp
+      · simp
+      · intro x hx
+        norm_num at hx
+        convert (hasDerivAt_mul_log hx.left.ne.symm).sub (hasDerivAt_id x) using 1
+        norm_num
+      · rw [← neg_neg log]
+        apply IntervalIntegrable.neg
+        apply intervalIntegrable_deriv_of_nonneg (g := fun x ↦ -(x * log x - x))
+        · exact (continuous_mul_log.continuousOn.sub continuous_id.continuousOn).neg
+        · intro x hx
+          norm_num at hx
+          convert ((hasDerivAt_mul_log hx.left.ne.symm).sub (hasDerivAt_id x)).neg using 1
+          norm_num
+        · intro x hx
+          norm_num at hx
+          rw [Pi.neg_apply, Left.nonneg_neg_iff]
+          exact (log_nonpos_iff hx.left).mpr hx.right.le
+      · have := tendsto_log_mul_rpow_nhds_zero zero_lt_one
+        simp_rw [rpow_one, mul_comm] at this
+        -- tendsto_nhdsWithin_of_tendsto_nhds should be under Tendsto namespace
+        convert this.sub (tendsto_nhdsWithin_of_tendsto_nhds tendsto_id)
+        norm_num
+      · rw [(by simp : -1 = 1 * log 1 - 1)]
+        apply tendsto_nhdsWithin_of_tendsto_nhds
+        exact (continuousAt_id.mul (continuousAt_log one_ne_zero)).sub continuousAt_id
+    · -- ∫ x in 1..t, log x = t * log t + 1, just use integral_log_of_pos
+      rw [integral_log_of_pos zero_lt_one ht]
+      norm_num
+  · abel
+  · -- log is integrable on [[0, 1]]
+    rw [← neg_neg log]
+    apply IntervalIntegrable.neg
+    apply intervalIntegrable_deriv_of_nonneg (g := fun x ↦ -(x * log x - x))
+    · exact (continuous_mul_log.continuousOn.sub continuous_id.continuousOn).neg
+    · intro x hx
+      norm_num at hx
+      convert ((hasDerivAt_mul_log hx.left.ne.symm).sub (hasDerivAt_id x)).neg using 1
+      norm_num
+    · intro x hx
+      norm_num at hx
+      rw [Pi.neg_apply, Left.nonneg_neg_iff]
+      exact (log_nonpos_iff hx.left).mpr hx.right.le
+  · -- log is integrable on [[0, t]]
+    simp [Set.mem_uIcc, ht]
+
+
+lemma integral_log_sin : ∫ (x : ℝ) in (0)..(π / 2), log (sin x) =  -log 2 * π/2 := by
+
+  have t₀ {x : ℝ} : sin (2 * x) = 2 * sin x * cos x := sin_two_mul x
+
+  have t₁ {x : ℝ} : log (sin (2 * x)) = log 2 + log (sin x) + log (cos x) := by
+    rw [sin_two_mul x, log_mul, log_mul]
+    exact Ne.symm (NeZero.ne' 2)
+    sorry
+    sorry
+    sorry
+
+  have t₂ {x : ℝ} : log (sin x) = log (sin (2 * x)) - log 2 - log (cos x) := by
+    rw [t₁]
+    ring
+
+  conv =>
+    left
+    arg 1
+    intro x
+    rw [t₂]
+
+  rw [intervalIntegral.integral_sub, intervalIntegral.integral_sub]
+  rw [intervalIntegral.integral_const]
+  rw [intervalIntegral.integral_comp_mul_left (c := 2) (f := fun x ↦ log (sin x))]
+  simp
+  have : 2 * (π / 2) = π := by linarith
+  rw [this]
+
+  have : ∫ (x : ℝ) in (0)..π, log (sin x) = 2 * ∫ (x : ℝ) in (0)..(π / 2), log (sin x) := by
+    sorry
+  rw [this]
+  have : ∫ (x : ℝ) in (0)..(π / 2), log (sin x) = ∫ (x : ℝ) in (0)..(π / 2), log (cos x) := by
+    sorry
+  rw [← this]
+  simp
+  linarith
+
+  exact Ne.symm (NeZero.ne' 2)
+  -- IntervalIntegrable (fun x => log (sin (2 * x))) volume 0 (π / 2)
+  sorry
+  -- IntervalIntegrable (fun x => log 2) volume 0 (π / 2)
+  simp
+  -- IntervalIntegrable (fun x => log (sin (2 * x)) - log 2) volume 0 (π / 2)
+  apply IntervalIntegrable.sub
+  -- -- IntervalIntegrable (fun x => log (sin (2 * x))) volume 0 (π / 2)
+  sorry
+  -- -- IntervalIntegrable (fun x => log 2) volume 0 (π / 2)
+  simp
+  -- -- IntervalIntegrable (fun x => log (cos x)) volume 0 (π / 2)
+  exact intervalIntegrable_log_cos
+
+
+lemma int₁₁ : ∫ (x : ℝ) in (0)..π, log (4 * sin x ^ 2) = 0 := by
+  sorry
+
+
+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

Nevanlinna/specialFunctions_Integral_log_sin.lean

@@ -0,0 +1,388 @@
+import Mathlib.Analysis.SpecialFunctions.Integrals
+import Mathlib.Analysis.SpecialFunctions.Log.NegMulLog
+import Mathlib.MeasureTheory.Integral.CircleIntegral
+import Mathlib.MeasureTheory.Measure.Restrict
+
+open scoped Interval Topology
+open Real Filter MeasureTheory intervalIntegral
+
+-- The following theorem was suggested by Gareth Ma on Zulip
+
+
+lemma logsinBound : ∀ x ∈ (Set.Icc 0 1), ‖(log ∘ sin) x‖ ≤ ‖log ((π / 2)⁻¹ * x)‖ := by
+
+  intro x hx
+  by_cases h'x : x = 0
+  · rw [h'x]; simp
+
+  -- Now handle the case where x ≠ 0
+  have l₀ : log ((π / 2)⁻¹ * x) ≤ 0 := by
+    apply log_nonpos
+    apply mul_nonneg
+    apply le_of_lt
+    apply inv_pos.2
+    apply div_pos
+    exact pi_pos
+    exact zero_lt_two
+    apply (Set.mem_Icc.1 hx).1
+    simp
+    apply mul_le_one
+    rw [div_le_one pi_pos]
+    exact two_le_pi
+    exact (Set.mem_Icc.1 hx).1
+    exact (Set.mem_Icc.1 hx).2
+
+  have l₁ : 0 ≤ sin x := by
+    apply sin_nonneg_of_nonneg_of_le_pi (Set.mem_Icc.1 hx).1
+    trans (1 : ℝ)
+    exact (Set.mem_Icc.1 hx).2
+    trans π / 2
+    exact one_le_pi_div_two
+    norm_num [pi_nonneg]
+  have l₂ : log (sin x) ≤ 0 := log_nonpos l₁ (sin_le_one x)
+
+  simp only [norm_eq_abs, Function.comp_apply]
+  rw [abs_eq_neg_self.2 l₀]
+  rw [abs_eq_neg_self.2 l₂]
+  simp only [neg_le_neg_iff, ge_iff_le]
+
+  have l₃ : x ∈ (Set.Ioi 0) := by
+    simp
+    exact lt_of_le_of_ne (Set.mem_Icc.1 hx).1 ( fun a => h'x (id (Eq.symm a)) )
+
+  have l₅ : 0 < (π / 2)⁻¹ * x := by
+    apply mul_pos
+    apply inv_pos.2
+    apply div_pos pi_pos zero_lt_two
+    exact l₃
+
+  have : ∀ x ∈ (Set.Icc 0 (π / 2)), (π / 2)⁻¹ * x ≤ sin x := by
+    intro x hx
+
+    have i₀ : 0 ∈ Set.Icc 0 π :=
+      Set.left_mem_Icc.mpr pi_nonneg
+    have i₁ : π / 2 ∈ Set.Icc 0 π :=
+      Set.mem_Icc.mpr ⟨div_nonneg pi_nonneg zero_le_two, half_le_self pi_nonneg⟩
+
+    have i₂ : 0 ≤ 1 - (π / 2)⁻¹ * x := by
+      rw [sub_nonneg]
+      calc (π / 2)⁻¹ * x
+      _ ≤ (π / 2)⁻¹ * (π / 2) := by
+        apply mul_le_mul_of_nonneg_left
+        exact (Set.mem_Icc.1 hx).2
+        apply inv_nonneg.mpr (div_nonneg pi_nonneg zero_le_two)
+      _ = 1 := by
+        apply inv_mul_cancel
+        apply div_ne_zero_iff.mpr
+        constructor
+        · exact pi_ne_zero
+        · exact Ne.symm (NeZero.ne' 2)
+
+    have i₃ : 0 ≤ (π / 2)⁻¹ * x := by
+      apply mul_nonneg
+      apply inv_nonneg.2
+      apply div_nonneg
+      exact pi_nonneg
+      exact zero_le_two
+      exact (Set.mem_Icc.1 hx).1
+
+    have i₄ : 1 - (π / 2)⁻¹ * x + (π / 2)⁻¹ * x = 1 := by ring
+
+    let B := strictConcaveOn_sin_Icc.concaveOn.2 i₀ i₁ i₂ i₃ i₄
+    simp [Real.sin_pi_div_two] at B
+    rw [(by ring_nf; rw [mul_inv_cancel pi_ne_zero, one_mul] : 2 / π * x * (π / 2) = x)] at B
+    simpa
+
+  apply log_le_log l₅
+  apply this
+  apply Set.mem_Icc.mpr
+  constructor
+  · exact le_of_lt l₃
+  · trans 1
+    exact (Set.mem_Icc.1 hx).2
+    exact one_le_pi_div_two
+
+
+lemma intervalIntegrable_log_sin₁ : IntervalIntegrable (log ∘ sin) volume 0 1 := by
+
+  have int_log : IntervalIntegrable (fun x ↦ ‖log x‖) volume 0 1 := by
+    apply IntervalIntegrable.norm
+    rw [← neg_neg log]
+    apply IntervalIntegrable.neg
+    apply intervalIntegrable_deriv_of_nonneg (g := fun x ↦ -(x * log x - x))
+    · exact (continuous_mul_log.continuousOn.sub continuous_id.continuousOn).neg
+    · intro x hx
+      norm_num at hx
+      convert ((hasDerivAt_mul_log hx.left.ne.symm).sub (hasDerivAt_id x)).neg using 1
+      norm_num
+    · intro x hx
+      norm_num at hx
+      rw [Pi.neg_apply, Left.nonneg_neg_iff]
+      exact (log_nonpos_iff hx.left).mpr hx.right.le
+
+
+  have int_log : IntervalIntegrable (fun x ↦ ‖log ((π / 2)⁻¹ * x)‖) volume 0 1 := by
+
+    have A := IntervalIntegrable.comp_mul_right int_log (π / 2)⁻¹
+    simp only [norm_eq_abs] at A
+    conv =>
+      arg 1
+      intro x
+      rw [mul_comm]
+    simp only [norm_eq_abs]
+    apply IntervalIntegrable.mono A
+    simp
+    trans Set.Icc 0 (π / 2)
+    exact Set.Icc_subset_Icc (Preorder.le_refl 0) one_le_pi_div_two
+    exact Set.Icc_subset_uIcc
+    exact Preorder.le_refl volume
+
+  apply IntervalIntegrable.mono_fun' (g := fun x ↦ ‖log ((π / 2)⁻¹ * x)‖)
+  exact int_log
+
+  -- AEStronglyMeasurable (log ∘ sin) (volume.restrict (Ι 0 1))
+  apply ContinuousOn.aestronglyMeasurable
+  apply ContinuousOn.comp (t := Ι 0 1)
+  apply ContinuousOn.mono (s := {0}ᶜ)
+  exact continuousOn_log
+  intro x hx
+  by_contra contra
+  simp at contra
+  rw [contra, Set.left_mem_uIoc] at hx
+  linarith
+  exact continuousOn_sin
+
+  -- Set.MapsTo sin (Ι 0 1) (Ι 0 1)
+  rw [Set.uIoc_of_le (zero_le_one' ℝ)]
+  exact fun x hx ↦ ⟨sin_pos_of_pos_of_le_one hx.1 hx.2, sin_le_one x⟩
+
+  -- MeasurableSet (Ι 0 1)
+  exact measurableSet_uIoc
+
+  -- (fun x => ‖(log ∘ sin) x‖) ≤ᶠ[ae (volume.restrict (Ι 0 1))] ‖log‖
+  dsimp [EventuallyLE]
+  rw [MeasureTheory.ae_restrict_iff]
+  apply MeasureTheory.ae_of_all
+  intro x hx
+  have : x ∈ Set.Icc 0 1 := by
+    simp
+    simp at hx
+    constructor
+    · exact le_of_lt hx.1
+    · exact hx.2
+  let A := logsinBound x this
+  simp only [Function.comp_apply, norm_eq_abs] at A
+  exact A
+
+  apply measurableSet_le
+  apply Measurable.comp'
+  exact continuous_abs.measurable
+  exact Measurable.comp' measurable_log continuous_sin.measurable
+  -- Measurable fun a => |log ((π / 2)⁻¹ * a)|
+  apply Measurable.comp'
+  exact continuous_abs.measurable
+  apply Measurable.comp'
+  exact measurable_log
+  exact measurable_const_mul (π / 2)⁻¹
+
+lemma intervalIntegrable_log_sin₂ : IntervalIntegrable (log ∘ sin) volume 0 (π / 2) := by
+
+  apply IntervalIntegrable.trans (b := 1)
+  exact intervalIntegrable_log_sin₁
+
+  -- IntervalIntegrable (log ∘ sin) volume 1 (π / 2)
+  apply ContinuousOn.intervalIntegrable
+  apply ContinuousOn.comp continuousOn_log continuousOn_sin
+  intro x hx
+  rw [Set.uIcc_of_le, Set.mem_Icc] at hx
+  have : 0 < sin x := by
+    apply Real.sin_pos_of_pos_of_lt_pi
+    · calc 0
+      _ < 1 := Real.zero_lt_one
+      _ ≤ x := hx.1
+    · calc x
+      _ ≤ π / 2 := hx.2
+      _ < π := div_two_lt_of_pos pi_pos
+  by_contra h₁x
+  simp at h₁x
+  rw [h₁x] at this
+  simp at this
+  exact one_le_pi_div_two
+
+lemma intervalIntegrable_log_sin : IntervalIntegrable (log ∘ sin) volume 0 π := by
+  apply IntervalIntegrable.trans (b := π / 2)
+  exact intervalIntegrable_log_sin₂
+  let A := IntervalIntegrable.comp_sub_left intervalIntegrable_log_sin₂ π
+  simp at A
+  let B := IntervalIntegrable.symm A
+  have : π - π / 2 = π / 2 := by linarith
+  rwa [this] at B
+
+lemma intervalIntegrable_log_cos : IntervalIntegrable (log ∘ cos) volume 0 (π / 2) := by
+  let A := IntervalIntegrable.comp_sub_left intervalIntegrable_log_sin₂ (π / 2)
+  simp only [Function.comp_apply, sub_zero, sub_self] at A
+  simp_rw [sin_pi_div_two_sub] at A
+  have : (fun x => log (cos x)) = log ∘ cos := rfl
+  apply IntervalIntegrable.symm
+  rwa [← this]
+
+
+theorem logInt
+  {t : ℝ}
+  (ht : 0 < t) :
+  ∫ x in (0 : ℝ)..t, log x = t * log t - t := by
+  rw [← integral_add_adjacent_intervals (b := 1)]
+  trans (-1) + (t * log t - t + 1)
+  · congr
+    · -- ∫ x in 0..1, log x = -1, same proof as before
+      rw [integral_eq_sub_of_hasDerivAt_of_tendsto (f := fun x ↦ x * log x - x) (fa := 0) (fb := -1)]
+      · simp
+      · simp
+      · intro x hx
+        norm_num at hx
+        convert (hasDerivAt_mul_log hx.left.ne.symm).sub (hasDerivAt_id x) using 1
+        norm_num
+      · rw [← neg_neg log]
+        apply IntervalIntegrable.neg
+        apply intervalIntegrable_deriv_of_nonneg (g := fun x ↦ -(x * log x - x))
+        · exact (continuous_mul_log.continuousOn.sub continuous_id.continuousOn).neg
+        · intro x hx
+          norm_num at hx
+          convert ((hasDerivAt_mul_log hx.left.ne.symm).sub (hasDerivAt_id x)).neg using 1
+          norm_num
+        · intro x hx
+          norm_num at hx
+          rw [Pi.neg_apply, Left.nonneg_neg_iff]
+          exact (log_nonpos_iff hx.left).mpr hx.right.le
+      · have := tendsto_log_mul_rpow_nhds_zero zero_lt_one
+        simp_rw [rpow_one, mul_comm] at this
+        -- tendsto_nhdsWithin_of_tendsto_nhds should be under Tendsto namespace
+        convert this.sub (tendsto_nhdsWithin_of_tendsto_nhds tendsto_id)
+        norm_num
+      · rw [(by simp : -1 = 1 * log 1 - 1)]
+        apply tendsto_nhdsWithin_of_tendsto_nhds
+        exact (continuousAt_id.mul (continuousAt_log one_ne_zero)).sub continuousAt_id
+    · -- ∫ x in 1..t, log x = t * log t + 1, just use integral_log_of_pos
+      rw [integral_log_of_pos zero_lt_one ht]
+      norm_num
+  · abel
+  · -- log is integrable on [[0, 1]]
+    rw [← neg_neg log]
+    apply IntervalIntegrable.neg
+    apply intervalIntegrable_deriv_of_nonneg (g := fun x ↦ -(x * log x - x))
+    · exact (continuous_mul_log.continuousOn.sub continuous_id.continuousOn).neg
+    · intro x hx
+      norm_num at hx
+      convert ((hasDerivAt_mul_log hx.left.ne.symm).sub (hasDerivAt_id x)).neg using 1
+      norm_num
+    · intro x hx
+      norm_num at hx
+      rw [Pi.neg_apply, Left.nonneg_neg_iff]
+      exact (log_nonpos_iff hx.left).mpr hx.right.le
+  · -- log is integrable on [[0, t]]
+    simp [Set.mem_uIcc, ht]
+
+
+lemma integral_log_sin : ∫ (x : ℝ) in (0)..(π / 2), log (sin x) =  -log 2 * π/2 := by
+
+  have t₀ {x : ℝ} : sin (2 * x) = 2 * sin x * cos x := sin_two_mul x
+
+  have t₁ {x : ℝ} : log (sin (2 * x)) = log 2 + log (sin x) + log (cos x) := by
+    rw [sin_two_mul x, log_mul, log_mul]
+    exact Ne.symm (NeZero.ne' 2)
+    sorry
+    sorry
+    sorry
+
+  have t₂ {x : ℝ} : log (sin x) = log (sin (2 * x)) - log 2 - log (cos x) := by
+    rw [t₁]
+    ring
+
+  conv =>
+    left
+    arg 1
+    intro x
+    rw [t₂]
+
+  rw [intervalIntegral.integral_sub, intervalIntegral.integral_sub]
+  rw [intervalIntegral.integral_const]
+  rw [intervalIntegral.integral_comp_mul_left (c := 2) (f := fun x ↦ log (sin x))]
+  simp
+  have : 2 * (π / 2) = π := by linarith
+  rw [this]
+
+  have : ∫ (x : ℝ) in (0)..π, log (sin x) = 2 * ∫ (x : ℝ) in (0)..(π / 2), log (sin x) := by
+    sorry
+  rw [this]
+  have : ∫ (x : ℝ) in (0)..(π / 2), log (sin x) = ∫ (x : ℝ) in (0)..(π / 2), log (cos x) := by
+    sorry
+  rw [← this]
+  simp
+  linarith
+
+  exact Ne.symm (NeZero.ne' 2)
+  -- IntervalIntegrable (fun x => log (sin (2 * x))) volume 0 (π / 2)
+  sorry
+  -- IntervalIntegrable (fun x => log 2) volume 0 (π / 2)
+  simp
+  -- IntervalIntegrable (fun x => log (sin (2 * x)) - log 2) volume 0 (π / 2)
+  apply IntervalIntegrable.sub
+  -- -- IntervalIntegrable (fun x => log (sin (2 * x))) volume 0 (π / 2)
+  sorry
+  -- -- IntervalIntegrable (fun x => log 2) volume 0 (π / 2)
+  simp
+  -- -- IntervalIntegrable (fun x => log (cos x)) volume 0 (π / 2)
+  exact intervalIntegrable_log_cos
+
+
+lemma int₁₁ : ∫ (x : ℝ) in (0)..π, log (4 * sin x ^ 2) = 0 := by
+  sorry
+
+
+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