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@ -125,6 +125,23 @@ theorem HolomorphicAt.analyticAt
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exact IsOpen.mem_nhds h₁s h₂s
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theorem AnalyticAt.holomorphicAt
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[CompleteSpace F]
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{f : ℂ → F}
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{x : ℂ} :
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AnalyticAt ℂ f x → HolomorphicAt f x := by
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intro hf
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rw [HolomorphicAt_iff]
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use {x : ℂ | AnalyticAt ℂ f x}
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constructor
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· exact isOpen_analyticAt ℂ f
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· constructor
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· simpa
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· intro z hz
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simp at hz
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exact differentiableAt hz
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theorem HolomorphicAt.contDiffAt
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[CompleteSpace F]
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{f : ℂ → F}
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@ -7,72 +7,12 @@ import Nevanlinna.specialFunctions_CircleIntegral_affine
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open Real
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/-
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noncomputable def Zeroset
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{f : ℂ → ℂ}
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{s : Set ℂ}
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(hf : ∀ z ∈ s, HolomorphicAt f z) :
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Set ℂ := by
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exact f⁻¹' {0} ∩ s
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lemma h₁U : IsPreconnected (Metric.closedBall (0 : ℂ) 1) :=
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(convex_closedBall (0 : ℂ) 1).isPreconnected
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lemma h₂U : IsCompact (Metric.closedBall (0 : ℂ) 1) :=
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isCompact_closedBall 0 1
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noncomputable def ZeroFinset
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{f : ℂ → ℂ}
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(h₁f : ∀ z ∈ Metric.closedBall (0 : ℂ) 1, HolomorphicAt f z)
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(h₂f : f 0 ≠ 0) :
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Finset ℂ := by
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let Z := f⁻¹' {0} ∩ Metric.closedBall (0 : ℂ) 1
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have hZ : Set.Finite Z := by
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dsimp [Z]
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rw [Set.inter_comm]
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apply finiteZeros
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-- Ball is preconnected
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apply IsConnected.isPreconnected
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apply Convex.isConnected
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exact convex_closedBall 0 1
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exact Set.nonempty_of_nonempty_subtype
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--
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exact isCompact_closedBall 0 1
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--
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intro x hx
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have A := (h₁f x hx)
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let B := HolomorphicAt_iff.1 A
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obtain ⟨s, h₁s, h₂s, h₃s⟩ := B
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apply DifferentiableOn.analyticAt (s := s)
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intro z hz
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apply DifferentiableAt.differentiableWithinAt
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apply h₃s
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exact hz
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exact IsOpen.mem_nhds h₁s h₂s
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--
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use 0
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constructor
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· simp
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· exact h₂f
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exact hZ.toFinset
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lemma ZeroFinset_mem_iff
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{f : ℂ → ℂ}
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(h₁f : ∀ z ∈ Metric.closedBall (0 : ℂ) 1, HolomorphicAt f z)
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{h₂f : f 0 ≠ 0}
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(z : ℂ) :
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z ∈ ↑(ZeroFinset h₁f h₂f) ↔ z ∈ Metric.closedBall 0 1 ∧ f z = 0 := by
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dsimp [ZeroFinset]; simp
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tauto
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noncomputable def order
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{f : ℂ → ℂ}
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{h₁f : ∀ z ∈ Metric.closedBall (0 : ℂ) 1, HolomorphicAt f z}
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{h₂f : f 0 ≠ 0} :
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ZeroFinset h₁f h₂f → ℕ := by
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intro i
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let A := ((ZeroFinset_mem_iff h₁f i).1 i.2).1
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let B := (h₁f i.1 A).analyticAt
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exact B.order.toNat
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-/
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theorem jensen_case_R_eq_one
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(f : ℂ → ℂ)
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@ -81,19 +21,15 @@ theorem jensen_case_R_eq_one
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(h₂f : f 0 ≠ 0) :
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log ‖f 0‖ = -∑ᶠ s, (h'₁f.order s).toNat * log (‖s.1‖⁻¹) + (2 * π)⁻¹ * ∫ (x : ℝ) in (0)..(2 * π), log ‖f (circleMap 0 1 x)‖ := by
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have h₁U : IsPreconnected (Metric.closedBall (0 : ℂ) 1) :=
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(convex_closedBall (0 : ℂ) 1).isPreconnected
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have h₂U : IsCompact (Metric.closedBall (0 : ℂ) 1) :=
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isCompact_closedBall 0 1
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have h'₂f : ∃ u ∈ (Metric.closedBall (0 : ℂ) 1), f u ≠ 0 := by
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use 0; simp; exact h₂f
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obtain ⟨F, h₁F, h₂F, h₃F⟩ := AnalyticOnCompact.eliminateZeros₂ h₁U h₂U h'₁f h'₂f
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have h'₁F : ∀ z ∈ Metric.closedBall (0 : ℂ) 1, HolomorphicAt F z := by
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sorry
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intro z h₁z
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apply AnalyticAt.holomorphicAt
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exact h₁F z h₁z
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let G := fun z ↦ log ‖F z‖ + ∑ s ∈ (finiteZeros h₁U h₂U h'₁f h'₂f).toFinset, (h'₁f.order s).toNat * log ‖z - s‖
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@ -149,7 +85,7 @@ theorem jensen_case_R_eq_one
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exact h₂F z h₁z
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have : ∫ (x : ℝ) in (0)..2 * π, log ‖f (circleMap 0 1 x)‖ = ∫ (x : ℝ) in (0)..2 * π, G (circleMap 0 1 x) := by
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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
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rw [intervalIntegral.integral_congr_ae]
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rw [MeasureTheory.ae_iff]
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@ -184,7 +120,7 @@ theorem jensen_case_R_eq_one
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-- This is hard. Need to invoke specialFunctions_CircleIntegral_affine.
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sorry
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have : ∫ (x : ℝ) in (0)..2 * π, G (circleMap 0 1 x)
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have decompose_int_G : ∫ (x : ℝ) in (0)..2 * π, G (circleMap 0 1 x)
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= (∫ (x : ℝ) in (0)..2 * π, log (Complex.abs (F (circleMap 0 1 x))))
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+ ∑ x ∈ (finiteZeros h₁U h₂U h'₁f h'₂f).toFinset, (h'₁f.order x).toNat * ∫ (x_1 : ℝ) in (0)..2 * π, log (Complex.abs (circleMap 0 1 x_1 - ↑x)) := by
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dsimp [G]
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@ -290,9 +226,65 @@ theorem jensen_case_R_eq_one
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exact h₂F z hz
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apply harmonic_meanValue₁ 1 Real.zero_lt_one t₀
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simp_rw [← Complex.norm_eq_abs] at this
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rw [t₁] at this
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simp_rw [← Complex.norm_eq_abs] at decompose_int_G
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rw [t₁] at decompose_int_G
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conv at decompose_int_G =>
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right
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right
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arg 2
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intro x
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right
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rw [int₃ x.2]
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simp at decompose_int_G
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sorry
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rw [int_logAbs_f_eq_int_G]
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rw [decompose_int_G]
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rw [h₃F]
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simp
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have {l : ℝ} : π⁻¹ * 2⁻¹ * (2 * π * l) = l := by
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calc π⁻¹ * 2⁻¹ * (2 * π * l)
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_ = π⁻¹ * (2⁻¹ * 2) * π * l := by ring
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_ = π⁻¹ * π * l := by ring
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_ = (π⁻¹ * π) * l := by ring
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_ = 1 * l := by
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rw [inv_mul_cancel₀]
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exact pi_ne_zero
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_ = l := by simp
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rw [this]
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rw [log_mul]
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rw [log_prod]
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simp
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rw [finsum_eq_sum_of_support_subset _ (s := (finiteZeros h₁U h₂U h'₁f h'₂f).toFinset)]
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simp
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simp
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intro x ⟨h₁x, h₂x⟩
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simp
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dsimp [AnalyticOn.order] at h₁x
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simp only [Function.mem_support, ne_eq, Nat.cast_eq_zero, not_or] at h₁x
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exact AnalyticAt.supp_order_toNat (AnalyticOn.order.proof_1 h'₁f x) h₁x
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--
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intro x hx
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simp at hx
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simp
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intro h₁x
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nth_rw 1 [← h₁x] at h₂f
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tauto
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--
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rw [Finset.prod_ne_zero_iff]
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intro x hx
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simp at hx
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simp
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intro h₁x
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nth_rw 1 [← h₁x] at h₂f
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tauto
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--
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simp
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apply h₂F
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simp
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@ -360,3 +360,16 @@ lemma int₂
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simp
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simp_rw [this]
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exact int₁
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lemma int₃
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{a : ℂ}
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(ha : a ∈ Metric.closedBall 0 1) :
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∫ x in (0)..(2 * π), log ‖circleMap 0 1 x - a‖ = 0 := by
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by_cases h₁a : a ∈ Metric.ball 0 1
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· exact int₀ h₁a
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· apply int₂
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simp at ha
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simp at h₁a
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simp
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linarith
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