Working…

2024-09-10 15:29:30 +02:00
parent 8e5ada9a01
commit 745e614016
3 changed files with 74 additions and 460 deletions
--- a/Nevanlinna/analyticOn_zeroSet.lean
+++ b/Nevanlinna/analyticOn_zeroSet.lean
@@ -357,6 +357,53 @@ theorem AnalyticOnCompact.eliminateZeros
    · exact inter


+
+
+theorem AnalyticOnCompact.eliminateZeros₂
+  {f : ℂ → ℂ}
+  {U : Set ℂ}
+  (h₁U : IsPreconnected U)
+  (h₂U : IsCompact U)
+  (h₁f : AnalyticOn ℂ f U)
+  (h₂f : ∃ u ∈ U, f u ≠ 0) :
+  ∃ (g : ℂ → ℂ), AnalyticOn ℂ g U ∧ (∀ z ∈ U, g z ≠ 0) ∧ ∀ z, f z = (∏ a ∈ (finiteZeros h₁U h₂U h₁f h₂f).toFinset, (z - a) ^ (h₁f.order a).toNat) • g z := by
+
+  let A := (finiteZeros h₁U h₂U h₁f h₂f).toFinset
+
+  let n : U → ℕ := fun z ↦ (h₁f z z.2).order.toNat
+
+  have hn : ∀ a ∈ A, (h₁f a a.2).order = n a := by
+    intro a _
+    dsimp [n, AnalyticOn.order]
+    rw [eq_comm]
+    apply XX h₁U
+    exact h₂f
+
+  obtain ⟨g, h₁g, h₂g, h₃g⟩ := AnalyticOn.eliminateZeros (A := A) h₁f n hn
+  use g
+
+  have inter : ∀ (z : ℂ), f z = (∏ a ∈ A, (z - ↑a) ^ (h₁f (↑a) a.property).order.toNat) • g z := by
+    intro z
+    rw [h₃g z]
+
+  constructor
+  · exact h₁g
+  · constructor
+    · intro z h₁z
+      by_cases h₂z : ⟨z, h₁z⟩  ∈ ↑A.toSet
+      · exact h₂g ⟨z, h₁z⟩ h₂z
+      · have : f z ≠ 0 := by
+          by_contra C
+          have : ⟨z, h₁z⟩ ∈ ↑A.toSet := by
+            dsimp [A]
+            simp
+            exact C
+          tauto
+        rw [inter z] at this
+        exact right_ne_zero_of_smul this
+    · exact h₃g
+
+
 theorem AnalyticOnCompact.eliminateZeros₁
  {f : ℂ → ℂ}
  {U : Set ℂ}
@@ -364,8 +411,7 @@ theorem AnalyticOnCompact.eliminateZeros₁
  (h₂U : IsCompact U)
  (h₁f : AnalyticOn ℂ f U)
  (h₂f : ∃ u ∈ U, f u ≠ 0) :
-  ∃ (g : ℂ → ℂ), AnalyticOn ℂ g U ∧ (∀ z ∈ U, g z ≠ 0) ∧ ∀ z, f z = (∏ᶠ a : U, (z - a) ^ (h₁f.order a).toNat) • g z := by
-
+  ∃ (g : ℂ → ℂ), AnalyticOn ℂ g U ∧ (∀ z ∈ U, g z ≠ 0) ∧ ∀ z, f z = (∏ᶠ a, (z - a) ^ (h₁f.order a).toNat) • g z := by

  let A := (finiteZeros h₁U h₂U h₁f h₂f).toFinset

--- a/Nevanlinna/analyticOn_zeroSet2.lean
+++ b/Nevanlinna/analyticOn_zeroSet2.lean
@@ -1,449 +0,0 @@
-import Mathlib.Analysis.Analytic.Constructions
-import Mathlib.Analysis.Analytic.IsolatedZeros
-import Mathlib.Analysis.Complex.Basic
-
-
-theorem AnalyticOn.order_eq_nat_iff
-  {f : ℂ → ℂ}
-  {U : Set ℂ}
-  {z₀ : ℂ}
-  (hf : AnalyticOn ℂ f U)
-  (hz₀ : z₀ ∈ U)
-  (n : ℕ) :
-  (hf z₀ hz₀).order = ↑n ↔ ∃ (g : ℂ → ℂ), AnalyticOn ℂ g U ∧ g z₀ ≠ 0 ∧ ∀ z, f z = (z - z₀) ^ n • g z := by
-
-  constructor
-  -- Direction →
-  intro hn
-  obtain ⟨gloc, h₁gloc, h₂gloc, h₃gloc⟩ := (AnalyticAt.order_eq_nat_iff (hf z₀ hz₀) n).1 hn
-
-  -- Define a candidate function; this is (f z) / (z - z₀) ^ n with the
-  -- removable singularity removed
-  let g : ℂ → ℂ := fun z ↦ if z = z₀ then gloc z₀ else (f z) / (z - z₀) ^ n
-
-  -- Describe g near z₀
-  have g_near_z₀ : ∀ᶠ (z : ℂ) in nhds z₀, g z = gloc z := by
-    rw [eventually_nhds_iff]
-    obtain ⟨t, h₁t, h₂t, h₃t⟩ := eventually_nhds_iff.1 h₃gloc
-    use t
-    constructor
-    · intro y h₁y
-      by_cases h₂y : y = z₀
-      · dsimp [g]; simp [h₂y]
-      · dsimp [g]; simp [h₂y]
-        rw [div_eq_iff_mul_eq, eq_comm, mul_comm]
-        exact h₁t y h₁y
-        norm_num
-        rw [sub_eq_zero]
-        tauto
-    · constructor
-      · assumption
-      · assumption
-
-  -- Describe g near points z₁ that are different from z₀
-  have g_near_z₁ {z₁ : ℂ} : z₁ ≠ z₀ → ∀ᶠ (z : ℂ) in nhds z₁, g z = f z / (z - z₀) ^ n := by
-    intro hz₁
-    rw [eventually_nhds_iff]
-    use {z₀}ᶜ
-    constructor
-    · intro y hy
-      simp at hy
-      simp [g, hy]
-    · exact ⟨isOpen_compl_singleton, hz₁⟩
-
-  -- Use g and show that it has all required properties
-  use g
-  constructor
-  · -- AnalyticOn ℂ g U
-    intro z h₁z
-    by_cases h₂z : z = z₀
-    · rw [h₂z]
-      apply AnalyticAt.congr h₁gloc
-      exact Filter.EventuallyEq.symm g_near_z₀
-    · simp_rw [eq_comm] at g_near_z₁
-      apply AnalyticAt.congr _ (g_near_z₁ h₂z)
-      apply AnalyticAt.div
-      exact hf z h₁z
-      apply AnalyticAt.pow
-      apply AnalyticAt.sub
-      apply analyticAt_id
-      apply analyticAt_const
-      simp
-      rw [sub_eq_zero]
-      tauto
-  · constructor
-    · simp [g]; tauto
-    · intro z
-      by_cases h₂z : z = z₀
-      · rw [h₂z, g_near_z₀.self_of_nhds]
-        exact h₃gloc.self_of_nhds
-      · rw [(g_near_z₁ h₂z).self_of_nhds]
-        simp [h₂z]
-        rw [div_eq_mul_inv, mul_comm, mul_assoc, inv_mul_cancel₀]
-        simp; norm_num
-        rw [sub_eq_zero]
-        tauto
-
-  -- direction ←
-  intro h
-  obtain ⟨g, h₁g, h₂g, h₃g⟩ := h
-  rw [AnalyticAt.order_eq_nat_iff]
-  use g
-  exact ⟨h₁g z₀ hz₀, ⟨h₂g, Filter.Eventually.of_forall h₃g⟩⟩
-
-
-theorem AnalyticAt.order_mul
-  {f₁ f₂ : ℂ → ℂ}
-  {z₀ : ℂ}
-  (hf₁ : AnalyticAt ℂ f₁ z₀)
-  (hf₂ : AnalyticAt ℂ f₂ z₀) :
-  (AnalyticAt.mul hf₁ hf₂).order = hf₁.order + hf₂.order := by
-  by_cases h₂f₁ : hf₁.order = ⊤
-  · simp [h₂f₁]
-    rw [AnalyticAt.order_eq_top_iff, eventually_nhds_iff]
-    rw [AnalyticAt.order_eq_top_iff, eventually_nhds_iff] at h₂f₁
-    obtain ⟨t, h₁t, h₂t, h₃t⟩ := h₂f₁
-    use t
-    constructor
-    · intro y hy
-      rw [h₁t y hy]
-      ring
-    · exact ⟨h₂t, h₃t⟩
-  · by_cases h₂f₂ : hf₂.order = ⊤
-    · simp [h₂f₂]
-      rw [AnalyticAt.order_eq_top_iff, eventually_nhds_iff]
-      rw [AnalyticAt.order_eq_top_iff, eventually_nhds_iff] at h₂f₂
-      obtain ⟨t, h₁t, h₂t, h₃t⟩ := h₂f₂
-      use t
-      constructor
-      · intro y hy
-        rw [h₁t y hy]
-        ring
-      · exact ⟨h₂t, h₃t⟩
-    · obtain ⟨g₁, h₁g₁, h₂g₁, h₃g₁⟩ := (AnalyticAt.order_eq_nat_iff hf₁ ↑hf₁.order.toNat).1 (eq_comm.1 (ENat.coe_toNat h₂f₁))
-      obtain ⟨g₂, h₁g₂, h₂g₂, h₃g₂⟩ := (AnalyticAt.order_eq_nat_iff hf₂ ↑hf₂.order.toNat).1 (eq_comm.1 (ENat.coe_toNat h₂f₂))
-      rw [← ENat.coe_toNat h₂f₁, ← ENat.coe_toNat h₂f₂, ← ENat.coe_add]
-      rw [AnalyticAt.order_eq_nat_iff (AnalyticAt.mul hf₁ hf₂) ↑(hf₁.order.toNat + hf₂.order.toNat)]
-      use g₁ * g₂
-      constructor
-      · exact AnalyticAt.mul h₁g₁ h₁g₂
-      · constructor
-        · simp; tauto
-        · obtain ⟨t₁, h₁t₁, h₂t₁, h₃t₁⟩ := eventually_nhds_iff.1 h₃g₁
-          obtain ⟨t₂, h₁t₂, h₂t₂, h₃t₂⟩ := eventually_nhds_iff.1 h₃g₂
-          rw [eventually_nhds_iff]
-          use t₁ ∩ t₂
-          constructor
-          · intro y hy
-            rw [h₁t₁ y hy.1, h₁t₂ y hy.2]
-            simp; ring
-          · constructor
-            · exact IsOpen.inter h₂t₁ h₂t₂
-            · exact Set.mem_inter h₃t₁ h₃t₂
-
-
-theorem AnalyticOn.eliminateZeros
-  {f : ℂ → ℂ}
-  {U : Set ℂ}
-  {A : Finset U}
-  (hf : AnalyticOn ℂ f U)
-  (n : ℂ → ℕ) :
-  (∀ a ∈ A, (hf a.1 a.2).order = n a) → ∃ (g : ℂ → ℂ), AnalyticOn ℂ g U ∧ (∀ a ∈ A, g a ≠ 0) ∧ ∀ z, f z = (∏ a ∈ A, (z - a) ^ (n a)) • g z := by
-
-  apply Finset.induction (α := U) (p := fun A ↦ (∀ a ∈ A, (hf a.1 a.2).order = n a) → ∃ (g : ℂ → ℂ), AnalyticOn ℂ g U ∧ (∀ a ∈ A, g a ≠ 0) ∧ ∀ z, f z = (∏ a ∈ A, (z - a) ^ (n a)) • g z)
-
-  -- case empty
-  simp
-  use f
-  simp
-  exact hf
-
-  -- case insert
-  intro b₀ B hb iHyp
-  intro hBinsert
-  obtain ⟨g₀, h₁g₀, h₂g₀, h₃g₀⟩ := iHyp (fun a ha ↦ hBinsert a (Finset.mem_insert_of_mem ha))
-
-  have : (h₁g₀ b₀ b₀.2).order = n b₀ := by
-
-    rw [← hBinsert b₀ (Finset.mem_insert_self b₀ B)]
-
-    let φ := fun z ↦ (∏ a ∈ B, (z - a.1) ^ n a.1)
-
-    have : f = fun z ↦ φ z * g₀ z := by
-      funext z
-      rw [h₃g₀ z]
-      rfl
-    simp_rw [this]
-
-    have h₁φ : AnalyticAt ℂ φ b₀ := by
-      dsimp [φ]
-      apply Finset.analyticAt_prod
-      intro b _
-      apply AnalyticAt.pow
-      apply AnalyticAt.sub
-      apply analyticAt_id ℂ
-      exact analyticAt_const
-
-    have h₂φ : h₁φ.order = (0 : ℕ) := by
-      rw [AnalyticAt.order_eq_nat_iff h₁φ 0]
-      use φ
-      constructor
-      · assumption
-      · constructor
-        · dsimp [φ]
-          push_neg
-          rw [Finset.prod_ne_zero_iff]
-          intro a ha
-          simp
-          have : ¬ (b₀.1 - a.1 = 0) := by
-            by_contra C
-            rw [sub_eq_zero] at C
-            rw [SetCoe.ext C] at hb
-            tauto
-          tauto
-        · simp
-
-    rw [AnalyticAt.order_mul h₁φ (h₁g₀ b₀ b₀.2)]
-
-    rw [h₂φ]
-    simp
-
-
-  obtain ⟨g₁, h₁g₁, h₂g₁, h₃g₁⟩ := (AnalyticOn.order_eq_nat_iff h₁g₀ b₀.2 (n b₀)).1 this
-
-  use g₁
-  constructor
-  · exact h₁g₁
-  · constructor
-    · intro a h₁a
-      by_cases h₂a : a = b₀
-      · rwa [h₂a]
-      · let A' := Finset.mem_of_mem_insert_of_ne h₁a h₂a
-        let B' := h₃g₁ a
-        let C' := h₂g₀ a A'
-        rw [B'] at C'
-        exact right_ne_zero_of_smul C'
-    · intro z
-      let A' := h₃g₀ z
-      rw [h₃g₁ z] at A'
-      rw [A']
-      rw [← smul_assoc]
-      congr
-      simp
-      rw [Finset.prod_insert]
-      ring
-      exact hb
-
-
-theorem XX
-  {f : ℂ → ℂ}
-  {U : Set ℂ}
-  (hU : IsPreconnected U)
-  (h₁f : AnalyticOn ℂ f U)
-  (h₂f : ∃ u ∈ U, f u ≠ 0) :
-  ∀ (hu : u ∈ U), (h₁f u hu).order.toNat = (h₁f u hu).order := by
-
-  intro hu
-  apply ENat.coe_toNat
-  by_contra C
-  rw [(h₁f u hu).order_eq_top_iff] at C
-  rw [← (h₁f u hu).frequently_zero_iff_eventually_zero] at C
-  obtain ⟨u₁, h₁u₁, h₂u₁⟩ := h₂f
-  rw [(h₁f.eqOn_zero_of_preconnected_of_frequently_eq_zero hU hu C) h₁u₁] at h₂u₁
-  tauto
-
-
-theorem discreteZeros
-  {f : ℂ → ℂ}
-  {U : Set ℂ}
-  (hU : IsPreconnected U)
-  (h₁f : AnalyticOn ℂ f U)
-  (h₂f : ∃ u ∈ U, f u ≠ 0) :
-  DiscreteTopology ↑(U ∩ f⁻¹' {0}) := by
-
-  simp_rw [← singletons_open_iff_discrete]
-  simp_rw [Metric.isOpen_singleton_iff]
-
-  intro z
-
-  let A := XX hU h₁f h₂f z.2.1
-  rw [eq_comm] at A
-  rw [AnalyticAt.order_eq_nat_iff] at A
-  obtain ⟨g, h₁g, h₂g, h₃g⟩ := A
-
-  rw [Metric.eventually_nhds_iff_ball] at h₃g
-  have : ∃ ε > 0, ∀ y ∈ Metric.ball (↑z) ε, g y ≠ 0 := by
-    have h₄g : ContinuousAt g z := AnalyticAt.continuousAt h₁g
-    have : {0}ᶜ ∈ nhds (g z) := by
-      exact compl_singleton_mem_nhds_iff.mpr h₂g
-
-    let F := h₄g.preimage_mem_nhds this
-    rw [Metric.mem_nhds_iff] at F
-    obtain ⟨ε, h₁ε, h₂ε⟩ := F
-    use ε
-    constructor; exact h₁ε
-    intro y hy
-    let G := h₂ε hy
-    simp at G
-    exact G
-  obtain ⟨ε₁, h₁ε₁⟩ := this
-
-  obtain ⟨ε₂, h₁ε₂, h₂ε₂⟩ := h₃g
-  use min ε₁ ε₂
-  constructor
-  · have : 0 < min ε₁ ε₂ := by
-      rw [lt_min_iff]
-      exact And.imp_right (fun _ => h₁ε₂) h₁ε₁
-    exact this
-
-  intro y
-  intro h₁y
-
-  have h₂y : ↑y ∈ Metric.ball (↑z) ε₂ := by
-    simp
-    calc dist y z
-    _ < min ε₁ ε₂ := by assumption
-    _ ≤ ε₂ := by exact min_le_right ε₁ ε₂
-
-  have h₃y : ↑y ∈ Metric.ball (↑z) ε₁ := by
-    simp
-    calc dist y z
-    _ < min ε₁ ε₂ := by assumption
-    _ ≤ ε₁ := by exact min_le_left ε₁ ε₂
-
-
-  have F := h₂ε₂ y.1 h₂y
-  rw [y.2.2] at F
-  simp at F
-
-  have : g y.1 ≠ 0 := by
-    exact h₁ε₁.2 y h₃y
-  simp [this] at F
-  ext
-  rw [sub_eq_zero] at F
-  tauto
-
-
-theorem finiteZeros
-  {f : ℂ → ℂ}
-  {U : Set ℂ}
-  (h₁U : IsPreconnected U)
-  (h₂U : IsCompact U)
-  (h₁f : AnalyticOn ℂ f U)
-  (h₂f : ∃ u ∈ U, f u ≠ 0) :
-  Set.Finite ↑(U ∩ f⁻¹' {0}) := by
-
-  have hinter : IsCompact ↑(U ∩ f⁻¹' {0}) := by
-    apply IsCompact.of_isClosed_subset h₂U
-    apply h₁f.continuousOn.preimage_isClosed_of_isClosed
-    exact IsCompact.isClosed h₂U
-    exact isClosed_singleton
-    exact Set.inter_subset_left
-
-  apply hinter.finite
-  apply DiscreteTopology.of_subset (s := ↑(U ∩ f⁻¹' {0}))
-  exact discreteZeros h₁U h₁f h₂f
-  rfl
-
-
-theorem AnalyticOnCompact.eliminateZeros
-  {f : ℂ → ℂ}
-  {U : Set ℂ}
-  (h₁U : IsPreconnected U)
-  (h₂U : IsCompact U)
-  (h₁f : AnalyticOn ℂ f U)
-  (h₂f : ∃ u ∈ U, f u ≠ 0) :
-  ∃ (g : ℂ → ℂ) (A : Finset U), AnalyticOn ℂ g U ∧ (∀ z ∈ U, g z ≠ 0) ∧ ∀ z, f z = (∏ a ∈ A, (z - a) ^ (h₁f a a.2).order.toNat) • g z := by
-
-  let ι : U → ℂ := Subtype.val
-
-  let A₁ := ι⁻¹' (U ∩ f⁻¹' {0})
-
-  have : A₁.Finite := by
-    apply Set.Finite.preimage
-    exact Set.injOn_subtype_val
-    exact finiteZeros h₁U h₂U h₁f h₂f
-  let A := this.toFinset
-
-  let n : ℂ → ℕ := by
-    intro z
-    by_cases hz : z ∈ U
-    · exact (h₁f z hz).order.toNat
-    · exact 0
-
-  have hn : ∀ a ∈ A, (h₁f a a.2).order = n a := by
-    intro a _
-    dsimp [n]
-    simp
-    rw [eq_comm]
-    apply XX h₁U
-    exact h₂f
-
-  obtain ⟨g, h₁g, h₂g, h₃g⟩ := AnalyticOn.eliminateZeros (A := A) h₁f n hn
-  use g
-  use A
-
-  have inter : ∀ (z : ℂ), f z = (∏ a ∈ A, (z - ↑a) ^ (h₁f (↑a) a.property).order.toNat) • g z := by
-    intro z
-    rw [h₃g z]
-    congr
-    funext a
-    congr
-    dsimp [n]
-    simp [a.2]
-
-  constructor
-  · exact h₁g
-  · constructor
-    · intro z h₁z
-      by_cases h₂z : ⟨z, h₁z⟩  ∈ ↑A.toSet
-      · exact h₂g ⟨z, h₁z⟩ h₂z
-      · have : f z ≠ 0 := by
-          by_contra C
-          have : ⟨z, h₁z⟩ ∈ ↑A₁ := by
-            dsimp [A₁, ι]
-            simp
-            exact C
-          have : ⟨z, h₁z⟩ ∈ ↑A.toSet := by
-            dsimp [A]
-            simp
-            exact this
-          tauto
-        rw [inter z] at this
-        exact right_ne_zero_of_smul this
-    · exact inter
-
-
-noncomputable def AnalyticOn.order
-  {f : ℂ → ℂ}
-  {U : Set ℂ}
-  (hf : AnalyticOn ℂ f U) :
-  ℂ → ℕ∞ := by
-  intro z
-  if hz : z ∈ U then
-    exact (hf z hz).order
-  else
-    exact 0
-
-
-theorem AnalyticOnCompact.eliminateZeros₁
-  {f : ℂ → ℂ}
-  {U : Set ℂ}
-  (h₁U : IsPreconnected U)
-  (h₂U : IsCompact U)
-  (h₁f : AnalyticOn ℂ f U)
-  (h₂f : ∃ u ∈ U, f u ≠ 0) :
-  ∃ (g : ℂ → ℂ), AnalyticOn ℂ g U ∧ (∀ z ∈ U, g z ≠ 0) ∧ ∀ z, f z = (∏ᶠ u, (z - u) ^ (h₁f.order u).toNat) • g z := by
-
-  obtain ⟨g, A, h₁g, h₂g, h₃g⟩ := AnalyticOnCompact.eliminateZeros h₁U h₂U h₁f h₂f
-
-  use g
-  constructor
-  · exact h₁g
-  · constructor
-    · exact h₂g
-    · intro z
-      rw [h₃g z]
-      congr
-
-      sorry
--- a/Nevanlinna/holomorphic_JensenFormula2.lean
+++ b/Nevanlinna/holomorphic_JensenFormula2.lean
@@ -77,16 +77,29 @@ noncomputable def order
 theorem jensen_case_R_eq_one
  (f : ℂ → ℂ)
  (h₁f : ∀ z ∈ Metric.closedBall (0 : ℂ) 1, HolomorphicAt f z)
-  (h'₁f : ∀ z ∈ Metric.closedBall (0 : ℂ) 1, AnalyticAt ℂ f z)
+  (h'₁f : AnalyticOn ℂ f (Metric.closedBall (0 : ℂ) 1))
  (h₂f : f 0 ≠ 0) :
-  log ‖f 0‖ = -∑ s : (ZeroFinset h₁f h₂f), order s * log (‖s.1‖⁻¹) + (2 * π )⁻¹ * ∫ (x : ℝ) in (0)..2 * π, log ‖f (circleMap 0 1 x)‖ := by
+  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

-  have F : ℂ → ℂ := by sorry
-  have h₁F : ∀ z ∈ Metric.closedBall (0 : ℂ) 1, HolomorphicAt F z := by sorry
-  have h₂F : ∀ z ∈ Metric.closedBall (0 : ℂ) 1, F z ≠ 0 := by sorry
-  have h₃F : f = fun z ↦ (F z) * ∏ s : ZeroFinset h₁f h₂f, (z - s) ^ (order s) := by sorry
+  have h₁U : IsPreconnected (Metric.closedBall (0 : ℂ) 1) := by
+    apply IsConnected.isPreconnected
+    apply Convex.isConnected
+    exact convex_closedBall 0 1
+    exact Set.nonempty_of_nonempty_subtype

-  let G := fun z ↦ log ‖F z‖ + ∑ s : ZeroFinset h₁f h₂f, (order s) * log ‖z - s‖
+  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 h'₁f h'₂f
+
+  have h'₁F : ∀ z ∈ Metric.closedBall (0 : ℂ) 1, HolomorphicAt F z := by
+    sorry
+
+  let G := fun z ↦ log ‖F z‖ + ∑ s ∈ (finiteZeros h₁U h₂U h'₁f h'₂f).toFinset, (h'₁f.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
@@ -95,9 +108,10 @@ theorem jensen_case_R_eq_one
      left
      arg 1
      rw [h₃F]
+      rw [smul_eq_mul]
      rw [norm_mul]
      rw [norm_prod]
-      right
+      left
      arg 2
      intro b
      rw [norm_pow]
@@ -106,13 +120,15 @@ theorem jensen_case_R_eq_one
    rw [Real.log_prod]
    conv =>
      left
-      right
+      left
      arg 2
      intro s
      rw [Real.log_pow]
    dsimp [G]
+    abel

    -- ∀ x ∈ (ZeroFinset h₁f).attach, Complex.abs (z - ↑x) ^ order x ≠ 0
+
    simp
    intro s hs
    rw [ZeroFinset_mem_iff h₁f s] at hs
@@ -153,6 +169,7 @@ theorem jensen_case_R_eq_one
    apply finiteZeros

    -- IsPreconnected (Metric.closedBall (0 : ℂ) 1)
+
    apply IsConnected.isPreconnected
    apply Convex.isConnected
    exact convex_closedBall 0 1