diff --git a/Nevanlinna/analyticOn_zeroSet2.lean b/Nevanlinna/analyticOn_zeroSet2.lean
new file mode 100644
index 0000000..94c65bf
--- /dev/null
+++ b/Nevanlinna/analyticOn_zeroSet2.lean
@@ -0,0 +1,449 @@
+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
diff --git a/Nevanlinna/laplace.lean b/Nevanlinna/laplace.lean
index 35c1128..d85b4d0 100644
--- a/Nevanlinna/laplace.lean
+++ b/Nevanlinna/laplace.lean
@@ -85,11 +85,11 @@ theorem laplace_add_ContDiffOn
   have t₁ : DifferentiableAt ℝ (partialDeriv ℝ 1 f₁) x := by
     let B₀ := (h₁ x hx).contDiffAt (IsOpen.mem_nhds hs hx)
     let A₀ := partialDeriv_contDiffAt ℝ B₀ 1
-    exact A₀.differentiableAt (Submonoid.oneLE.proof_2 ℕ∞)
+    apply A₀.differentiableAt (Preorder.le_refl 1)
   have t₂ : DifferentiableAt ℝ (partialDeriv ℝ 1 f₂) x := by
     let B₀ := (h₂ x hx).contDiffAt (IsOpen.mem_nhds hs hx)
     let A₀ := partialDeriv_contDiffAt ℝ B₀ 1
-    exact A₀.differentiableAt (Submonoid.oneLE.proof_2 ℕ∞)
+    exact A₀.differentiableAt (Preorder.le_refl 1)
   rw [partialDeriv_add₂_differentiableAt ℝ t₁ t₂]
 
   have : partialDeriv ℝ Complex.I (f₁ + f₂) =ᶠ[nhds x] (partialDeriv ℝ Complex.I f₁) + (partialDeriv ℝ Complex.I f₂) := by
@@ -105,11 +105,11 @@ theorem laplace_add_ContDiffOn
   have t₃ : DifferentiableAt ℝ (partialDeriv ℝ Complex.I f₁) x := by
     let B₀ := (h₁ x hx).contDiffAt (IsOpen.mem_nhds hs hx)
     let A₀ := partialDeriv_contDiffAt ℝ B₀ Complex.I
-    exact A₀.differentiableAt (Submonoid.oneLE.proof_2 ℕ∞)
+    exact A₀.differentiableAt (Preorder.le_refl 1)
   have t₄ : DifferentiableAt ℝ (partialDeriv ℝ Complex.I f₂) x := by
     let B₀ := (h₂ x hx).contDiffAt (IsOpen.mem_nhds hs hx)
     let A₀ := partialDeriv_contDiffAt ℝ B₀ Complex.I
-    exact A₀.differentiableAt (Submonoid.oneLE.proof_2 ℕ∞)
+    exact A₀.differentiableAt (Preorder.le_refl 1)
   rw [partialDeriv_add₂_differentiableAt ℝ t₃ t₄]
 
   -- I am super confused at this point because the tactic 'ring' does not work.
diff --git a/Nevanlinna/periodic_integrability.lean b/Nevanlinna/periodic_integrability.lean
index bd382dd..d55165d 100644
--- a/Nevanlinna/periodic_integrability.lean
+++ b/Nevanlinna/periodic_integrability.lean
@@ -76,7 +76,7 @@ theorem periodic_integrability4
     obtain ⟨n₁, hn₁⟩ := exists_nat_ge ((t -min a₁ a₂) / T)
     use n₁
     rw [sub_le_iff_le_add]
-    rw [div_le_iff hT] at hn₁
+    rw [div_le_iff₀ hT] at hn₁
     rw [sub_le_iff_le_add] at hn₁
     rw [add_comm]
     exact hn₁
@@ -84,7 +84,7 @@ theorem periodic_integrability4
     obtain ⟨n₂, hn₂⟩ := exists_nat_ge ((max a₁ a₂ - t) / T)
     use n₂
     rw [← sub_le_iff_le_add]
-    rw [div_le_iff hT] at hn₂
+    rw [div_le_iff₀ hT] at hn₂
     linarith
 
   have : Set.uIcc a₁ a₂ ⊆ Set.uIcc (t - n₁ * T) (t + n₂ * T) := by
diff --git a/Nevanlinna/specialFunctions_Integral_log_sin.lean b/Nevanlinna/specialFunctions_Integral_log_sin.lean
index ca37f2f..e70488b 100644
--- a/Nevanlinna/specialFunctions_Integral_log_sin.lean
+++ b/Nevanlinna/specialFunctions_Integral_log_sin.lean
@@ -1,5 +1,6 @@
 import Mathlib.Analysis.SpecialFunctions.Integrals
 import Mathlib.Analysis.SpecialFunctions.Log.NegMulLog
+import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
 
 open scoped Interval Topology
 open Real Filter MeasureTheory intervalIntegral