From aa79fdb9ebc5898d766d3f6d1b9ec1e55ecc45bb Mon Sep 17 00:00:00 2001
From: Stefan Kebekus <kebekus@users.noreply.github.com>
Date: Tue, 10 Sep 2024 10:55:53 +0200
Subject: [PATCH] Update analyticOn_zeroSet.lean

---
 Nevanlinna/analyticOn_zeroSet.lean | 72 ++++++------------------------
 1 file changed, 13 insertions(+), 59 deletions(-)

diff --git a/Nevanlinna/analyticOn_zeroSet.lean b/Nevanlinna/analyticOn_zeroSet.lean
index 4039eed..95d38d8 100644
--- a/Nevanlinna/analyticOn_zeroSet.lean
+++ b/Nevanlinna/analyticOn_zeroSet.lean
@@ -1,21 +1,25 @@
 import Mathlib.Analysis.Analytic.Constructions
 import Mathlib.Analysis.Analytic.IsolatedZeros
 import Mathlib.Analysis.Complex.Basic
+import Nevanlinna.analyticAt
+
+
+noncomputable def AnalyticOn.order
+  {f : ℂ → ℂ} {U : Set ℂ} (hf : AnalyticOn ℂ f U) : U → ℕ∞ := fun u ↦ (hf u u.2).order
 
 
 theorem AnalyticOn.order_eq_nat_iff
   {f : ℂ → ℂ}
   {U : Set ℂ}
-  {z₀ : ℂ}
+  {z₀ : U}
   (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
+  hf.order z₀ = ↑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
+  obtain ⟨gloc, h₁gloc, h₂gloc, h₃gloc⟩ := (AnalyticAt.order_eq_nat_iff (hf z₀ z₀.2) n).1 hn
 
   -- Define a candidate function; this is (f z) / (z - z₀) ^ n with the
   -- removable singularity removed
@@ -44,7 +48,7 @@ theorem AnalyticOn.order_eq_nat_iff
   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
+    use {z₀.1}ᶜ
     constructor
     · intro y hy
       simp at hy
@@ -87,59 +91,10 @@ theorem AnalyticOn.order_eq_nat_iff
   -- direction ←
   intro h
   obtain ⟨g, h₁g, h₂g, h₃g⟩ := h
+  dsimp [AnalyticOn.order]
   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
+  exact ⟨h₁g z₀ z₀.2, ⟨h₂g, Filter.Eventually.of_forall h₃g⟩⟩
 
 
 theorem AnalyticOn.eliminateZeros
@@ -148,7 +103,7 @@ theorem AnalyticOn.eliminateZeros
   {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
+  (∀ a ∈ A, hf.order a = 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)
 
@@ -208,8 +163,7 @@ theorem AnalyticOn.eliminateZeros
     rw [h₂φ]
     simp
 
-
-  obtain ⟨g₁, h₁g₁, h₂g₁, h₃g₁⟩ := (AnalyticOn.order_eq_nat_iff h₁g₀ b₀.2 (n b₀)).1 this
+  obtain ⟨g₁, h₁g₁, h₂g₁, h₃g₁⟩ := (AnalyticOn.order_eq_nat_iff h₁g₀ (n b₀)).1 this
 
   use g₁
   constructor