From 9410087ddf1b35fb5a77c363e41551fee9de1afc Mon Sep 17 00:00:00 2001
From: Stefan Kebekus <kebekus@users.noreply.github.com>
Date: Fri, 16 Aug 2024 10:43:57 +0200
Subject: [PATCH] Update holomorphic_zero.lean

---
 Nevanlinna/holomorphic_zero.lean | 75 ++++++++++++++++++++++++++++++++
 1 file changed, 75 insertions(+)

diff --git a/Nevanlinna/holomorphic_zero.lean b/Nevanlinna/holomorphic_zero.lean
index 282eed3..8362bf5 100644
--- a/Nevanlinna/holomorphic_zero.lean
+++ b/Nevanlinna/holomorphic_zero.lean
@@ -12,9 +12,84 @@ noncomputable def zeroDivisor
     exact 0
 
 
+theorem analyticAtZeroDivisorSupport
+  {f : ℂ → ℂ}
+  {z : ℂ}
+  (h : z ∈ Function.support (zeroDivisor f)) :
+  AnalyticAt ℂ f z := by
+  by_contra h₁f
+  simp at h
+  dsimp [zeroDivisor] at h
+  simp [h₁f] at h
+
+
+lemma toNatEqSelf_iff {n : ℕ∞} : n.toNat = n ↔ ∃ m : ℕ, m = n := by
+  constructor
+  · intro H₁
+    rw [← ENat.some_eq_coe, ← WithTop.ne_top_iff_exists]
+    by_contra H₂
+    rw [H₂] at H₁
+    simp at H₁
+  · intro H
+    obtain ⟨m, hm⟩ := H
+    rw [← hm]
+    simp
+
+
 theorem discreteZeros
   {f : ℂ → ℂ} :
   DiscreteTopology (Function.support (zeroDivisor f)) := by
+  apply singletons_open_iff_discrete.mp
+  intro z
+
+  let A := analyticAtZeroDivisorSupport z.2
+  let c : WithTop ℕ := A.order
+  let B := AnalyticAt.order_eq_nat_iff A
+  let n := zeroDivisor f z.1
+
+
+
+  have : ∃ a : ℕ, a = A.order := by
+    rw [← ENat.some_eq_coe]
+    rw [← WithTop.ne_top_iff_exists]
+    by_contra H
+    rw [AnalyticAt.order_eq_top_iff] at H
+
+
+
+    dsimp [n, zeroDivisor]
+    simp [A]
+
+
+
+
+
+    sorry
+  let C := (B n).1 this
+
+  apply Metric.isOpen_singleton_iff.mpr
+
+
+
+  /-
+
+Try this: refine Metric.isOpen_singleton_iff.mpr ?_
+Remaining subgoals:
+⊢ ∃ ε > 0, ∀ (y : ↑(Function.support (zeroDivisor f))), dist y z < ε → y = z
+Suggestions
+Try this: refine isClosed_compl_iff.mp ?_
+Remaining subgoals:
+⊢ IsClosed {z}ᶜ
+Suggestions
+Try this: refine disjoint_frontier_iff_isOpen.mp ?_
+Remaining subgoals:
+⊢ Disjoint (frontier {z}) {z}
+Suggestions
+Try this: refine isOpen_iff_forall_mem_open.mpr ?_
+Remaining subgoals:
+⊢ ∀ x ∈ {z}, ∃ t ⊆ {z}, IsOpen t ∧ x ∈ t
+
+  -/
   sorry