Update analyticOn_zeroSet.lean

Nevanlinna/analyticOn_zeroSet.lean

@@ -1,3 +1,4 @@
+import Mathlib.Analysis.Analytic.Linear
 import Init.Classical
 import Mathlib.Analysis.Analytic.Meromorphic
 import Mathlib.Topology.ContinuousOn
@@ -146,7 +147,7 @@ theorem AnalyticOn.order_of_mul
             · exact Set.mem_inter h₃t₁ h₃t₂
 
 
-theorem AnalyticOn.order_eq_nat_iff'
+theorem AnalyticOn.eliminateZeros
   {f : ℂ → ℂ}
   {U : Set ℂ}
   {A : Finset U}
@@ -169,9 +170,45 @@ theorem AnalyticOn.order_eq_nat_iff'
 
   have : (h₁g₀ b₀ b₀.2).order = n b₀ := by
 
-    let A := hBinsert b₀ (Finset.mem_insert_self b₀ B)
-    exact A
-    sorry
+    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 hb
+      apply AnalyticAt.pow
+      apply AnalyticAt.sub
+      apply analyticAt_id
+
+    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
+          have AA : b₀.1 - a ≠ 0 := by
+            sorry
+          simp [AA]
+
+        · simp
+
+    rw [AnalyticOn.order_of_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