Update analyticOn_zeroSet.lean

Nevanlinna/analyticOn_zeroSet.lean

@@ -101,9 +101,9 @@ theorem AnalyticOn.order_eq_nat_iff'
   {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 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)
+  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
@@ -115,3 +115,29 @@ theorem AnalyticOn.order_eq_nat_iff'
   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 sorry
+  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