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Nevanlinna/analyticAt.lean

@@ -50,3 +50,27 @@ theorem AnalyticAt.order_mul
           · constructor
             · exact IsOpen.inter h₂t₁ h₂t₂
             · exact Set.mem_inter h₃t₁ h₃t₂
+
+
+theorem AnalyticAt.order_eq_zero_iff
+  {f : ℂ → ℂ}
+  {z₀ : ℂ}
+  (hf : AnalyticAt ℂ f z₀) :
+  hf.order = 0 ↔ f z₀ ≠ 0 := by
+
+  have : (0 : ENat) = (0 : Nat) := by rfl
+  rw [this, AnalyticAt.order_eq_nat_iff hf 0]
+
+  constructor
+  · intro hz
+    obtain ⟨g, _, h₂g, h₃g⟩ := hz
+    simp at h₃g
+    rw [Filter.Eventually.self_of_nhds h₃g]
+    tauto
+  · intro hz
+    use f
+    constructor
+    · exact hf
+    · constructor
+      · exact hz
+      · simp

Nevanlinna/analyticOn_zeroSet.lean

@@ -97,6 +97,26 @@ theorem AnalyticOn.order_eq_nat_iff
   exact ⟨h₁g z₀ z₀.2, ⟨h₂g, Filter.Eventually.of_forall h₃g⟩⟩
 
 
+theorem AnalyticOn.support_of_order₁
+  {f : ℂ → ℂ}
+  {U : Set ℂ}
+  (hf : AnalyticOn ℂ f U) :
+  Function.support hf.order = U.restrict f⁻¹' {0} := by
+  ext u
+  simp [AnalyticOn.order]
+  rw [not_iff_comm, (hf u u.2).order_eq_zero_iff]
+
+
+theorem AnalyticOn.support_of_order₂
+  {f : ℂ → ℂ}
+  {U : Set ℂ}
+  (h₁U : IsPreconnected U)
+  (h₁f : AnalyticOn ℂ f U)
+  (h₂f : ∃ u ∈ U, f u ≠ 0) :
+  Function.support (ENat.toNat ∘ h₁f.order) = U.restrict f⁻¹' {0} := by
+  sorry
+
+
 theorem AnalyticOn.eliminateZeros
   {f : ℂ → ℂ}
   {U : Set ℂ}
@@ -358,16 +378,11 @@ theorem AnalyticOnCompact.eliminateZeros₁
 
   let A := (finiteZeros h₁U h₂U h₁f h₂f).toFinset
 
-  let n : ℂ → ℕ := by
+  let n : U → ℕ := fun z ↦ (h₁f z z.2).order.toNat
-    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]
+    dsimp [n, AnalyticOn.order]
-    simp
     rw [eq_comm]
     apply XX h₁U
     exact h₂f
@@ -375,14 +390,9 @@ theorem AnalyticOnCompact.eliminateZeros₁
   obtain ⟨g, h₁g, h₂g, h₃g⟩ := AnalyticOn.eliminateZeros (A := A) h₁f n hn
   use g
 
-  have inter : ∀ (z : ℂ), f z = (∏ a ∈ A, (z - ↑a) ^ (h₁f.order a).toNat) • g z := by
+  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
@@ -400,5 +410,15 @@ theorem AnalyticOnCompact.eliminateZeros₁
           tauto
         rw [inter z] at this
         exact right_ne_zero_of_smul this
-    ·
+    · intro z
-      exact inter
+      let φ : U → ℂ := fun a ↦ (z - ↑a) ^ (h₁f.order a).toNat
+      have hφ : Function.mulSupport φ ⊆ A := by
+        intro x hx
+        simp [φ] at hx
+        have : (h₁f.order x).toNat ≠ 0 := by
+          sorry
+
+        sorry
+      rw [finprod_eq_prod_of_mulSupport_subset φ hφ]
+      rw [inter z]
+      rfl