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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

@@ -97,12 +97,32 @@ 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}
   {A : Finset U}
   (hf : AnalyticOn ℂ f U)
-  (n : ℂ → ℕ) :
+  (n : U → ℕ) :
   (∀ 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)
@@ -122,7 +142,7 @@ theorem AnalyticOn.eliminateZeros
 
     rw [← hBinsert b₀ (Finset.mem_insert_self b₀ B)]
 
-    let φ := fun z ↦ (∏ a ∈ B, (z - a.1) ^ n a.1)
+    let φ := fun z ↦ (∏ a ∈ B, (z - a.1) ^ n a)
 
     have : f = fun z ↦ φ z * g₀ z := by
       funext z
@@ -307,24 +327,20 @@ theorem AnalyticOnCompact.eliminateZeros
   (h₂U : IsCompact U)
   (h₁f : AnalyticOn ℂ f U)
   (h₂f : ∃ u ∈ U, f u ≠ 0) :
-  ∃ (g : ℂ → ℂ) (A : Finset U), AnalyticOn ℂ g U ∧ (∀ z ∈ U, g z ≠ 0) ∧ ∀ z, f z = (∏ a ∈ A, (z - a) ^ (h₁f a a.2).order.toNat) • g z := by
+  ∃ (g : ℂ → ℂ) (A : Finset U), AnalyticOn ℂ g U ∧ (∀ z ∈ U, g z ≠ 0) ∧ ∀ z, f z = (∏ a ∈ A, (z - a) ^ (h₁f.order a).toNat) • g z := by
 
   let A := (finiteZeros h₁U h₂U h₁f h₂f).toFinset
 
-  let n : ℂ → ℕ := by
-    intro z
-    by_cases hz : z ∈ U
-    · exact (h₁f z hz).order.toNat
-    · exact 0
+  let n : U → ℕ := fun z ↦ (h₁f z z.2).order.toNat
 
   have hn : ∀ a ∈ A, (h₁f a a.2).order = n a := by
     intro a _
-    dsimp [n]
-    simp
+    dsimp [n, AnalyticOn.order]
     rw [eq_comm]
     apply XX h₁U
     exact h₂f
 
+
   obtain ⟨g, h₁g, h₂g, h₃g⟩ := AnalyticOn.eliminateZeros (A := A) h₁f n hn
   use g
   use A
@@ -332,11 +348,6 @@ theorem AnalyticOnCompact.eliminateZeros
   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
   · exact h₁g
@@ -354,3 +365,60 @@ theorem AnalyticOnCompact.eliminateZeros
         rw [inter z] at this
         exact right_ne_zero_of_smul this
     · exact inter
+
+
+theorem AnalyticOnCompact.eliminateZeros₁
+  {f : ℂ → ℂ}
+  {U : Set ℂ}
+  (h₁U : IsPreconnected U)
+  (h₂U : IsCompact U)
+  (h₁f : AnalyticOn ℂ f U)
+  (h₂f : ∃ u ∈ U, f u ≠ 0) :
+  ∃ (g : ℂ → ℂ), AnalyticOn ℂ g U ∧ (∀ z ∈ U, g z ≠ 0) ∧ ∀ z, f z = (∏ᶠ a : U, (z - a) ^ (h₁f.order a).toNat) • g z := by
+
+  let A := (finiteZeros h₁U h₂U h₁f h₂f).toFinset
+
+  let n : U → ℕ := fun z ↦ (h₁f z z.2).order.toNat
+
+  have hn : ∀ a ∈ A, (h₁f a a.2).order = n a := by
+    intro a _
+    dsimp [n, AnalyticOn.order]
+    rw [eq_comm]
+    apply XX h₁U
+    exact h₂f
+
+  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 (↑a) a.property).order.toNat) • g z := by
+    intro z
+    rw [h₃g z]
+
+
+  constructor
+  · exact h₁g
+  · constructor
+    · intro z h₁z
+      by_cases h₂z : ⟨z, h₁z⟩  ∈ ↑A.toSet
+      · exact h₂g ⟨z, h₁z⟩ h₂z
+      · have : f z ≠ 0 := by
+          by_contra C
+          have : ⟨z, h₁z⟩ ∈ ↑A.toSet := by
+            dsimp [A]
+            simp
+            exact C
+          tauto
+        rw [inter z] at this
+        exact right_ne_zero_of_smul this
+    · intro z
+      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