Nevanlinna/analyticOn_zeroSet.lean

@@ -346,6 +346,7 @@ theorem finiteZeros
   rfl
 
 
+
 theorem AnalyticOnCompact.eliminateZeros
   {f : ℂ → ℂ}
   {U : Set ℂ}
@@ -353,60 +354,95 @@ 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), AnalyticOn ℂ g U ∧ (∀ a ∈ U, g a ≠ 0) ∧ ∀ z, f z = (∏ a ∈ A, (z - a) ^ (n a)) • g z := by
 
-  let ι : U → ℂ := Subtype.val
 
-  let A₁ := ι⁻¹' (U ∩ f⁻¹' {0})
+  let A := U ∩ f ⁻¹' {0}
 
-  have t₁ : (U ∩ f⁻¹' {0}).Finite := by
-    sorry
+  by sorry -- (finiteZeros h₁U h₂U h₁f h₂f).toFinset
+  let B := AnalyticOn.eliminateZeros h₁f
 
-  have : A₁.Finite := by
-    apply Set.Finite.preimage
-    exact Set.injOn_subtype_val
-    exact t₁
-  let A := this.toFinset
 
-  let n : ℂ → ℕ := by
-    intro z
-    by_cases hz : z ∈ U
-    · exact (h₁f z hz).order.toNat
-    · exact 0
+  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)
 
-  have hn : ∀ a ∈ A, (h₁f a a.2).order = n a := by
-    sorry
+  -- case empty
+  simp
+  use f
+  simp
+  exact hf
 
-  obtain ⟨g, h₁g, h₂g, h₃g⟩ := AnalyticOn.eliminateZeros (A := A) h₁f n hn
-  use g
-  use A
+  -- case insert
+  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 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]
+  have : (h₁g₀ b₀ b₀.2).order = n b₀ := by
 
+    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 _
+      apply AnalyticAt.pow
+      apply AnalyticAt.sub
+      apply analyticAt_id ℂ
+      exact analyticAt_const
+
+    have h₂φ : h₁φ.order = (0 : ℕ) := by
+      rw [AnalyticAt.order_eq_nat_iff h₁φ 0]
+      use φ
       constructor
-  · exact h₁g
+      · assumption
       · constructor
-    · intro z h₁z
-      by_cases h₂z : ⟨z, h₁z⟩  ∈ ↑A.toSet
-      · exact h₂g ⟨z, h₁z⟩ h₂z
-      · have : f z ≠ 0 := by
+        · dsimp [φ]
+          push_neg
+          rw [Finset.prod_ne_zero_iff]
+          intro a ha
+          simp
+          have : ¬ (b₀.1 - a.1 = 0) := by
             by_contra C
-          have : ⟨z, h₁z⟩ ∈ ↑A₁ := by
-            dsimp [A₁, ι]
-            simp
-            exact C
-          have : ⟨z, h₁z⟩ ∈ ↑A.toSet := by
-            dsimp [A]
-            simp
-            exact this
+            rw [sub_eq_zero] at C
+            rw [SetCoe.ext C] at hb
             tauto
-        rw [inter z] at this
-        exact right_ne_zero_of_smul this
-    · exact inter
+          tauto
+        · simp
+
+    rw [AnalyticAt.order_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
+
+  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