Update holomorphic_zero.lean

Nevanlinna/holomorphic_zero.lean

@@ -1,3 +1,5 @@
+import Init.Classical
+import Mathlib.Analysis.Analytic.Meromorphic
 import Mathlib.Topology.ContinuousOn
 import Mathlib.Analysis.Analytic.IsolatedZeros
 import Nevanlinna.holomorphic
@@ -247,7 +249,7 @@ theorem zeroDivisor_finiteOnCompact
   {U : Set ℂ}
   (hU : IsPreconnected U)
   (h₁f : AnalyticOn ℂ f U)
-  (h₂f : ∃ z ∈ U, f z ≠ 0)
+  (h₂f : ∃ z ∈ U, f z ≠ 0) -- not needed!
   (h₂U : IsCompact U) :
   Set.Finite (U ∩ Function.support (zeroDivisor f)) := by
 
@@ -268,26 +270,120 @@ theorem zeroDivisor_finiteOnCompact
   exact Set.inter_subset_right
 
 
+theorem AnalyticOn.order_eq_nat_iff
+  {f : ℂ → ℂ}
+  {U : Set ℂ}
+  {z₀ : ℂ}
+  (hf : AnalyticOn ℂ f U)
+  (hz₀ : z₀ ∈ U)
+  (n : ℕ) :
+  (hf z₀ hz₀).order = ↑n → ∃ (g : ℂ → ℂ), AnalyticOn ℂ g U ∧ g z₀ ≠ 0 ∧ ∀ z ∈ U, f z = (z - z₀) ^ n • g z := by
+
+  intro hn
+
+  obtain ⟨gloc, h₁gloc, h₂gloc, h₃gloc⟩ := (AnalyticAt.order_eq_nat_iff (hf z₀ hz₀) n).1 hn
+
+  let g : ℂ → ℂ := fun z ↦ if hz : z = z₀ then gloc z₀ else (f z) / (z - z₀) ^ n
+
+  have t₀ : ∀ᶠ (z : ℂ) in nhds z₀, g z = gloc z := by
+    rw [eventually_nhds_iff]
+    rw [eventually_nhds_iff] at h₃gloc
+    obtain ⟨t, h₁t, h₂t, h₃t⟩ := h₃gloc
+    use t
+    constructor
+    · intro y h₁y
+      by_cases h₂y : y = z₀
+      · dsimp [g]; simp [h₂y]
+      · dsimp [g]; simp [h₂y]
+        rw [div_eq_iff_mul_eq, eq_comm, mul_comm]
+        exact h₁t y h₁y
+        norm_num
+        rw [sub_eq_zero]
+        tauto
+    · constructor
+      · assumption
+      · assumption
+
+  have t₁ {z₁ : ℂ} : z₁ ≠ z₀ → ∀ᶠ (z : ℂ) in nhds z₁, g z = f z / (z - z₀) ^ n := by
+    intro hz₁
+    rw [eventually_nhds_iff]
+    use {z₀}ᶜ
+    constructor
+    · intro y hy
+      simp at hy
+      dsimp [g]
+      simp [hy]
+    · constructor
+      · exact isOpen_compl_singleton
+      · tauto
+
+  use g
+  constructor
+  · -- AnalyticOn ℂ g U
+    intro z h₁z
+    by_cases h₂z : z = z₀
+    · rw [h₂z]
+      apply AnalyticAt.congr h₁gloc
+      exact Filter.EventuallyEq.symm t₀
+    · simp_rw [eq_comm] at t₁
+      apply AnalyticAt.congr _ (t₁ h₂z)
+      apply AnalyticAt.div
+      exact hf z h₁z
+      apply AnalyticAt.pow
+      apply AnalyticAt.sub
+      apply analyticAt_id
+      exact analyticAt_const
+      simp
+      rw [sub_eq_zero]
+      tauto
+
+
 theorem eliminatingZeros
   {f : ℂ → ℂ}
-  {z₀ : ℂ}
+  {U : Set ℂ}
-  {R : ℝ}
+  (h₁U : IsPreconnected U)
-  (h₁f : ∀ z ∈ Metric.closedBall z₀ R, HolomorphicAt f z)
+  (h₂U : IsCompact U)
-  (h₂f : ∃ z ∈ Metric.ball z₀ R, f z ≠ 0) :
+  (h₁f : AnalyticOn ℂ f U)
-  ∃ F : ℂ → ℂ, ∀ z ∈ Metric.ball z₀ R, (HolomorphicAt F z) ∧ (f z = (F z) * ∏ᶠ a ∈ Metric.ball z₀ R, (z - a) ^ (zeroDivisor f a) ) := by
+  (h₂f : ∃ z ∈ U, f z ≠ 0) :
+  ∃ F : ℂ → ℂ, (AnalyticOn ℂ F U) ∧ (f = F * ∏ᶠ a ∈ (U ∩ (zeroDivisor f).support), fun z ↦ (z - a) ^ (zeroDivisor f a)) := by
+
+  have hs := zeroDivisor_finiteOnCompact h₁U h₁f h₂f h₂U
+  rw [finprod_mem_eq_finite_toFinset_prod _ hs]
+
+  let A := hs.card_eq
+
+
 
   let F : ℂ → ℂ := by
     intro z
-    if hz : z ∈ (Metric.closedBall z₀ R) ∩ Function.support (zeroDivisor f) then
+    if hz : z ∈ U ∩ (zeroDivisor f).support then
-      exact 0
+      exact (Classical.choose (zeroDivisor_support_iff.1 hz.2).2.2) z
     else
-      exact f z * (∏ᶠ a ∈ Metric.ball z₀ R, (z - a) ^ (zeroDivisor f a))⁻¹
+      exact (f z) / ∏ i ∈ hs.toFinset, (z - i) ^ zeroDivisor f i
 
   use F
-  intro z hz
+  constructor
-  by_cases h₂z : z ∈ (Metric.closedBall z₀ R) ∩ Function.support (zeroDivisor f)
+  · intro z h₁z
-  · -- Positive case
+    by_cases h₂z : z ∈ (zeroDivisor f).support
-    sorry
+    · -- case: z ∈ Function.support (zeroDivisor f)
+      have : z ∈ U ∩ (zeroDivisor f).support := by exact Set.mem_inter h₁z h₂z
 
-  · -- Negative case
+
-    sorry
+      sorry
+  · sorry
+
+  have : MeromorphicOn F U := by
+    apply MeromorphicOn.div
+    exact AnalyticOn.meromorphicOn h₁f
+    apply AnalyticOn.meromorphicOn
+    apply Finset.analyticOn_prod
+    intro n hn
+    apply AnalyticOn.pow
+    apply AnalyticOn.sub
+    exact analyticOn_id ℂ
+    exact analyticOn_const
+
+  --use F
+
+
+  sorry