working

Nevanlinna/meromorphicOn.lean

@@ -93,11 +93,14 @@ theorem MeromorphicOn.open_of_order_neq_top
       · exact h₃t'
 
 
-theorem MeromorphicOn.order_ne_top
+
+theorem MeromorphicOn.clopen_of_order_eq_top
   {f : ℂ → ℂ}
   {U : Set ℂ}
-  (h₁U : IsConnected U)
   (h₁f : MeromorphicOn f U) :
-  (∃ z₀ : U, (h₁f z₀.1 z₀.2).order = ⊤) ↔ (∀ z : U, (h₁f z.1 z.2).order = ⊤) := by
+  IsClopen { u : U | (h₁f u.1 u.2).order = ⊤ } := by
 
-  sorry
+  constructor
+  · rw [← isOpen_compl_iff]
+    exact open_of_order_neq_top h₁f
+  · exact open_of_order_eq_top h₁f

Nevanlinna/stronglyMeromorphicOn_eliminate.lean

@@ -2,6 +2,7 @@ import Mathlib.Analysis.Analytic.Meromorphic
 import Nevanlinna.analyticAt
 import Nevanlinna.divisor
 import Nevanlinna.meromorphicAt
+import Nevanlinna.meromorphicOn
 import Nevanlinna.meromorphicOn_divisor
 import Nevanlinna.stronglyMeromorphicOn
 import Nevanlinna.mathlibAddOn
@@ -290,3 +291,50 @@ theorem MeromorphicOn.decompose₂
           rw [this]
           --
           simpa
+
+
+theorem MeromorphicOn.decompose₃
+  {f : ℂ → ℂ}
+  {U : Set ℂ}
+  (h₁U : IsCompact U)
+  (h₂U : IsConnected U)
+  (h₁f : StronglyMeromorphicOn f U)
+  (h₂f : ∃ u : U, f u ≠ 0) :
+  ∃ g : ℂ → ℂ, (MeromorphicOn g U)
+    ∧ (AnalyticOn ℂ g U)
+    ∧ (∀ u : U, g u ≠ 0)
+    ∧ (f = g * ∏ᶠ u : U, fun z ↦ (z - u.1) ^ (h₁f.meromorphicOn.divisor u.1)) := by
+
+  have h₃f : ∀ u : U, (h₁f u u.2).meromorphicAt.order ≠ ⊤ := by
+    let A := h₁f.meromorphicOn.clopen_of_order_eq_top
+    have : PreconnectedSpace U := by
+      apply isPreconnected_iff_preconnectedSpace.mp
+      exact IsConnected.isPreconnected h₂U
+    rw [isClopen_iff] at A
+    rcases A with h|h
+    · intro u
+      have : u ∉ (∅ : Set U) := by exact fun a => a
+      rw [← h] at this
+      simp at this
+      tauto
+    · obtain ⟨u, hu⟩ := h₂f
+      let A := (h₁f u u.2).order_eq_zero_iff.2 hu
+      have : u ∈ (Set.univ : Set U) := by trivial
+      rw [← h] at this
+      simp at this
+      rw [A] at this
+      tauto
+
+  have h₄f : Finite (Function.support h₁f.meromorphicOn.divisor) := by
+    exact h₁f.meromorphicOn.divisor.finiteSupport h₁U
+
+  let P' : Set U := Subtype.val ⁻¹' Function.support h₁f.meromorphicOn.divisor
+  have : Finite P' := by
+    unfold P'
+    refine Finite.of_injective ?f ?H
+    simp
+    apply Finite.of_injective
+    sorry
+
+
+  sorry