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Nevanlinna/meromorphicOn.lean

@@ -54,6 +54,7 @@ theorem MeromorphicOn.open_of_order_eq_top
     · exact isOpen_induced h₂t'
     · exact h₃t'
 
+
 theorem MeromorphicOn.open_of_order_neq_top
   {f : ℂ → ℂ}
   {U : Set ℂ}
@@ -99,15 +100,4 @@ theorem MeromorphicOn.order_ne_top
   (h₁f : MeromorphicOn f U) :
   (∃ z₀ : U, (h₁f z₀.1 z₀.2).order = ⊤) ↔ (∀ z : U, (h₁f z.1 z.2).order = ⊤) := by
 
-  constructor
-  · intro h
-    obtain ⟨h₁z₀, h₂z₀⟩ := h
-    intro hz
-
-
-    sorry
-
-  · intro h
-    obtain ⟨w, hw⟩ := h₁U.nonempty
-    use ⟨w, hw⟩
-    exact h ⟨w, hw⟩
+  sorry

Nevanlinna/stronglyMeromorphicOn_eliminate.lean

@@ -11,6 +11,53 @@ open scoped Interval Topology
 open Real Filter MeasureTheory intervalIntegral
 
 
+theorem MeromorphicOn.decompose₁
+  {f : ℂ → ℂ}
+  {U : Set ℂ}
+  {z₀ : ℂ}
+  (hz₀ : z₀ ∈ U)
+  (h₁f : MeromorphicOn f U)
+  (h₂f : StronglyMeromorphicAt f z₀) :
+  ∃ g : ℂ → ℂ, (MeromorphicOn g U)
+    ∧ (AnalyticAt ℂ g z₀)
+    ∧ (g z₀ ≠ 0)
+    ∧ (f = g * fun z ↦ (z - z₀) ^ (h₁f.divisor z₀)) := by
+
+  let h₁ := fun z ↦ (z - z₀) ^ (-h₁f.divisor z₀)
+  have h₁h₁ : MeromorphicOn h₁ U := by
+    apply MeromorphicOn.zpow
+    apply AnalyticOnNhd.meromorphicOn
+    apply AnalyticOnNhd.sub
+    exact analyticOnNhd_id
+    exact analyticOnNhd_const
+  have h₂h₁ : (h₁h₁ z₀ hz₀).order = -h₁f.divisor z₀ := by
+
+    sorry
+
+  let g₁ := f * h₁
+  have h₁g₁ : MeromorphicOn g₁ U := by
+    apply h₁f.mul h₁h₁
+  have h₂g₁ : (h₁g₁ z₀ hz₀).order = 0 := by
+    let A := (h₁g₁ z₀ hz₀).order_mul (h₁h₁ z₀ hz₀)
+
+    sorry
+  let g := (h₁g₁ z₀ hz₀).makeStronglyMeromorphicAt
+  have h₁g : MeromorphicOn g U := by
+    sorry
+  have h₂g : StronglyMeromorphicAt g z₀ := by
+    sorry
+  use g
+  constructor
+  · exact h₁g
+  · constructor
+    · apply h₂g.analytic
+
+      sorry
+    · constructor
+      · sorry
+      · sorry
+
+
 
 theorem MeromorphicOn.decompose
   {f : ℂ → ℂ}