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Nevanlinna/meromorphicOn_divisor.lean

@@ -74,3 +74,29 @@ noncomputable def MeromorphicOn.divisor
           simp
         · simp [h₃y]
       · tauto
+
+
+theorem MeromorphicOn.divisor_def₁
+  {f : ℂ → ℂ}
+  {U : Set ℂ}
+  {z : ℂ}
+  (hf : MeromorphicOn f U)
+  (hz : z ∈ U) :
+  hf.divisor z = ((hf z hz).order.untop' 0 : ℤ) := by
+  unfold MeromorphicOn.divisor
+  simp [hz]
+
+
+theorem MeromorphicOn.divisor_def₂
+  {f : ℂ → ℂ}
+  {U : Set ℂ}
+  {z : ℂ}
+  (hf : MeromorphicOn f U)
+  (hz : z ∈ U)
+  (h₂f : (hf z hz).order ≠ ⊤) :
+  hf.divisor z = (hf z hz).order.untop h₂f := by
+  unfold MeromorphicOn.divisor
+  simp [hz]
+  rw [WithTop.untop'_eq_iff]
+  left
+  exact Eq.symm (WithTop.coe_untop (hf z hz).order h₂f)

Nevanlinna/stronglyMeromorphicOn_eliminate.lean

@@ -10,6 +10,13 @@ import Nevanlinna.mathlibAddOn
 open scoped Interval Topology
 open Real Filter MeasureTheory intervalIntegral
 
+lemma untop_eq_untop'
+  {n : WithTop ℤ}
+  (hn : n ≠ ⊤) :
+  n.untop' 0 = n.untop hn := by
+  rw [WithTop.untop'_eq_iff]
+  simp
+
 
 theorem MeromorphicOn.decompose₁
   {f : ℂ → ℂ}
@@ -17,7 +24,8 @@ theorem MeromorphicOn.decompose₁
   {z₀ : ℂ}
   (hz₀ : z₀ ∈ U)
   (h₁f : MeromorphicOn f U)
-  (h₂f : StronglyMeromorphicAt f z₀) :
+  (h₂f : StronglyMeromorphicAt f z₀)
+  (h₃f : h₂f.meromorphicAt.order ≠ ⊤) :
   ∃ g : ℂ → ℂ, (MeromorphicOn g U)
     ∧ (AnalyticAt ℂ g z₀)
     ∧ (g z₀ ≠ 0)
@@ -46,20 +54,52 @@ theorem MeromorphicOn.decompose₁
   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)
+    rw [(h₁f z₀ hz₀).order_mul (h₁h₁ z₀ hz₀)]
+    rw [h₂h₁]
+    unfold n
+    rw [MeromorphicOn.divisor_def₂ h₁f hz₀ h₃f]
+    conv =>
+      left
+      left
+      rw [Eq.symm (WithTop.coe_untop (h₁f z₀ hz₀).order h₃f)]
+    have
+      (a b c : ℤ)
+      (h : a + b = c) :
+      (a : WithTop ℤ) + (b : WithTop ℤ) = (c : WithTop ℤ) := by
+      rw [← h]
+      simp
+    rw [this ((h₁f z₀ hz₀).order.untop h₃f) (-(h₁f z₀ hz₀).order.untop h₃f) 0]
+    simp
+    ring
 
-    sorry
   let g := (h₁g₁ z₀ hz₀).makeStronglyMeromorphicAt
-  have h₁g : MeromorphicOn g U := by
-    sorry
   have h₂g : StronglyMeromorphicAt g z₀ := by
-    sorry
+    exact StronglyMeromorphicAt_of_makeStronglyMeromorphic (h₁g₁ z₀ hz₀)
+  have h₁g : MeromorphicOn g U := by
+    intro z hz
+    by_cases h₁z : z = z₀
+    · rw [h₁z]
+      apply h₂g.meromorphicAt
+    · apply (h₁g₁ z hz).congr
+      rw [eventuallyEq_nhdsWithin_iff]
+      rw [eventually_nhds_iff]
+      use {z₀}ᶜ
+      constructor
+      · intro y h₁y h₂y
+        let A := m₁ (h₁g₁ z₀ hz₀) y h₁y
+        unfold g
+        rw [← A]
+      · constructor
+        · exact isOpen_compl_singleton
+        · exact h₁z
+
   use g
   constructor
   · exact h₁g
   · constructor
     · apply h₂g.analytic
 
+
       sorry
     · constructor
       · sorry