Update, working, …

Nevanlinna/stronglyMeromorphic.lean

@@ -81,41 +81,49 @@ theorem AnalyticAt.stronglyMeromorphicAt
     tauto
 
 
-theorem MeromorphicAt.order_neq_top_iff
-  {f : ℂ → ℂ}
-  {z₀ : ℂ}
-  (hf : MeromorphicAt f z₀) :
-  hf.order ≠ ⊤ ↔ ∃ (g : ℂ → ℂ), AnalyticAt ℂ g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ (z : ℂ) in nhdsWithin z₀ {z₀}ᶜ, f z = (z - z₀) ^ (hf.order.untop' 0) • g z := by
-
-  rw [← hf.order_eq_int_iff (hf.order.untop' 0)]
-  constructor
-  · intro h₁f
-    apply?
-    exact Eq.symm (ENat.coe_toNat h₁f)
-  · intro h₁f
-    exact ENat.coe_toNat_eq_self.mp (id (Eq.symm h₁f))
-
-
 /- Make strongly MeromorphicAt -/
 noncomputable def MeromorphicAt.makeStronglyMeromorphicAt
   {f : ℂ → ℂ}
   {z₀ : ℂ}
   (hf : MeromorphicAt f z₀) :
   ℂ → ℂ := by
-
-  by_cases h₁f : hf.order = ⊤
-  · exact 0
-  ·
-
   intro z
   by_cases z = z₀
-  · by_cases h₂f : hf.order = 0
-    · have : (0 : WithTop ℤ) = (0 : ℤ) := rfl
-      rw [this, hf.order_eq_int_iff] at h₂f
-      exact (Classical.choose h₂f) z₀
+  · by_cases h₁f : hf.order = (0 : ℤ)
+    · rw [hf.order_eq_int_iff] at h₁f
+      exact (Classical.choose h₁f) z₀
     · exact 0
   · exact f z
 
+lemma m₁
+  {f : ℂ → ℂ}
+  {z₀ : ℂ}
+  (hf : MeromorphicAt f z₀) :
+  ∀ z ≠ z₀, f z = hf.makeStronglyMeromorphicAt z := by
+  intro z hz
+  unfold MeromorphicAt.makeStronglyMeromorphicAt
+  simp [hz]
+
+lemma m₂
+  {f : ℂ → ℂ}
+  {z₀ : ℂ}
+  (hf : MeromorphicAt f z₀) :
+  ∀ᶠ (z : ℂ) in nhdsWithin z₀ {z₀}ᶜ, f z = hf.makeStronglyMeromorphicAt z := by
+  apply eventually_nhdsWithin_of_forall
+  exact fun x a => m₁ hf x a
+
+
+open Topology
+
+lemma Mnhds
+  {f g : ℂ → ℂ}
+  {z₀ : ℂ}
+  (h₁ : ∀ᶠ (z : ℂ) in nhdsWithin z₀ {z₀}ᶜ, f z = g z)
+  (h₂ : f z₀ = g z₀) :
+  ∀ᶠ (z : ℂ) in nhds z₀, f z = g z := by
+  rw [eventually_nhds_iff]
+  rw [eventually_nhdsWithin_iff] at h₁
+  sorry
 
 theorem StronglyMeromorphicAt_of_makeStronglyMeromorphic
   {f : ℂ → ℂ}
@@ -123,6 +131,26 @@ theorem StronglyMeromorphicAt_of_makeStronglyMeromorphic
   (hf : MeromorphicAt f z₀) :
   StronglyMeromorphicAt hf.makeStronglyMeromorphicAt z₀ := by
 
+  by_cases h₂f : hf.order = ⊤
+  · rw [MeromorphicAt.order_eq_top_iff] at h₂f
+    let Z : ℂ → ℂ := fun z ↦ 0
+    have : hf.makeStronglyMeromorphicAt =ᶠ[𝓝 z₀] 0 := by
+      --unfold Filter.EventuallyEq
+      apply Mnhds
+      · apply eventually_nhdsWithin_of_forall
+        intro x hx
+
+        sorry
+
+
+      sorry
+
+
+
+    apply AnalyticAt.stronglyMeromorphicAt
+    rw [analyticAt_congr this]
+    apply analyticAt_const
+
 
 
   by_cases h₂f : hf.order = 0