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Nevanlinna/analyticAt.lean

@@ -118,10 +118,6 @@ theorem AnalyticAt.supp_order_toNat
   intro h₁f
   simp [hf.order_eq_zero_iff.2 h₁f]
 
-/-
-theorem ContinuousLinearEquiv.analyticAt
-  (ℓ : ℂ ≃L[ℂ] ℂ) (z₀ : ℂ) : AnalyticAt ℂ (⇑ℓ) z₀ := ℓ.toContinuousLinearMap.analyticAt z₀
--/
 
 theorem eventually_nhds_comp_composition
   {f₁ f₂ ℓ : ℂ → ℂ}
@@ -260,3 +256,13 @@ theorem AnalyticAt.localIdentity
       let _ := Filter.Eventually.exists A
       tauto
   exact Filter.eventuallyEq_iff_sub.mpr this
+
+
+theorem AnalyticAt.mul₁
+  {f g : ℂ → ℂ}
+  {z : ℂ}
+  (hf : AnalyticAt ℂ f z)
+  (hg : AnalyticAt ℂ g z) :
+  AnalyticAt ℂ (f * g) z := by
+  rw [(by rfl : f * g = (fun x ↦ f x * g x))]
+  exact mul hf hg

Nevanlinna/meromorphicOn_decompose.lean

@@ -20,7 +20,7 @@ theorem MeromorphicOn.decompose
   (h₂f : ∃ z₀ ∈ U, f z₀ ≠ 0) :
   ∃ g : ℂ → ℂ, (AnalyticOnNhd ℂ g U)
     ∧ (∀ z ∈ U, g z ≠ 0)
-    ∧ (Set.EqOn h₁f.makeStronglyMeromorphicOn (fun z ↦ ∏ᶠ p, (z - p) ^ (h₁f.divisor p) * g z ) U) := by
+    ∧ (Set.EqOn h₁f.makeStronglyMeromorphicOn ((∏ᶠ p, fun z ↦ (z - p) ^ (h₁f.divisor p)) * g) U) := by
 
   let g₁ : ℂ → ℂ := f * (fun z ↦ ∏ᶠ p, (z - p) ^ (h₁f.divisor p))
   have h₁g₁ : MeromorphicOn g₁ U := by
@@ -65,5 +65,18 @@ theorem MeromorphicOn.decompose
         sorry
       apply Filter.EventuallyEq.eq_of_nhds
       apply StronglyMeromorphicAt.localIdentity
+      · exact StronglyMeromorphicOn_of_makeStronglyMeromorphic h₁f z hz
+      · right
+        use h₁f.divisor z
+        use (∏ᶠ p ≠ z, (fun x ↦ (x - p) ^ h₁f.divisor p)) * g
+        constructor
+        · apply AnalyticAt.mul₁
+          ·
+            sorry
+          · apply (h₃g z hz).analytic
+            rw [h₂g ⟨z, hz⟩]
+        · constructor
+          · sorry
+          · sorry
 
       sorry

Nevanlinna/stronglyMeromorphicOn.lean

@@ -81,9 +81,9 @@ theorem makeStronglyMeromorphicOn_changeDiscrete
 
 theorem StronglyMeromorphicOn_of_makeStronglyMeromorphic
   {f : ℂ → ℂ}
-  {z₀ : ℂ}
   (hf : MeromorphicOn f U) :
   StronglyMeromorphicOn hf.makeStronglyMeromorphicOn U := by
+
   sorry