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Nevanlinna/stronglyMeromorphicAt.lean

@@ -289,7 +289,7 @@ theorem StronglyMeromorphicAt_of_makeStronglyMeromorphic
             rwa [WithTop.untop_eq_iff h₂f]
 
 
-theorem makeStronglyMeromorphic_id
+theorem StronglyMeromorphicAt.makeStronglyMeromorphic_id
   {f : ℂ → ℂ}
   {z₀ : ℂ}
   (hf : StronglyMeromorphicAt f z₀) :

Nevanlinna/stronglyMeromorphicOn.lean

@@ -74,7 +74,7 @@ theorem makeStronglyMeromorphicOn_changeDiscrete
     unfold MeromorphicOn.makeStronglyMeromorphicOn
     by_cases h₂v : v ∈ U
     · simp [h₂v]
-      rw [← makeStronglyMeromorphic_id]
+      rw [← StronglyMeromorphicAt.makeStronglyMeromorphic_id]
       exact AnalyticAt.stronglyMeromorphicAt (h₂V v hv)
     · simp [h₂v]
 

Nevanlinna/stronglyMeromorphicOn_eliminate.lean

@@ -92,90 +92,60 @@ theorem MeromorphicOn.decompose₁
       · constructor
         · exact isOpen_compl_singleton
         · exact h₁z
+  have h₃g : (h₁g z₀ hz₀).order = 0 := by
+    unfold g
+    let B := m₂ (h₁g₁ z₀ hz₀)
+    let A := (h₁g₁ z₀ hz₀).order_congr B
+    rw [← A]
+    rw [h₂g₁]
+  have h₄g : AnalyticAt ℂ g z₀ := by
+    apply h₂g.analytic
+    rw [h₃g]
 
   use g
   constructor
   · exact h₁g
   · constructor
-    · apply h₂g.analytic
-
-
-      sorry
+    · exact h₄g
     · constructor
-      · sorry
-      · sorry
-
-
-
-theorem MeromorphicOn.decompose
-  {f : ℂ → ℂ}
-  {U : Set ℂ}
-  (h₁U : IsConnected U)
-  (h₂U : IsCompact U)
-  (h₁f : MeromorphicOn f U)
-  (h₂f : ∃ z₀ ∈ U, f z₀ ≠ 0) :
-  ∃ g : ℂ → ℂ, (AnalyticOnNhd ℂ g U)
-    ∧ (∀ z ∈ U, g z ≠ 0)
-    ∧ (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
-    sorry
-  let g := h₁g₁.makeStronglyMeromorphicOn
-  have h₁g : MeromorphicOn g U := by
-    sorry
-  have h₂g : ∀ z : U, (h₁g z.1 z.2).order = 0 := by
-    sorry
-  have h₃g : StronglyMeromorphicOn g U := by
-    sorry
-  have h₄g : AnalyticOnNhd ℂ g U := by
-    intro z hz
-    apply StronglyMeromorphicAt.analytic (h₃g z hz)
-    rw [h₂g ⟨z, hz⟩]
-  use g
-  constructor
-  · exact h₄g
-  · constructor
-    · intro z hz
-      rw [← (h₄g z hz).order_eq_zero_iff]
-      have A := (h₄g z hz).meromorphicAt_order
-      rw [h₂g ⟨z, hz⟩] at A
-      have t₀ : (h₄g z hz).order ≠ ⊤ := by
-        by_contra hC
-        rw [hC] at A
-        tauto
-      have t₁ : ∃ n : ℕ, (h₄g z hz).order = n := by
-        exact Option.ne_none_iff_exists'.mp t₀
-      obtain ⟨n, hn⟩ := t₁
-      rw [hn] at A
-      apply WithTopCoe
-      rw [eq_comm]
-      rw [hn]
-      exact A
-    · intro z hz
-      have t₀ : ∀ᶠ x in 𝓝[≠] z, AnalyticAt ℂ f x := by
-        sorry
-      have t₂ : ∀ᶠ x in 𝓝[≠] z, h₁f.divisor z = 0 := by
-        sorry
-      have t₁ : ∀ᶠ x in 𝓝[≠] z, AnalyticAt ℂ (fun z => ∏ᶠ (p : ℂ), (z - p) ^ h₁f.divisor p * g z) x := by
-        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}ᶜ : Set ℂ), (fun x ↦ (x - p.1) ^ h₁f.divisor p.1)) * g
-        constructor
-        · apply AnalyticAt.mul₁
-          · apply analyticAt_finprod
-            intro w
-
-
-            sorry
-          · apply (h₃g z hz).analytic
-            rw [h₂g ⟨z, hz⟩]
-        · constructor
-          · sorry
-          · sorry
-
-      sorry
+      · exact (h₂g.order_eq_zero_iff).mp h₃g
+      · funext z
+        by_cases hz : z = z₀
+        · rw [hz]
+          simp
+          by_cases h : h₁f.divisor z₀ = 0
+          · simp [h]
+            have h₂h₁ : h₁ = 1 := by
+              funext w
+              unfold h₁
+              simp [h]
+            have h₃g₁ : g₁ = f := by
+              unfold g₁
+              rw [h₂h₁]
+              simp
+            have h₄g₁ : StronglyMeromorphicAt g₁ z₀ := by
+              rwa [h₃g₁]
+            let A := h₄g₁.makeStronglyMeromorphic_id
+            unfold g
+            rw [← A, h₃g₁]
+          · have : (0 : ℂ) ^ h₁f.divisor z₀ = (0 : ℂ) := by
+              exact zero_zpow (h₁f.divisor z₀) h
+            rw [this]
+            simp
+            let A := h₂f.order_eq_zero_iff.not
+            simp at A
+            rw [← A]
+            unfold MeromorphicOn.divisor at h
+            simp [hz₀] at h
+            exact h.1
+        · simp
+          let B := m₁ (h₁g₁ z₀ hz₀) z hz
+          unfold g
+          rw [← B]
+          unfold g₁ h₁
+          simp [hz]
+          rw [mul_assoc]
+          rw [inv_mul_cancel₀]
+          simp
+          apply zpow_ne_zero
+          rwa [sub_ne_zero]