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@ -289,7 +289,7 @@ theorem StronglyMeromorphicAt_of_makeStronglyMeromorphic
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rwa [WithTop.untop_eq_iff h₂f]
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theorem makeStronglyMeromorphic_id
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theorem StronglyMeromorphicAt.makeStronglyMeromorphic_id
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{f : ℂ → ℂ}
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{z₀ : ℂ}
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(hf : StronglyMeromorphicAt f z₀) :
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@ -74,7 +74,7 @@ theorem makeStronglyMeromorphicOn_changeDiscrete
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unfold MeromorphicOn.makeStronglyMeromorphicOn
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by_cases h₂v : v ∈ U
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· simp [h₂v]
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rw [← makeStronglyMeromorphic_id]
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rw [← StronglyMeromorphicAt.makeStronglyMeromorphic_id]
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exact AnalyticAt.stronglyMeromorphicAt (h₂V v hv)
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· simp [h₂v]
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@ -92,90 +92,60 @@ theorem MeromorphicOn.decompose₁
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· constructor
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· exact isOpen_compl_singleton
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· exact h₁z
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have h₃g : (h₁g z₀ hz₀).order = 0 := by
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unfold g
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let B := m₂ (h₁g₁ z₀ hz₀)
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let A := (h₁g₁ z₀ hz₀).order_congr B
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rw [← A]
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rw [h₂g₁]
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have h₄g : AnalyticAt ℂ g z₀ := by
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apply h₂g.analytic
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rw [h₃g]
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use g
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constructor
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· exact h₁g
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· constructor
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· apply h₂g.analytic
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sorry
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· exact h₄g
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· constructor
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· sorry
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· sorry
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theorem MeromorphicOn.decompose
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{f : ℂ → ℂ}
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{U : Set ℂ}
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(h₁U : IsConnected U)
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(h₂U : IsCompact U)
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(h₁f : MeromorphicOn f U)
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(h₂f : ∃ z₀ ∈ U, f z₀ ≠ 0) :
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∃ g : ℂ → ℂ, (AnalyticOnNhd ℂ g U)
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∧ (∀ z ∈ U, g z ≠ 0)
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∧ (Set.EqOn h₁f.makeStronglyMeromorphicOn ((∏ᶠ p, fun z ↦ (z - p) ^ (h₁f.divisor p)) * g) U) := by
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let g₁ : ℂ → ℂ := f * (fun z ↦ ∏ᶠ p, (z - p) ^ (h₁f.divisor p))
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have h₁g₁ : MeromorphicOn g₁ U := by
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sorry
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let g := h₁g₁.makeStronglyMeromorphicOn
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have h₁g : MeromorphicOn g U := by
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sorry
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have h₂g : ∀ z : U, (h₁g z.1 z.2).order = 0 := by
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sorry
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have h₃g : StronglyMeromorphicOn g U := by
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sorry
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have h₄g : AnalyticOnNhd ℂ g U := by
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intro z hz
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apply StronglyMeromorphicAt.analytic (h₃g z hz)
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rw [h₂g ⟨z, hz⟩]
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use g
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constructor
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· exact h₄g
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· constructor
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· intro z hz
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rw [← (h₄g z hz).order_eq_zero_iff]
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have A := (h₄g z hz).meromorphicAt_order
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rw [h₂g ⟨z, hz⟩] at A
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have t₀ : (h₄g z hz).order ≠ ⊤ := by
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by_contra hC
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rw [hC] at A
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tauto
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have t₁ : ∃ n : ℕ, (h₄g z hz).order = n := by
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exact Option.ne_none_iff_exists'.mp t₀
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obtain ⟨n, hn⟩ := t₁
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rw [hn] at A
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apply WithTopCoe
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rw [eq_comm]
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rw [hn]
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exact A
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· intro z hz
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have t₀ : ∀ᶠ x in 𝓝[≠] z, AnalyticAt ℂ f x := by
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sorry
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have t₂ : ∀ᶠ x in 𝓝[≠] z, h₁f.divisor z = 0 := by
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sorry
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have t₁ : ∀ᶠ x in 𝓝[≠] z, AnalyticAt ℂ (fun z => ∏ᶠ (p : ℂ), (z - p) ^ h₁f.divisor p * g z) x := by
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sorry
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apply Filter.EventuallyEq.eq_of_nhds
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apply StronglyMeromorphicAt.localIdentity
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· exact StronglyMeromorphicOn_of_makeStronglyMeromorphic h₁f z hz
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· right
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use h₁f.divisor z
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use (∏ᶠ p : ({z}ᶜ : Set ℂ), (fun x ↦ (x - p.1) ^ h₁f.divisor p.1)) * g
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constructor
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· apply AnalyticAt.mul₁
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· apply analyticAt_finprod
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intro w
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sorry
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· apply (h₃g z hz).analytic
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rw [h₂g ⟨z, hz⟩]
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· constructor
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· sorry
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· sorry
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sorry
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· exact (h₂g.order_eq_zero_iff).mp h₃g
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· funext z
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by_cases hz : z = z₀
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· rw [hz]
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simp
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by_cases h : h₁f.divisor z₀ = 0
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· simp [h]
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have h₂h₁ : h₁ = 1 := by
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funext w
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unfold h₁
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simp [h]
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have h₃g₁ : g₁ = f := by
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unfold g₁
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rw [h₂h₁]
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simp
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have h₄g₁ : StronglyMeromorphicAt g₁ z₀ := by
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rwa [h₃g₁]
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let A := h₄g₁.makeStronglyMeromorphic_id
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unfold g
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rw [← A, h₃g₁]
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· have : (0 : ℂ) ^ h₁f.divisor z₀ = (0 : ℂ) := by
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exact zero_zpow (h₁f.divisor z₀) h
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rw [this]
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simp
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let A := h₂f.order_eq_zero_iff.not
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simp at A
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rw [← A]
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unfold MeromorphicOn.divisor at h
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simp [hz₀] at h
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exact h.1
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· simp
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let B := m₁ (h₁g₁ z₀ hz₀) z hz
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unfold g
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rw [← B]
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unfold g₁ h₁
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simp [hz]
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rw [mul_assoc]
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rw [inv_mul_cancel₀]
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simp
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apply zpow_ne_zero
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rwa [sub_ne_zero]
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