82 lines
2.4 KiB
Lean4
82 lines
2.4 KiB
Lean4
import Mathlib.Analysis.Analytic.Meromorphic
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import Nevanlinna.analyticAt
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import Nevanlinna.divisor
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import Nevanlinna.meromorphicAt
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import Nevanlinna.meromorphicOn_divisor
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import Nevanlinna.stronglyMeromorphicOn
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open scoped Interval Topology
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open Real Filter MeasureTheory intervalIntegral
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lemma WithTopCoe
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{n : WithTop ℕ} :
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WithTop.map (Nat.cast : ℕ → ℤ) n = 0 → n = 0 := by
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rcases n with h|h
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· intro h
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contradiction
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· intro h₁
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simp only [WithTop.map, Option.map] at h₁
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have : (h : ℤ) = 0 := by
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exact WithTop.coe_eq_zero.mp h₁
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have : h = 0 := by
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exact Int.ofNat_eq_zero.mp this
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rw [this]
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rfl
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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 (fun z ↦ ∏ᶠ p, (z - p) ^ (h₁f.divisor p) * g z ) 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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sorry
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