122 lines
3.2 KiB
Plaintext
122 lines
3.2 KiB
Plaintext
import Mathlib.Analysis.Analytic.Meromorphic
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import Nevanlinna.analyticAt
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import Nevanlinna.divisor
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open scoped Interval Topology
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open Real Filter MeasureTheory intervalIntegral
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theorem MeromorphicAt.eventually_eq_zero_or_eventually_ne_zero
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{f : ℂ → ℂ}
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{z₀ : ℂ}
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(hf : MeromorphicAt f z₀) :
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(∀ᶠ (z : ℂ) in nhdsWithin z₀ {z₀}ᶜ, f z = 0) ∨ ∀ᶠ (z : ℂ) in nhdsWithin z₀ {z₀}ᶜ, f z ≠ 0 := by
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obtain ⟨n, h⟩ := hf
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let A := h.eventually_eq_zero_or_eventually_ne_zero
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rw [eventually_nhdsWithin_iff]
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rw [eventually_nhds_iff]
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rcases A with h₁|h₂
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· rw [eventually_nhds_iff] at h₁
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obtain ⟨N, h₁N, h₂N, h₃N⟩ := h₁
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left
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use N
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constructor
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· intro y h₁y h₂y
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let A := h₁N y h₁y
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simp at A
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rcases A with h₃|h₄
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· let B := h₃.1
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simp at h₂y
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let C := sub_eq_zero.1 B
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tauto
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· assumption
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· constructor
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· exact h₂N
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· exact h₃N
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· right
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rw [eventually_nhdsWithin_iff]
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rw [eventually_nhds_iff]
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rw [eventually_nhdsWithin_iff] at h₂
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rw [eventually_nhds_iff] at h₂
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obtain ⟨N, h₁N, h₂N, h₃N⟩ := h₂
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use N
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constructor
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· intro y h₁y h₂y
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by_contra h
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let A := h₁N y h₁y h₂y
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rw [h] at A
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simp at A
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· constructor
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· exact h₂N
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· exact h₃N
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noncomputable def MeromorphicOn.divisor
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{f : ℂ → ℂ}
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{U : Set ℂ}
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(hf : MeromorphicOn f U) :
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Divisor U where
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toFun := by
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intro z
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if hz : z ∈ U then
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exact ((hf z hz).order.untop' 0 : ℤ)
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else
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exact 0
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supportInU := by
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intro z hz
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simp at hz
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by_contra h₂z
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simp [h₂z] at hz
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locallyFiniteInU := by
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intro z hz
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apply eventually_nhdsWithin_iff.2
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rw [eventually_nhds_iff]
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rcases MeromorphicAt.eventually_eq_zero_or_eventually_ne_zero (hf z hz) with h|h
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· rw [eventually_nhdsWithin_iff] at h
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rw [eventually_nhds_iff] at h
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obtain ⟨N, h₁N, h₂N, h₃N⟩ := h
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use N
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constructor
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· intro y h₁y h₂y
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by_cases h₃y : y ∈ U
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· simp [h₃y]
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right
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rw [MeromorphicAt.order_eq_top_iff (hf y h₃y)]
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rw [eventually_nhdsWithin_iff]
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rw [eventually_nhds_iff]
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use N ∩ {z}ᶜ
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constructor
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· intro x h₁x _
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exact h₁N x h₁x.1 h₁x.2
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· constructor
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· exact IsOpen.inter h₂N isOpen_compl_singleton
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· exact Set.mem_inter h₁y h₂y
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· simp [h₃y]
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· tauto
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· let A := (hf z hz).eventually_analyticAt
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let B := Filter.eventually_and.2 ⟨h, A⟩
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rw [eventually_nhdsWithin_iff, eventually_nhds_iff] at B
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obtain ⟨N, h₁N, h₂N, h₃N⟩ := B
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use N
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constructor
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· intro y h₁y h₂y
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by_cases h₃y : y ∈ U
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· simp [h₃y]
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left
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rw [(h₁N y h₁y h₂y).2.meromorphicAt_order]
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let D := (h₁N y h₁y h₂y).2.order_eq_zero_iff.2
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let C := (h₁N y h₁y h₂y).1
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let E := D C
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rw [E]
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simp
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· simp [h₃y]
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· tauto
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