27 lines
884 B
Lean4
27 lines
884 B
Lean4
import Mathlib.Analysis.Meromorphic.Basic
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import Mathlib.Analysis.Meromorphic.Order
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open MeromorphicOn Metric Real Set Classical
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variable
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{𝕜 : Type*} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜]
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/-- Derivatives of meromorphic functions are meromorphic. -/
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@[fun_prop]
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theorem meromorphicAt_deriv {f : 𝕜 → 𝕜} {x : 𝕜}
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(h : MeromorphicAt f x) (h₁ : h.order ≠ ⊤) :
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MeromorphicAt (deriv f) x := by
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obtain ⟨g, h₁g, h₂g, h₃⟩ := h.order_ne_top_iff.1 h₁
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lift h.order to ℤ using h₁ with n hn
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have : (n : WithTop ℤ).untop₀ = n := by
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sorry
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simp_all [this]
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have : deriv f =ᶠ[nhdsWithin x {x}ᶜ] deriv (fun z ↦ (z - x) ^ n * g z) := by
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sorry
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have : deriv f =ᶠ[nhdsWithin x {x}ᶜ] fun z ↦ n * (z - x) ^ (n - 1) * g z + (z - x) ^ n * deriv g z := by
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sorry
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apply MeromorphicAt.congr _ this.symm
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sorry
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