import Mathlib.Analysis.Meromorphic.Basic
import Mathlib.Analysis.Meromorphic.Order

open MeromorphicOn Metric Real Set Classical

variable
  {𝕜 : Type*} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜]

/-- Derivatives of meromorphic functions are meromorphic. -/
@[fun_prop]
theorem meromorphicAt_deriv {f : 𝕜 → 𝕜} {x : 𝕜}
    (h : MeromorphicAt f x) (h₁ : h.order ≠ ⊤) :
    MeromorphicAt (deriv f) x := by

  obtain ⟨g, h₁g, h₂g, h₃⟩ := h.order_ne_top_iff.1 h₁
  lift h.order to ℤ using h₁ with n hn

  have : (n : WithTop ℤ).untop₀ = n := by
    sorry
  simp_all [this]
  have : deriv f =ᶠ[nhdsWithin x {x}ᶜ] deriv (fun z ↦ (z - x) ^ n * g z) := by
    sorry
  have : deriv f =ᶠ[nhdsWithin x {x}ᶜ] fun z ↦ n * (z - x) ^ (n - 1) * g z + (z - x) ^ n * deriv g z := by
    sorry
  apply MeromorphicAt.congr _ this.symm
  sorry