nevanlinna/Nevanlinna/stronglyMeromorphicOn.lean

import Nevanlinna.stronglyMeromorphicAt


open Topology


/- Strongly MeromorphicOn -/
def StronglyMeromorphicOn
  (f : ℂ → ℂ)
  (U : Set ℂ) :=
  ∀ z ∈ U, StronglyMeromorphicAt f z


/- Strongly MeromorphicAt is Meromorphic -/
theorem StronglyMeromorphicOn.meromorphicOn
  {f : ℂ → ℂ}
  {U : Set ℂ}
  (hf : StronglyMeromorphicOn f U) :
  MeromorphicOn f U := by
  intro z hz
  exact StronglyMeromorphicAt.meromorphicAt (hf z hz)


/- Strongly MeromorphicOn of non-negative order is analytic -/
theorem StronglyMeromorphicOn.analytic
  {f : ℂ → ℂ}
  {U : Set ℂ}
  (h₁f : StronglyMeromorphicOn f U)
  (h₂f : ∀ x, (hx : x ∈ U) → 0 ≤ (h₁f x hx).meromorphicAt.order):
  ∀ z ∈ U, AnalyticAt ℂ f z := by
  intro z hz
  apply StronglyMeromorphicAt.analytic
  exact h₂f z hz
  exact h₁f z hz


/- Analytic functions are strongly meromorphic -/
theorem AnalyticOn.stronglyMeromorphicOn
  {f : ℂ → ℂ}
  {U : Set ℂ}
  (h₁f : AnalyticOnNhd ℂ f U) :
  StronglyMeromorphicOn f U := by
  intro z hz
  apply AnalyticAt.stronglyMeromorphicAt
  exact h₁f z hz


/- Make strongly MeromorphicAt -/
noncomputable def MeromorphicOn.makeStronglyMeromorphicOn
  {f : ℂ → ℂ}
  {U : Set ℂ}
  (hf : MeromorphicOn f U) :
  ℂ → ℂ := by
  intro z
  by_cases hz : z ∈ U
  · exact (hf z hz).makeStronglyMeromorphicAt z
  · exact f z


theorem StronglyMeromorphicOn_of_makeStronglyMeromorphicOn
  {f : ℂ → ℂ}
  {U : Set ℂ}
  (hf : MeromorphicOn f U) :
  StronglyMeromorphicOn hf.makeStronglyMeromorphicOn U := by
  sorry