nevanlinna/Nevanlinna/meromorphicOn_decompose.lean

import Mathlib.Analysis.Analytic.Meromorphic
import Nevanlinna.analyticAt
import Nevanlinna.divisor
import Nevanlinna.meromorphicAt
import Nevanlinna.meromorphicOn_divisor
import Nevanlinna.stronglyMeromorphicOn


open scoped Interval Topology
open Real Filter MeasureTheory intervalIntegral


theorem MeromorphicOn.decompose
  {f : ℂ → ℂ}
  {U : Set ℂ}
  (h₁U : IsConnected U)
  (h₂U : IsCompact U)
  (h₁f : MeromorphicOn f U)
  (h₂f : ∃ z₀ ∈ U, f z₀ ≠ 0) :
  ∃ g : ℂ → ℂ, (AnalyticOnNhd ℂ g U)
    ∧ (∀ z ∈ U, g z ≠ 0)
    ∧ (Set.EqOn h₁f.makeStronglyMeromorphicOn (fun z ↦ ∏ᶠ p, (z - p) ^ (h₁f.divisor p) * g z ) U) := by

  let g₁ : ℂ → ℂ := f * (fun z ↦ ∏ᶠ p, (z - p) ^ (h₁f.divisor p))
  have h₁g₁ : MeromorphicOn g₁ U := by sorry
  let g := h₁g₁.makeStronglyMeromorphicOn
  have h₁g : MeromorphicOn g U := by sorry
  have h₂g : ∀ z : U, (h₁g z.1 z.2).order = 0 := by sorry
  have h₃g : StronglyMeromorphicOn g U := by sorry
  have h₄g : AnalyticOnNhd ℂ g U := by
    intro z hz
    apply StronglyMeromorphicAt.analytic (h₃g z hz)
    rw [h₂g ⟨z, hz⟩]
  use g
  constructor
  · exact h₄g
  · constructor
    · intro z hz
      rw [← (h₄g z hz).order_eq_zero_iff]

      have A := (h₄g z hz).meromorphicAt_order
      rw [h₂g ⟨z, hz⟩] at A

      have t₀ : (0 : WithTop ℤ) = WithTop.map Nat.cast (0 : WithTop ℕ) := by
        sorry
      --rw [← this] at A

      rw [WithTop.map_coe] at A

      sorry
    · intro z hz

      sorry