nevanlinna/Nevanlinna/meromorphicOn_divisor.lean

import Mathlib.Analysis.Analytic.Meromorphic
import Nevanlinna.analyticAt
import Nevanlinna.divisor


open scoped Interval Topology
open Real Filter MeasureTheory intervalIntegral

theorem MeromorphicAt.eventually_eq_zero_or_eventually_ne_zero
  {f : ℂ → ℂ}
  {z₀ : ℂ}
  (hf : MeromorphicAt f z₀) :
  (∀ᶠ (z : ℂ) in nhdsWithin z₀ {z₀}ᶜ, f z = 0) ∨ ∀ᶠ (z : ℂ) in nhdsWithin z₀ {z₀}ᶜ, f z ≠ 0 := by

  obtain ⟨n, h⟩ := hf
  let A := h.eventually_eq_zero_or_eventually_ne_zero

  rw [eventually_nhdsWithin_iff]
  rw [eventually_nhds_iff]
  rcases A with h₁|h₂
  · rw [eventually_nhds_iff] at h₁
    obtain ⟨N, h₁N, h₂N, h₃N⟩ := h₁
    left
    use N
    constructor
    · intro y h₁y h₂y
      let A := h₁N y h₁y
      simp at A
      rcases A with h₃|h₄
      · let B := h₃.1
        simp at h₂y
        let C := sub_eq_zero.1 B
        tauto
      · assumption
    · constructor
      · exact h₂N
      · exact h₃N
  · right
    rw [eventually_nhdsWithin_iff]
    rw [eventually_nhds_iff]
    rw [eventually_nhdsWithin_iff] at h₂
    rw [eventually_nhds_iff] at h₂
    obtain ⟨N, h₁N, h₂N, h₃N⟩ := h₂
    use N
    constructor
    · intro y h₁y h₂y
      by_contra h
      let A := h₁N y h₁y h₂y
      rw [h] at A
      simp at A
    · constructor
      · exact h₂N
      · exact h₃N


noncomputable def MeromorphicOn.divisor
  {f : ℂ → ℂ}
  {U : Set ℂ}
  (hf : MeromorphicOn f U) :
  Divisor U where

  toFun := by
    intro z
    if hz : z ∈ U then
      exact ((hf z hz).order.untop' 0 : ℤ)
    else
      exact 0

  supportInU := by
    intro z hz
    simp at hz
    by_contra h₂z
    simp [h₂z] at hz

  locallyFiniteInU := by
    intro z hz

    apply eventually_nhdsWithin_iff.2
    rw [eventually_nhds_iff]

    rcases MeromorphicAt.eventually_eq_zero_or_eventually_ne_zero (hf z hz) with h|h
    · rw [eventually_nhdsWithin_iff] at h
      rw [eventually_nhds_iff] at h
      obtain ⟨N, h₁N, h₂N, h₃N⟩ := h
      use N
      constructor
      · intro y h₁y h₂y
        by_cases h₃y : y ∈ U
        · simp [h₃y]
          right
          rw [MeromorphicAt.order_eq_top_iff (hf y h₃y)]
          rw [eventually_nhdsWithin_iff]
          rw [eventually_nhds_iff]
          use N ∩ {z}ᶜ
          constructor
          · intro x h₁x _
            exact h₁N x h₁x.1 h₁x.2
          · constructor
            · exact IsOpen.inter h₂N isOpen_compl_singleton
            · exact Set.mem_inter h₁y h₂y
        · simp [h₃y]
      · tauto

    · let A := (hf z hz).eventually_analyticAt
      let B := Filter.eventually_and.2 ⟨h, A⟩
      rw [eventually_nhdsWithin_iff, eventually_nhds_iff] at B
      obtain ⟨N, h₁N, h₂N, h₃N⟩ := B
      use N
      constructor
      · intro y h₁y h₂y
        by_cases h₃y : y ∈ U
        · simp [h₃y]
          left
          rw [(h₁N y h₁y h₂y).2.meromorphicAt_order]
          let D := (h₁N y h₁y h₂y).2.order_eq_zero_iff.2
          let C := (h₁N y h₁y h₂y).1
          let E := D C
          rw [E]
          simp
        · simp [h₃y]
      · tauto