nevanlinna/Nevanlinna/meromorphicOn_divisor.lean

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


open scoped Interval Topology
open Real Filter MeasureTheory intervalIntegral


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


theorem MeromorphicOn.divisor_def₁
  {f : ℂ → ℂ}
  {U : Set ℂ}
  {z : ℂ}
  (hf : MeromorphicOn f U)
  (hz : z ∈ U) :
  hf.divisor z = ((hf z hz).order.untop' 0 : ℤ) := by
  unfold MeromorphicOn.divisor
  simp [hz]


theorem MeromorphicOn.divisor_def₂
  {f : ℂ → ℂ}
  {U : Set ℂ}
  {z : ℂ}
  (hf : MeromorphicOn f U)
  (hz : z ∈ U)
  (h₂f : (hf z hz).order ≠ ⊤) :
  hf.divisor z = (hf z hz).order.untop h₂f := by
  unfold MeromorphicOn.divisor
  simp [hz]
  rw [WithTop.untop'_eq_iff]
  left
  exact Eq.symm (WithTop.coe_untop (hf z hz).order h₂f)


theorem MeromorphicOn.divisor_mul₀
  {f₁ f₂ : ℂ → ℂ}
  {U : Set ℂ}
  {z : ℂ}
  (hz : z ∈ U)
  (h₁f₁ : MeromorphicOn f₁ U)
  (h₂f₁ : (h₁f₁ z hz).order ≠ ⊤)
  (h₁f₂ : MeromorphicOn f₂ U)
  (h₂f₂ : (h₁f₂ z hz).order ≠ ⊤) :
  (h₁f₁.mul h₁f₂).divisor.toFun z = h₁f₁.divisor.toFun z + h₁f₂.divisor.toFun z := by
  by_cases h₁z : z ∈ U
  · rw [MeromorphicOn.divisor_def₂ h₁f₁ hz h₂f₁]
    rw [MeromorphicOn.divisor_def₂ h₁f₂ hz h₂f₂]
    have B : ((h₁f₁.mul h₁f₂) z hz).order ≠ ⊤ := by
      rw [MeromorphicAt.order_mul (h₁f₁ z hz) (h₁f₂ z hz)]
      simp; tauto
    rw [MeromorphicOn.divisor_def₂ (h₁f₁.mul h₁f₂) hz B]
    simp_rw [MeromorphicAt.order_mul (h₁f₁ z hz) (h₁f₂ z hz)]
    rw [untop_add]
  · unfold MeromorphicOn.divisor
    simp [h₁z]


theorem MeromorphicOn.divisor_mul
  {f₁ f₂ : ℂ → ℂ}
  {U : Set ℂ}
  (h₁f₁ : MeromorphicOn f₁ U)
  (h₂f₁ : ∀ z, (hz : z ∈ U) → (h₁f₁ z hz).order ≠ ⊤)
  (h₁f₂ : MeromorphicOn f₂ U)
  (h₂f₂ : ∀ z, (hz : z ∈ U) → (h₁f₂ z hz).order ≠ ⊤) :
  (h₁f₁.mul h₁f₂).divisor.toFun = h₁f₁.divisor.toFun + h₁f₂.divisor.toFun := by
  funext z
  by_cases hz : z ∈ U
  · rw [MeromorphicOn.divisor_mul₀ hz h₁f₁ (h₂f₁ z hz) h₁f₂ (h₂f₂ z hz)]
    simp
  · simp
    rw [Function.nmem_support.mp (fun a => hz (h₁f₁.divisor.supportInU a))]
    rw [Function.nmem_support.mp (fun a => hz (h₁f₂.divisor.supportInU a))]
    rw [Function.nmem_support.mp (fun a => hz ((h₁f₁.mul h₁f₂).divisor.supportInU a))]
    simp