nevanlinna/Nevanlinna/analyticAt.lean

import Mathlib.Analysis.Analytic.IsolatedZeros
import Mathlib.Analysis.Complex.Basic


theorem AnalyticAt.order_mul
  {f₁ f₂ : ℂ → ℂ}
  {z₀ : ℂ}
  (hf₁ : AnalyticAt ℂ f₁ z₀)
  (hf₂ : AnalyticAt ℂ f₂ z₀) :
  (AnalyticAt.mul hf₁ hf₂).order = hf₁.order + hf₂.order := by
  by_cases h₂f₁ : hf₁.order = ⊤
  · simp [h₂f₁]
    rw [AnalyticAt.order_eq_top_iff, eventually_nhds_iff]
    rw [AnalyticAt.order_eq_top_iff, eventually_nhds_iff] at h₂f₁
    obtain ⟨t, h₁t, h₂t, h₃t⟩ := h₂f₁
    use t
    constructor
    · intro y hy
      rw [h₁t y hy]
      ring
    · exact ⟨h₂t, h₃t⟩
  · by_cases h₂f₂ : hf₂.order = ⊤
    · simp [h₂f₂]
      rw [AnalyticAt.order_eq_top_iff, eventually_nhds_iff]
      rw [AnalyticAt.order_eq_top_iff, eventually_nhds_iff] at h₂f₂
      obtain ⟨t, h₁t, h₂t, h₃t⟩ := h₂f₂
      use t
      constructor
      · intro y hy
        rw [h₁t y hy]
        ring
      · exact ⟨h₂t, h₃t⟩
    · obtain ⟨g₁, h₁g₁, h₂g₁, h₃g₁⟩ := (AnalyticAt.order_eq_nat_iff hf₁ ↑hf₁.order.toNat).1 (eq_comm.1 (ENat.coe_toNat h₂f₁))
      obtain ⟨g₂, h₁g₂, h₂g₂, h₃g₂⟩ := (AnalyticAt.order_eq_nat_iff hf₂ ↑hf₂.order.toNat).1 (eq_comm.1 (ENat.coe_toNat h₂f₂))
      rw [← ENat.coe_toNat h₂f₁, ← ENat.coe_toNat h₂f₂, ← ENat.coe_add]
      rw [AnalyticAt.order_eq_nat_iff (AnalyticAt.mul hf₁ hf₂) ↑(hf₁.order.toNat + hf₂.order.toNat)]
      use g₁ * g₂
      constructor
      · exact AnalyticAt.mul h₁g₁ h₁g₂
      · constructor
        · simp; tauto
        · obtain ⟨t₁, h₁t₁, h₂t₁, h₃t₁⟩ := eventually_nhds_iff.1 h₃g₁
          obtain ⟨t₂, h₁t₂, h₂t₂, h₃t₂⟩ := eventually_nhds_iff.1 h₃g₂
          rw [eventually_nhds_iff]
          use t₁ ∩ t₂
          constructor
          · intro y hy
            rw [h₁t₁ y hy.1, h₁t₂ y hy.2]
            simp; ring
          · constructor
            · exact IsOpen.inter h₂t₁ h₂t₂
            · exact Set.mem_inter h₃t₁ h₃t₂


theorem AnalyticAt.order_eq_zero_iff
  {f : ℂ → ℂ}
  {z₀ : ℂ}
  (hf : AnalyticAt ℂ f z₀) :
  hf.order = 0 ↔ f z₀ ≠ 0 := by

  have : (0 : ENat) = (0 : Nat) := by rfl
  rw [this, AnalyticAt.order_eq_nat_iff hf 0]

  constructor
  · intro hz
    obtain ⟨g, _, h₂g, h₃g⟩ := hz
    simp at h₃g
    rw [Filter.Eventually.self_of_nhds h₃g]
    tauto
  · intro hz
    use f
    constructor
    · exact hf
    · constructor
      · exact hz
      · simp