nevanlinna/Nevanlinna/analyticAt.lean

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


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


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

  contrapose
  intro h₁f
  simp [hf.order_eq_zero_iff.2 h₁f]


theorem ContinuousLinearEquiv.analyticAt
  (ℓ : ℂ ≃L[ℂ] ℂ) (z₀ : ℂ) : AnalyticAt ℂ (⇑ℓ) z₀ := ℓ.toContinuousLinearMap.analyticAt z₀


theorem eventually_nhds_comp_composition
  {f₁ f₂ ℓ : ℂ → ℂ}
  {z₀ : ℂ}
  (hf : ∀ᶠ (z : ℂ) in nhds (ℓ z₀), f₁ z = f₂ z)
  (hℓ : Continuous ℓ) :
  ∀ᶠ (z : ℂ) in nhds z₀, (f₁ ∘ ℓ) z = (f₂ ∘ ℓ) z := by
  obtain ⟨t, h₁t, h₂t, h₃t⟩ := eventually_nhds_iff.1 hf
  apply eventually_nhds_iff.2
  use ℓ⁻¹' t
  constructor
  · intro y hy
    exact h₁t (ℓ y) hy
  · constructor
    · apply IsOpen.preimage
      exact ContinuousLinearEquiv.continuous ℓ
      exact h₂t
    · exact h₃t


theorem AnalyticAt.order_congr
  {f₁ f₂ : ℂ → ℂ}
  {z₀ : ℂ}
  (hf₁ : AnalyticAt ℂ f₁ z₀)
  (hf₂ : AnalyticAt ℂ f₂ z₀)
  (hf : ∀ᶠ (z : ℂ) in nhds z₀, f₁ z = f₂ z) :
  hf₁.order = hf₂.order := by

  sorry


theorem AnalyticAt.order_comp_CLE
  (ℓ : ℂ ≃L[ℂ] ℂ)
  {f : ℂ → ℂ}
  {z₀ : ℂ}
  (hf : AnalyticAt ℂ f (ℓ z₀)) :
  hf.order = (hf.comp (ℓ.analyticAt z₀)).order := by

  by_cases h₁f : hf.order = ⊤
  · rw [h₁f]
    rw [AnalyticAt.order_eq_top_iff] at h₁f
    let A := eventually_nhds_comp_composition h₁f ℓ.continuous
    simp at A
    have : AnalyticAt ℂ (0 : ℂ → ℂ) z₀ := by apply analyticAt_const
    rw [AnalyticAt.order_congr (hf.comp (ℓ.analyticAt z₀)) this A]

    have : this.order = ⊤ := by
      rw [AnalyticAt.order_eq_top_iff]
      simp
    rw [this]
  · let n := hf.order.toNat
    have hn : hf.order = n := Eq.symm (ENat.coe_toNat h₁f)
    rw [hn]
    rw [AnalyticAt.order_eq_nat_iff] at hn
    obtain ⟨g, h₁g, h₂g, h₃g⟩ := hn
    have A := eventually_nhds_comp_composition h₃g ℓ.continuous
    simp only [Function.comp_apply] at A
    have : AnalyticAt ℂ (fun z ↦ (ℓ z - ℓ z₀) ^ n • g (ℓ z) : ℂ → ℂ) z₀ := by apply analyticAt_const
    rw [AnalyticAt.order_congr (hf.comp (ℓ.analyticAt z₀)) this A]
    simp
    rw [AnalyticAt.order_mul]




    --rw [hn, AnalyticAt.order_eq_nat_iff]
    rw [AnalyticAt.order_eq_nat_iff] at hn
    obtain ⟨g, h₁g, h₂g, h₃g⟩ := hn
    use g ∘ ℓ
    constructor
    · exact h₁g.comp (ℓ.analyticAt z₀)
    · constructor
      · exact h₂g
      · rw [eventually_nhds_iff]
        rw [eventually_nhds_iff] at h₃g
        obtain ⟨t, h₁t, h₂t, h₃t⟩ := h₃g
        use ℓ⁻¹' t
        constructor
        · intro y hy
          simp
          rw [h₁t (ℓ y) hy]


          sorry
        · constructor
          · apply IsOpen.preimage
            exact ContinuousLinearEquiv.continuous ℓ
            exact h₂t
          · exact h₃t