nevanlinna/Nevanlinna/complexHarmonic.lean

import Mathlib.Data.Fin.Tuple.Basic
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Complex.TaylorSeries
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Symmetric
import Nevanlinna.cauchyRiemann
import Nevanlinna.partialDeriv

theorem CauchyRiemann₄ {f : ℂ → ℂ} : (Differentiable ℂ f)
  → Real.partialDeriv Complex.I f = Complex.I • Real.partialDeriv 1 f := by
  intro h
  unfold Real.partialDeriv

  conv =>
    left
    intro w
    rw [DifferentiableAt.fderiv_restrictScalars ℝ (h w)]
    simp
    rw [← mul_one Complex.I]
    rw [← smul_eq_mul]
    rw [ContinuousLinearMap.map_smul_of_tower (fderiv ℂ f w) Complex.I 1]
  conv =>
    right
    right
    intro w
    rw [DifferentiableAt.fderiv_restrictScalars ℝ (h w)]


noncomputable def Complex.laplace : (ℂ → ℂ) → (ℂ → ℂ) := by
  intro f
  let fx  := Real.partialDeriv 1 f
  let fxx := Real.partialDeriv 1 fx
  let fy  := Real.partialDeriv Complex.I f
  let fyy := Real.partialDeriv Complex.I fy
  exact fxx + fyy


def Harmonic (f : ℂ → ℂ) : Prop :=
  (ContDiff ℝ 2 f) ∧ (∀ z, Complex.laplace f z = 0)


lemma derivSymm (f : ℂ → ℂ) (hf : ContDiff ℝ 2 f) :
  ∀ z a b : ℂ, (fderiv ℝ (fun w => fderiv ℝ f w) z) a b = (fderiv ℝ (fun w => fderiv ℝ f w) z) b a := by
  intro z a b

  let f' := fderiv ℝ f
  have h₀ : ∀ y, HasFDerivAt f (f' y) y := by
    have h : Differentiable ℝ f := by
      exact (contDiff_succ_iff_fderiv.1 hf).left
    exact fun y => DifferentiableAt.hasFDerivAt (h y)

  let f'' := (fderiv ℝ f' z)
  have h₁ : HasFDerivAt f' f'' z := by
    apply DifferentiableAt.hasFDerivAt
    let A := (contDiff_succ_iff_fderiv.1 hf).right
    let B := (contDiff_succ_iff_fderiv.1 A).left
    simp at B
    exact B z

  let A := second_derivative_symmetric h₀ h₁ a b
  dsimp [f'', f'] at A
  apply A


lemma l₂ {f : ℂ → ℂ} (hf : ContDiff ℝ 2 f) (z a b : ℂ) :
    fderiv ℝ (fderiv ℝ f) z b a = fderiv ℝ (fun w ↦ fderiv ℝ f w a) z b := by
  rw [fderiv_clm_apply]
  · simp
  · exact (contDiff_succ_iff_fderiv.1 hf).2.differentiable le_rfl z
  · simp


theorem holomorphic_is_harmonic {f : ℂ → ℂ} (h : Differentiable ℂ f) :
  Harmonic f := by

  -- f is real C²
  have f_is_real_C2 : ContDiff ℝ 2 f :=
    ContDiff.restrict_scalars ℝ (Differentiable.contDiff h)

  have fI_is_real_differentiable : Differentiable ℝ (Real.partialDeriv 1 f) := by
    exact (partialDeriv_contDiff f_is_real_C2 1).differentiable (Submonoid.oneLE.proof_2 ℕ∞)

  constructor
  · -- f is two times real continuously differentiable
    exact f_is_real_C2

  · -- Laplace of f is zero
    unfold Complex.laplace
    rw [CauchyRiemann₄ h]
    rw [partialDeriv_smul fI_is_real_differentiable]
    rw [partialDeriv_comm f_is_real_C2 Complex.I 1]
    rw [CauchyRiemann₄ h]
    rw [partialDeriv_smul fI_is_real_differentiable]
    rw [← smul_assoc]
    simp