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

noncomputable def Complex.laplace : (ℂ → ℂ) → (ℂ → ℂ) := by
  intro f

  let fx  := fun w ↦ (fderiv ℝ f w) 1
  let fxx := fun w ↦ (fderiv ℝ fx w) 1
  let fy  := fun w ↦ (fderiv ℝ f w) Complex.I
  let fyy := fun w ↦ (fderiv ℝ fy w) Complex.I
  exact fun z ↦ (fxx z) + (fyy z)


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' := fun w => (fderiv ℝ f w)
  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)

  -- f' is real C¹
  have f'_is_real_C1 : ContDiff ℝ 1 (fderiv ℝ f) :=
    (contDiff_succ_iff_fderiv.1 f_is_real_C2).right

  -- f' is real differentiable
  have f'_is_differentiable : Differentiable ℝ (fderiv ℝ f) :=
    (contDiff_succ_iff_fderiv.1 f'_is_real_C1).left


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

  · -- Laplace of f is zero
    intro z
    unfold Complex.laplace

    simp

    conv =>
      left
      right
      arg 1
      arg 2
      intro z
      rw [CauchyRiemann₁ (h z)]


    have t₀ : ∀ z, DifferentiableAt ℝ (fun w ↦ (fderiv ℝ f w) 1) z := by
      intro z
      let A := f'_is_differentiable
      fun_prop

    have t₁ : ∀ x, (fderiv ℝ (fun w ↦ Complex.I * (fderiv ℝ f w) 1) z) x
      = Complex.I * ((fderiv ℝ (fun w ↦ (fderiv ℝ f w) 1) z) x) := by
      intro x
      rw [fderiv_const_mul]
      simp
      exact t₀ z
    rw [t₁]

    have t₂ : (fderiv ℝ (fun w => (fderiv ℝ f w) 1) z) Complex.I
      = (fderiv ℝ (fun w => (fderiv ℝ f w) Complex.I) z) 1 := by
      let A := derivSymm f f_is_real_C2 z 1 Complex.I
      let B := l₂ f_is_real_C2 z Complex.I  1
      rw [← B]
      rw [A]
      let C := l₂ f_is_real_C2 z 1 Complex.I
      rw [C]
    rw [t₂]

    conv =>
      left
      right
      arg 2
      arg 1
      arg 2
      intro z
      rw [CauchyRiemann₁ (h z)]

    rw [t₁]

    rw [← mul_assoc]
    simp