nevanlinna/Nevanlinna/holomorphic.primitive.lean

import Mathlib.Analysis.Complex.TaylorSeries
import Mathlib.MeasureTheory.Integral.DivergenceTheorem
import Mathlib.MeasureTheory.Function.LocallyIntegrable
import Nevanlinna.cauchyRiemann
import Nevanlinna.partialDeriv


theorem MeasureTheory.integral2_divergence₃
  {E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
  (f g : ℝ × ℝ → E)
  (h₁f : ContDiff ℝ 1 f)
  (h₁g : ContDiff ℝ 1 g)
  (a₁ : ℝ)
  (a₂ : ℝ)
  (b₁ : ℝ)
  (b₂ : ℝ) :
  ∫ (x : ℝ) in a₁..b₁, ∫ (y : ℝ) in a₂..b₂, ((fderiv ℝ f) (x, y)) (1, 0) + ((fderiv ℝ g) (x, y)) (0, 1) = (((∫ (x : ℝ) in a₁..b₁, g (x, b₂)) - ∫ (x : ℝ) in a₁..b₁, g (x, a₂)) + ∫ (y : ℝ) in a₂..b₂, f (b₁, y)) - ∫ (y : ℝ) in a₂..b₂, f (a₁, y) := by

  apply integral2_divergence_prod_of_hasFDerivWithinAt_off_countable f g (fderiv ℝ f) (fderiv ℝ g) a₁ a₂ b₁ b₂ ∅
  exact Set.countable_empty
  -- ContinuousOn f (Set.uIcc a₁ b₁ ×ˢ Set.uIcc a₂ b₂)
  exact h₁f.continuous.continuousOn
  --
  exact h₁g.continuous.continuousOn
  --
  rw [Set.diff_empty]
  intro x _
  exact DifferentiableAt.hasFDerivAt ((h₁f.differentiable le_rfl) x)
  --
  rw [Set.diff_empty]
  intro y _
  exact DifferentiableAt.hasFDerivAt ((h₁g.differentiable le_rfl) y)
  --
  apply ContinuousOn.integrableOn_compact
  apply IsCompact.prod
  exact isCompact_uIcc
  exact isCompact_uIcc
  apply ContinuousOn.add
  apply Continuous.continuousOn
  exact Continuous.clm_apply (ContDiff.continuous_fderiv h₁f le_rfl) continuous_const
  apply Continuous.continuousOn
  exact Continuous.clm_apply (ContDiff.continuous_fderiv h₁g le_rfl) continuous_const


theorem integral_divergence₄
  {E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
  (f g : ℂ  → E)
  (h₁f : ContDiff ℝ 1 f)
  (h₁g : ContDiff ℝ 1 g)
  (a₁ : ℝ)
  (a₂ : ℝ)
  (b₁ : ℝ)
  (b₂ : ℝ) :
  ∫ (x : ℝ) in a₁..b₁, ∫ (y : ℝ) in a₂..b₂, ((fderiv ℝ f) ⟨x, y⟩ ) 1 + ((fderiv ℝ g) ⟨x, y⟩) Complex.I = (((∫ (x : ℝ) in a₁..b₁, g ⟨x, b₂⟩) - ∫ (x : ℝ) in a₁..b₁, g ⟨x, a₂⟩) + ∫ (y : ℝ) in a₂..b₂, f ⟨b₁, y⟩) - ∫ (y : ℝ) in a₂..b₂, f ⟨a₁, y⟩ := by

  let fr : ℝ × ℝ → E := f ∘ Complex.equivRealProdCLM.symm
  let gr : ℝ × ℝ → E := g ∘ Complex.equivRealProdCLM.symm

  have sfr {x y : ℝ} : f { re := x, im := y } = fr (x, y) := by exact rfl
  have sgr {x y : ℝ} : g { re := x, im := y } = gr (x, y) := by exact rfl
  repeat (conv in f { re := _, im := _ } => rw [sfr])
  repeat (conv in g { re := _, im := _ } => rw [sgr])

  have sfr' {x y : ℝ} {z : ℂ} : (fderiv ℝ f { re := x, im := y }) z = fderiv ℝ fr (x, y) (Complex.equivRealProdCLM z) := by
    rw [fderiv.comp]
    rw [Complex.equivRealProdCLM.symm.fderiv]
    tauto
    apply Differentiable.differentiableAt
    exact h₁f.differentiable le_rfl
    exact Complex.equivRealProdCLM.symm.differentiableAt
  conv in ⇑(fderiv ℝ f { re := _, im := _ }) _ => rw [sfr']

  have sgr' {x y : ℝ} {z : ℂ} : (fderiv ℝ g { re := x, im := y }) z = fderiv ℝ gr (x, y) (Complex.equivRealProdCLM z) := by
    rw [fderiv.comp]
    rw [Complex.equivRealProdCLM.symm.fderiv]
    tauto
    apply Differentiable.differentiableAt
    exact h₁g.differentiable le_rfl
    exact Complex.equivRealProdCLM.symm.differentiableAt
  conv in ⇑(fderiv ℝ g { re := _, im := _ }) _ => rw [sgr']

  apply MeasureTheory.integral2_divergence₃ fr gr _ _ a₁ a₂ b₁ b₂
  -- ContDiff ℝ 1 fr
  exact (ContinuousLinearEquiv.contDiff_comp_iff (ContinuousLinearEquiv.symm Complex.equivRealProdCLM)).mpr h₁f
  -- ContDiff ℝ 1 gr
  exact (ContinuousLinearEquiv.contDiff_comp_iff (ContinuousLinearEquiv.symm Complex.equivRealProdCLM)).mpr h₁g


theorem integral_divergence₅
  {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
  (F : ℂ  → E)
  (hF : Differentiable ℂ F)
  (lowerLeft upperRight : ℂ) :
  (∫ (x : ℝ) in lowerLeft.re..upperRight.re, F ⟨x, lowerLeft.im⟩)  + Complex.I • ∫ (x : ℝ) in lowerLeft.im..upperRight.im, F ⟨upperRight.re, x⟩ =
  (∫ (x : ℝ) in lowerLeft.re..upperRight.re, F ⟨x, upperRight.im⟩) + Complex.I • ∫ (x : ℝ) in lowerLeft.im..upperRight.im, F ⟨lowerLeft.re,  x⟩ := by

  let h₁f : ContDiff ℝ 1 F :=  (hF.contDiff : ContDiff ℂ 1 F).restrict_scalars ℝ

  let h₁g : ContDiff ℝ 1 (-Complex.I • F) := by
    have : -Complex.I • F = fun x ↦ -Complex.I • F x := by rfl
    rw [this]
    apply ContDiff.comp
    exact contDiff_const_smul _
    exact h₁f

  let A := integral_divergence₄ (-Complex.I • F) F h₁g h₁f lowerLeft.re upperRight.im upperRight.re lowerLeft.im

  have {z : ℂ} : fderiv ℝ F z Complex.I = partialDeriv ℝ _ F z := by rfl
  conv at A in (fderiv ℝ F _) _ => rw [this]
  have {z : ℂ} : fderiv ℝ (-Complex.I • F) z 1 = partialDeriv ℝ _ (-Complex.I • F) z := by rfl
  conv at A in (fderiv ℝ (-Complex.I • F) _) _ => rw [this]
  conv at A =>
    left
    arg 1
    intro x
    arg 1
    intro y
    rw [CauchyRiemann₄ hF]
    rw [partialDeriv_smul'₂]
    simp
  simp at A

  have {t₁ t₂ t₃ t₄ : E} : 0 = (t₁ - t₂) + t₃ + t₄ → t₁ + t₃ = t₂ - t₄ := by
    intro hyp
    calc
      t₁ + t₃ = t₁ + t₃ - 0 := by rw [sub_zero (t₁ + t₃)]
      _ = t₁ + t₃ - (t₁ - t₂ + t₃ + t₄) := by rw [hyp]
      _ = t₂ - t₄ := by abel
  let B := this A
  repeat
    rw [intervalIntegral.integral_symm lowerLeft.im upperRight.im] at B
  simp at B
  exact B