nevanlinna/Nevanlinna/complexHarmonic.examples.lean

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 Mathlib.Analysis.RCLike.Basic
import Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv
import Mathlib.Data.Complex.Module
import Mathlib.Data.Complex.Order
import Mathlib.Data.Complex.Exponential
import Mathlib.Data.Fin.Tuple.Basic
import Mathlib.Topology.Algebra.InfiniteSum.Module
import Mathlib.Topology.Defs.Filter
import Mathlib.Topology.Instances.RealVectorSpace
import Nevanlinna.cauchyRiemann
import Nevanlinna.laplace
import Nevanlinna.complexHarmonic
import Nevanlinna.holomorphic

variable {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
variable {F₁ : Type*} [NormedAddCommGroup F₁] [NormedSpace ℂ F₁] [CompleteSpace F₁]
variable {G : Type*} [NormedAddCommGroup G] [NormedSpace ℝ G]
variable {G₁ : Type*} [NormedAddCommGroup G₁] [NormedSpace ℂ G₁] [CompleteSpace G₁]

#check DifferentiableOn.contDiffOn

theorem holomorphicAt_is_harmonicAt
  {f : ℂ → F₁}
  {z : ℂ}
  (hf : HolomorphicAt f z) :
  HarmonicAt f z := by

  obtain ⟨s, hs, hz, h'f⟩ := HolomorphicAt_iff.1 hf

  constructor
  · -- ContDiffAt ℝ 2 f z
    apply ContDiffOn.contDiffAt _ ((IsOpen.mem_nhds_iff hs).2 hz)
    suffices h : ContDiffOn ℂ 2 f s from by
      apply ContDiffOn.restrict_scalars ℝ h
    apply DifferentiableOn.contDiffOn _ hs
    intro w hw
    apply DifferentiableAt.differentiableWithinAt
    exact h'f w hw
  · -- Δ f =ᶠ[nhds z] 0
    obtain ⟨t, ht, hz, h'f⟩ := HolomorphicAt_isOpen.1 hf
    apply Filter.eventuallyEq_iff_exists_mem.2
    use t
    constructor
    · exact IsOpen.mem_nhds ht hz
    · intro w hw
      unfold Complex.laplace
      simp

      rw [partialDeriv_eventuallyEq ℝ (CauchyRiemann'₆ (h'f w hw)) Complex.I]
      rw [partialDeriv_smul'₂]
      simp

      have f_is_real_C2 : ContDiffOn ℝ 2 f t :=
        ContDiffOn.restrict_scalars ℝ (DifferentiableOn.contDiffOn h'f ht)
      rw [partialDeriv_commOn ht f_is_real_C2 Complex.I 1 w hw]
      rw [partialDeriv_eventuallyEq ℝ (CauchyRiemann'₆ (h'f w hw)) 1]
      rw [partialDeriv_smul'₂]
      simp

      rw [← smul_assoc]
      simp


theorem holomorphic_is_harmonic {f : ℂ → F₁} (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 (partialDeriv ℝ 1 f) := by
    exact (partialDeriv_contDiff ℝ 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]

    -- This lemma says that partial derivatives commute with complex scalar
    -- multiplication. This is a consequence of partialDeriv_compContLin once we
    -- note that complex scalar multiplication is continuous ℝ-linear.
    have : ∀ v, ∀ s : ℂ, ∀ g : ℂ → F₁, Differentiable ℝ g → partialDeriv ℝ v (s • g) = s • (partialDeriv ℝ v g) := by
      intro v s g hg

      -- Present scalar multiplication as a continuous ℝ-linear map. This is
      -- horrible, there must be better ways to do that.
      let sMuls : F₁ →L[ℝ] F₁ :=
      {
        toFun := fun x ↦ s • x
        map_add' := by exact fun x y => DistribSMul.smul_add s x y
        map_smul' := by exact fun m x => (smul_comm ((RingHom.id ℝ) m) s x).symm
        cont := continuous_const_smul s
      }

      -- Bring the goal into a form that is recognized by
      -- partialDeriv_compContLin.
      have : s • g = sMuls ∘ g := by rfl
      rw [this]

      rw [partialDeriv_compContLin ℝ hg]
      rfl

    rw [this]
    rw [partialDeriv_comm f_is_real_C2 Complex.I 1]
    rw [CauchyRiemann₄ h]
    rw [this]
    rw [← smul_assoc]
    simp

    -- Subgoals coming from the application of 'this'
    -- Differentiable ℝ (Real.partialDeriv 1 f)
    exact fI_is_real_differentiable
    -- Differentiable ℝ (Real.partialDeriv 1 f)
    exact fI_is_real_differentiable


theorem holomorphicOn_is_harmonicOn {f : ℂ → F₁} {s : Set ℂ} (hs : IsOpen s) (h : DifferentiableOn ℂ f s) :
  HarmonicOn f s := by

  -- f is real C²
  have f_is_real_C2 : ContDiffOn ℝ 2 f s :=
    ContDiffOn.restrict_scalars ℝ (DifferentiableOn.contDiffOn h hs)

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

  · -- Laplace of f is zero
    unfold Complex.laplace
    intro z hz
    simp
    have : partialDeriv ℝ Complex.I f =ᶠ[nhds z] Complex.I • partialDeriv ℝ 1 f := by
      unfold Filter.EventuallyEq
      unfold Filter.Eventually
      simp
      refine mem_nhds_iff.mpr ?_
      use s
      constructor
      · intro x hx
        simp
        apply CauchyRiemann₅
        apply DifferentiableOn.differentiableAt h
        exact IsOpen.mem_nhds hs hx
      · constructor
        · exact hs
        · exact hz
    rw [partialDeriv_eventuallyEq ℝ this Complex.I]
    rw [partialDeriv_smul'₂]

    simp
    rw [partialDeriv_commOn hs f_is_real_C2 Complex.I 1 z hz]

    have : partialDeriv ℝ Complex.I f =ᶠ[nhds z] Complex.I • partialDeriv ℝ 1 f := by
      unfold Filter.EventuallyEq
      unfold Filter.Eventually
      simp
      refine mem_nhds_iff.mpr ?_
      use s
      constructor
      · intro x hx
        simp
        apply CauchyRiemann₅
        apply DifferentiableOn.differentiableAt h
        exact IsOpen.mem_nhds hs hx
      · constructor
        · exact hs
        · exact hz
    rw [partialDeriv_eventuallyEq ℝ this 1]
    rw [partialDeriv_smul'₂]
    simp
    rw [← smul_assoc]
    simp


theorem re_of_holomorphic_is_harmonic {f : ℂ → ℂ} (h : Differentiable ℂ f) :
  Harmonic (Complex.reCLM ∘ f) := by
  apply harmonic_comp_CLM_is_harmonic
  exact holomorphic_is_harmonic h


theorem im_of_holomorphic_is_harmonic {f : ℂ → ℂ} (h : Differentiable ℂ f) :
  Harmonic (Complex.imCLM ∘ f) := by
  apply harmonic_comp_CLM_is_harmonic
  exact holomorphic_is_harmonic h


theorem antiholomorphic_is_harmonic {f : ℂ → ℂ} (h : Differentiable ℂ f) :
  Harmonic (Complex.conjCLE ∘ f) := by
  apply harmonic_iff_comp_CLE_is_harmonic.1
  exact holomorphic_is_harmonic h


theorem log_normSq_of_holomorphicOn_is_harmonicOn'
  {f : ℂ → ℂ}
  {s : Set ℂ}
  (hs : IsOpen s)
  (h₁ : DifferentiableOn ℂ f s)
  (h₂ : ∀ z ∈ s, f z ≠ 0)
  (h₃ : ∀ z ∈ s, f z ∈ Complex.slitPlane) :
  HarmonicOn (Real.log ∘ Complex.normSq ∘ f) s := by

  suffices hyp : HarmonicOn (⇑Complex.ofRealCLM ∘ Real.log ∘ Complex.normSq ∘ f) s from
    (harmonicOn_comp_CLM_is_harmonicOn hs hyp : HarmonicOn (Complex.reCLM ∘ Complex.ofRealCLM ∘ Real.log ∘ Complex.normSq ∘ f) s)

  suffices hyp : HarmonicOn (Complex.log ∘ (((starRingEnd ℂ) ∘ f) * f)) s from by
    have : Complex.ofRealCLM ∘ Real.log ∘ Complex.normSq ∘ f = Complex.log ∘ (((starRingEnd ℂ) ∘ f) * f) := by
      funext z
      simp
      rw [Complex.ofReal_log (Complex.normSq_nonneg (f z))]
      rw [Complex.normSq_eq_conj_mul_self]
    rw [this]
    exact hyp


  -- Suffices to show Harmonic (Complex.log ∘ ⇑(starRingEnd ℂ) ∘ f + Complex.log ∘ f)
  -- THIS IS WHERE WE USE h₃
  have : ∀ z ∈ s, (Complex.log ∘ (⇑(starRingEnd ℂ) ∘ f * f)) z = (Complex.log ∘ ⇑(starRingEnd ℂ) ∘ f + Complex.log ∘ f) z := by
    intro z hz
    unfold Function.comp
    simp
    rw [Complex.log_mul_eq_add_log_iff]

    have : Complex.arg ((starRingEnd ℂ) (f z)) = - Complex.arg (f z) := by
      rw [Complex.arg_conj]
      have : ¬ Complex.arg (f z) = Real.pi := by
        exact Complex.slitPlane_arg_ne_pi (h₃ z hz)
      simp
      tauto
    rw [this]
    simp
    constructor
    · exact Real.pi_pos
    · exact Real.pi_nonneg
    exact (AddEquivClass.map_ne_zero_iff starRingAut).mpr (h₂ z hz)
    exact h₂ z hz

  rw [HarmonicOn_congr hs this]
  simp

  apply harmonicOn_add_harmonicOn_is_harmonicOn hs

  have : (fun x => Complex.log ((starRingEnd ℂ) (f x))) = (Complex.log ∘ ⇑(starRingEnd ℂ) ∘ f) := by
    rfl
  rw [this]

  -- HarmonicOn (Complex.log ∘ ⇑(starRingEnd ℂ) ∘ f) s
  have : ∀ z ∈ s, (Complex.log ∘ ⇑(starRingEnd ℂ) ∘ f) z = (Complex.conjCLE ∘ Complex.log ∘ f) z := by
    intro z hz
    unfold Function.comp
    rw [Complex.log_conj]
    rfl
    exact Complex.slitPlane_arg_ne_pi (h₃ z hz)
  rw [HarmonicOn_congr hs this]

  rw [← harmonicOn_iff_comp_CLE_is_harmonicOn]

  apply holomorphicOn_is_harmonicOn
  exact hs

  intro z hz
  apply DifferentiableAt.differentiableWithinAt
  apply DifferentiableAt.comp



  exact Complex.differentiableAt_log (h₃ z hz)
  apply DifferentiableOn.differentiableAt h₁ -- (h₁ z hz)
  exact IsOpen.mem_nhds hs hz
  exact hs

  -- HarmonicOn (Complex.log ∘ ⇑(starRingEnd ℂ) ∘ f) s
  apply holomorphicOn_is_harmonicOn hs
  exact DifferentiableOn.clog h₁ h₃


theorem log_normSq_of_holomorphicOn_is_harmonicOn
  {f : ℂ → ℂ}
  {s : Set ℂ}
  (hs : IsOpen s)
  (h₁ : DifferentiableOn ℂ f s)
  (h₂ : ∀ z ∈ s, f z ≠ 0) :
  HarmonicOn (Real.log ∘ Complex.normSq ∘ f) s := by

  have slitPlaneLemma {z : ℂ} (hz : z ≠ 0) : z ∈ Complex.slitPlane ∨ -z ∈ Complex.slitPlane := by
    rw [Complex.mem_slitPlane_iff]
    rw [Complex.mem_slitPlane_iff]
    simp at hz
    rw [Complex.ext_iff] at hz
    push_neg at hz
    simp at hz
    simp
    by_contra contra
    push_neg at contra
    exact hz (le_antisymm contra.1.1 contra.2.1) contra.1.2

  let s₁ : Set ℂ := { z | f z ∈ Complex.slitPlane} ∩ s

  have hs₁ : IsOpen s₁ := by
    let A := DifferentiableOn.continuousOn h₁
    let B := continuousOn_iff'.1 A
    obtain ⟨u, hu₁, hu₂⟩ := B Complex.slitPlane Complex.isOpen_slitPlane
    have : u ∩ s = s₁ := by
      rw [← hu₂]
      tauto
    rw [← this]
    apply IsOpen.inter hu₁ hs

  have harm₁ : HarmonicOn (Real.log ∘ Complex.normSq ∘ f) s₁ := by
    apply log_normSq_of_holomorphicOn_is_harmonicOn'
    exact hs₁
    apply DifferentiableOn.mono h₁ (Set.inter_subset_right {z | f z ∈ Complex.slitPlane} s)
    -- ∀ z ∈ s₁, f z ≠ 0
    exact fun z hz ↦ h₂ z (Set.mem_of_mem_inter_right hz)
    -- ∀ z ∈ s₁, f z ∈ Complex.slitPlane
    intro z hz
    apply hz.1

  let s₂ : Set ℂ := { z | -f z ∈ Complex.slitPlane} ∩ s

  have h₁' : DifferentiableOn ℂ (-f) s := by
    rw [← differentiableOn_neg_iff]
    simp
    exact h₁

  have hs₂ : IsOpen s₂ := by
    let A := DifferentiableOn.continuousOn h₁'
    let B := continuousOn_iff'.1 A
    obtain ⟨u, hu₁, hu₂⟩ := B Complex.slitPlane Complex.isOpen_slitPlane
    have : u ∩ s = s₂ := by
      rw [← hu₂]
      tauto
    rw [← this]
    apply IsOpen.inter hu₁ hs

  have harm₂ : HarmonicOn (Real.log ∘ Complex.normSq ∘ (-f)) s₂ := by
    apply log_normSq_of_holomorphicOn_is_harmonicOn'
    exact hs₂
    apply DifferentiableOn.mono h₁' (Set.inter_subset_right {z | -f z ∈ Complex.slitPlane} s)
    -- ∀ z ∈ s₁, f z ≠ 0
    intro z hz
    simp
    exact h₂ z (Set.mem_of_mem_inter_right hz)
    -- ∀ z ∈ s₁, f z ∈ Complex.slitPlane
    intro z hz
    apply hz.1

  apply HarmonicOn_of_locally_HarmonicOn
  intro z hz
  by_cases hfz : f z ∈ Complex.slitPlane
  · use s₁
    constructor
    · exact hs₁
    · constructor
      · tauto
      · have : s₁ = s ∩ s₁ := by
          apply Set.right_eq_inter.mpr
          exact Set.inter_subset_right {z | f z ∈ Complex.slitPlane} s
        rw [← this]
        exact harm₁
  · use s₂
    constructor
    · exact hs₂
    · constructor
      · constructor
        · apply Or.resolve_left (slitPlaneLemma (h₂ z hz)) hfz
        · exact hz
      · have : s₂ = s ∩ s₂ := by
          apply Set.right_eq_inter.mpr
          exact Set.inter_subset_right {z | -f z ∈ Complex.slitPlane} s
        rw [← this]
        have : Real.log ∘ ⇑Complex.normSq ∘ f = Real.log ∘ ⇑Complex.normSq ∘ (-f) := by
          funext x
          simp
        rw [this]
        exact harm₂


theorem log_normSq_of_holomorphic_is_harmonic
  {f : ℂ → ℂ}
  (h₁ : Differentiable ℂ f)
  (h₂ : ∀ z, f z ≠ 0)
  (h₃ : ∀ z, f z ∈ Complex.slitPlane) :
  Harmonic (Real.log ∘ Complex.normSq ∘ f) := by

  suffices hyp : Harmonic (⇑Complex.ofRealCLM ∘ Real.log ∘ Complex.normSq ∘ f) from
    (harmonic_comp_CLM_is_harmonic hyp : Harmonic (Complex.reCLM ∘ Complex.ofRealCLM ∘ Real.log ∘ Complex.normSq ∘ f))

  suffices hyp : Harmonic (Complex.log ∘ (((starRingEnd ℂ) ∘ f) * f)) from by
    have : Complex.ofRealCLM ∘ Real.log ∘ Complex.normSq ∘ f = Complex.log ∘ (((starRingEnd ℂ) ∘ f) * f) := by
      funext z
      simp
      rw [Complex.ofReal_log (Complex.normSq_nonneg (f z))]
      rw [Complex.normSq_eq_conj_mul_self]
    rw [this]
    exact hyp


  -- Suffices to show Harmonic (Complex.log ∘ ⇑(starRingEnd ℂ) ∘ f + Complex.log ∘ f)
  -- THIS IS WHERE WE USE h₃
  have : Complex.log ∘ (⇑(starRingEnd ℂ) ∘ f * f) = Complex.log ∘ ⇑(starRingEnd ℂ) ∘ f + Complex.log ∘ f := by
    unfold Function.comp
    funext z
    simp
    rw [Complex.log_mul_eq_add_log_iff]

    have : Complex.arg ((starRingEnd ℂ) (f z)) = - Complex.arg (f z) := by
      rw [Complex.arg_conj]
      have : ¬ Complex.arg (f z) = Real.pi := by
        exact Complex.slitPlane_arg_ne_pi (h₃ z)
      simp
      tauto
    rw [this]
    simp
    constructor
    · exact Real.pi_pos
    · exact Real.pi_nonneg
    exact (AddEquivClass.map_ne_zero_iff starRingAut).mpr (h₂ z)
    exact h₂ z
  rw [this]

  apply harmonic_add_harmonic_is_harmonic
  have : Complex.log ∘ ⇑(starRingEnd ℂ) ∘ f = Complex.conjCLE ∘ Complex.log ∘ f := by
    funext z
    unfold Function.comp
    rw [Complex.log_conj]
    rfl
    exact Complex.slitPlane_arg_ne_pi (h₃ z)
  rw [this]
  rw [← harmonic_iff_comp_CLE_is_harmonic]

  repeat
    apply holomorphic_is_harmonic
    intro z
    apply DifferentiableAt.comp
    exact Complex.differentiableAt_log (h₃ z)
    exact h₁ z


theorem logabs_of_holomorphic_is_harmonic
  {f : ℂ → ℂ}
  (h₁ : Differentiable ℂ f)
  (h₂ : ∀ z, f z ≠ 0)
  (h₃ : ∀ z, f z ∈ Complex.slitPlane) :
  Harmonic (fun z ↦ Real.log ‖f z‖) := by

  -- Suffices: Harmonic (2⁻¹ • Real.log ∘ ⇑Complex.normSq ∘ f)
  have : (fun z ↦ Real.log ‖f z‖) = (2 : ℝ)⁻¹ • (Real.log ∘ Complex.normSq ∘ f) := by
    funext z
    simp
    unfold Complex.abs
    simp
    rw [Real.log_sqrt]
    rw [div_eq_inv_mul (Real.log (Complex.normSq (f z))) 2]
    exact Complex.normSq_nonneg (f z)
  rw [this]

  -- Suffices: Harmonic (Real.log ∘ ⇑Complex.normSq ∘ f)
  apply (harmonic_iff_smul_const_is_harmonic (inv_ne_zero two_ne_zero)).1

  exact log_normSq_of_holomorphic_is_harmonic h₁ h₂ h₃