lean/OAI/Analysis/DirectCrouzeix/DividedDifferences.lean

import OAI.Analysis.DirectCrouzeix.CompactIntegrals

namespace OAI

noncomputable section

open scoped Matrix Matrix.Norms.L2Operator Kronecker

noncomputable section

open MeasureTheory Set Filter Metric

open scoped Topology Interval ENNReal NNReal ComplexConjugate

namespace DirectCrouzeix.Faber

def divided (g : ℂ → ℂ) (u v : ℂ) : ℂ := chordIntegral (deriv g) (fun _ => 1) u v

def dividedLeft (g : ℂ → ℂ) (u v : ℂ) : ℂ :=
  chordIntegral (deriv (deriv g)) (fun t => (1-(t:ℝ) : ℝ)) u v

def chordDenominator (c : ℂ) (g : ℂ → ℂ) (u v : ℂ) : ℂ := c-u*v*divided g u v

def correction (c : ℂ) (g : ℂ → ℂ) (u v : ℂ) : ℂ :=
  u*v*(divided g u v+u*dividedLeft g u v) / chordDenominator c g u v

theorem divided_eq_dslope {s : Set ℂ} (hconv : Convex ℝ s)
    {g : ℂ → ℂ} (hg : AnalyticOnNhd ℂ g s) {u v : ℂ} (hu : u ∈ s) (hv : v ∈ s) :
    divided g u v = dslope g v u := chordIntegral_deriv_eq_dslope hconv hg hu hv

theorem hasDerivAt_divided_left {s : Set ℂ} (hs : IsOpen s) (hconv : Convex ℝ s)
    {g : ℂ → ℂ} (hg : AnalyticOnNhd ℂ g s) {u v : ℂ} (hu : u ∈ s) (hv : v ∈ s) :
    HasDerivAt (fun z => divided g z v) (dividedLeft g u v) u := by
  simpa only [divided, dividedLeft, one_mul] using
    hasDerivAt_chordIntegral_left hs hconv hg.deriv (continuous_const (y := (1:ℂ))) hu hv

theorem continuousOn_correction {s : Set ℂ} (hs : IsOpen s) (hconv : Convex ℝ s)
    {g : ℂ → ℂ} (hg : AnalyticOnNhd ℂ g s) {c : ℂ}
    (hH : ∀ u ∈ s, ∀ v ∈ s, chordDenominator c g u v ≠ 0) :
    ContinuousOn (fun p : ℂ × ℂ => correction c g p.1 p.2) (s ×ˢ s) := by
  have hD := continuousOn_chordIntegral hs hconv hg.deriv.continuousOn (continuous_const (y := (1:ℂ)))
  have hE := continuousOn_chordIntegral hs hconv hg.deriv.deriv.continuousOn
    (show Continuous (fun t : Parameter => ((1-(t:ℝ) : ℝ):ℂ)) by fun_prop)
  exact ((continuous_fst.mul continuous_snd).continuousOn.mul
    (hD.add (continuous_fst.continuousOn.mul hE))).div
    (continuousOn_const.sub ((continuous_fst.mul continuous_snd).continuousOn.mul hD))
    (fun p hp => hH p.1 hp.1 p.2 hp.2)

theorem analyticOnNhd_correction_left {s : Set ℂ} (hs : IsOpen s) (hconv : Convex ℝ s)
    {g : ℂ → ℂ} (hg : AnalyticOnNhd ℂ g s) {c v : ℂ} (hv : v ∈ s)
    (hH : ∀ u ∈ s, chordDenominator c g u v ≠ 0) :
    AnalyticOnNhd ℂ (fun u => correction c g u v) s := by
  have hD := analyticOnNhd_chordIntegral_left hs hconv hg.deriv (continuous_const (y := (1:ℂ))) hv
  have hE := analyticOnNhd_chordIntegral_left hs hconv hg.deriv.deriv
    (show Continuous (fun t : Parameter => ((1-(t:ℝ) : ℝ):ℂ)) by fun_prop) hv
  intro u hu
  exact (((analyticAt_id.mul analyticAt_const).mul ((hD u hu).add (analyticAt_id.mul (hE u hu)))).div
    (analyticAt_const.sub ((analyticAt_id.mul analyticAt_const).mul (hD u hu))) (hH u hu))

theorem analyticOnNhd_correction_right {s : Set ℂ} (hs : IsOpen s) (hconv : Convex ℝ s)
    {g : ℂ → ℂ} (hg : AnalyticOnNhd ℂ g s) {c u : ℂ} (hu : u ∈ s)
    (hH : ∀ v ∈ s, chordDenominator c g u v ≠ 0) :
    AnalyticOnNhd ℂ (correction c g u) s := by
  have hD := analyticOnNhd_chordIntegral_right hs hconv hg.deriv (continuous_const (y := (1:ℂ))) hu
  have hE := analyticOnNhd_chordIntegral_right hs hconv hg.deriv.deriv
    (show Continuous (fun t : Parameter => ((1-(t:ℝ) : ℝ):ℂ)) by fun_prop) hu
  intro v hv
  exact (((analyticAt_const.mul analyticAt_id).mul ((hD v hv).add (analyticAt_const.mul (hE v hv)))).div
    (analyticAt_const.sub ((analyticAt_const.mul analyticAt_id).mul (hD v hv))) (hH v hv))

@[simp] theorem correction_zero_left (c : ℂ) (g : ℂ → ℂ) (v : ℂ) : correction c g 0 v = 0 := by
  simp [correction]

@[simp] theorem correction_zero_right (c : ℂ) (g : ℂ → ℂ) (u : ℂ) : correction c g u 0 = 0 := by
  simp [correction]

def exterior (c : ℂ) (g : ℂ → ℂ) (ζ : ℂ) := c*ζ+g ζ⁻¹

theorem hasDerivAt_exterior {s : Set ℂ} {g : ℂ → ℂ} (hg : AnalyticOnNhd ℂ g s)
    (c : ℂ) {ζ : ℂ} (hζ : ζ ≠ 0) (hζs : ζ⁻¹ ∈ s) :
    HasDerivAt (exterior c g) (c-ζ⁻¹^2*deriv g ζ⁻¹) ζ := by
  have hd := ((hasDerivAt_id ζ).const_mul c).add
    ((hg ζ⁻¹ hζs).differentiableAt.hasDerivAt.comp ζ (hasDerivAt_inv hζ))
  change HasDerivAt (fun z => exterior c g z) _ ζ
  simpa only [exterior, Function.comp_def, id_eq, mul_one, Pi.add_apply, inv_pow,
    div_eq_mul_inv, sub_eq_add_neg, mul_neg, mul_comm, mul_left_comm] using! hd

theorem chordDenominator_off_diagonal {s : Set ℂ} (hconv : Convex ℝ s)
    {g : ℂ → ℂ} (hg : AnalyticOnNhd ℂ g s) (c : ℂ) {u v : ℂ}
    (hu : u ∈ s) (hv : v ∈ s) (hu0 : u ≠ 0) (hv0 : v ≠ 0) (huv : u ≠ v) :
    chordDenominator c g u v = (exterior c g u⁻¹ - exterior c g v⁻¹) / (u⁻¹-v⁻¹) := by
  rw [chordDenominator, divided_eq_dslope hconv hg hu hv, dslope_of_ne g huv]
  simp only [slope, smul_eq_mul, vsub_eq_sub, exterior, inv_inv]
  field_simp
  ring

theorem chordDenominator_diagonal {s : Set ℂ} (hconv : Convex ℝ s)
    {g : ℂ → ℂ} (hg : AnalyticOnNhd ℂ g s) (c : ℂ) {u : ℂ} (hu : u ∈ s) :
    chordDenominator c g u u = c-u^2*deriv g u := by
  rw [chordDenominator, divided_eq_dslope hconv hg hu hu, dslope_same]
  ring

theorem chordDenominator_ne_zero {s : Set ℂ} (hconv : Convex ℝ s)
    {g : ℂ → ℂ} (hg : AnalyticOnNhd ℂ g s) {c : ℂ} (hc : c ≠ 0)
    (hinj : InjOn (exterior c g) (Inv.inv ⁻¹' s \ {0}))
    (hd : ∀ ζ, ζ ≠ 0 → ζ⁻¹ ∈ s → deriv (exterior c g) ζ ≠ 0)
    {u v : ℂ} (hu : u ∈ s) (hv : v ∈ s) : chordDenominator c g u v ≠ 0 := by
  by_cases hu0 : u = 0
  · simp [chordDenominator, hu0, hc]
  by_cases hv0 : v = 0
  · simp [chordDenominator, hv0, hc]
  by_cases huv : u = v
  · subst v
    rw [chordDenominator_diagonal hconv hg c hu]
    have hder := (hasDerivAt_exterior hg c (inv_ne_zero hu0) (by simpa using hu)).deriv
    have hn := hd u⁻¹ (inv_ne_zero hu0) (by simpa using hu)
    rwa [hder,inv_inv] at hn
  · rw [chordDenominator_off_diagonal hconv hg c hu hv hu0 hv0 huv]
    exact div_ne_zero (sub_ne_zero.mpr (fun he => huv (inv_injective (hinj
      ⟨by simpa using hu, by simpa using hu0⟩ ⟨by simpa using hv, by simpa using hv0⟩ he))))
      (sub_ne_zero.mpr (fun he => huv (inv_injective he)))

theorem divided_fundamental {s : Set ℂ} (hconv : Convex ℝ s)
    {g : ℂ → ℂ} (hg : AnalyticOnNhd ℂ g s) {u v : ℂ} (hu : u ∈ s) (hv : v ∈ s) :
    (u-v)*divided g u v = g u-g v := by
  have hh := chordIntegral_deriv_mul hconv hg hu hv
  change divided g u v * (v-u) = _ at hh
  linear_combination -hh

theorem divided_derivative_relation {s : Set ℂ} (hs : IsOpen s) (hconv : Convex ℝ s)
    {g : ℂ → ℂ} (hg : AnalyticOnNhd ℂ g s) {u v : ℂ} (hu : u ∈ s) (hv : v ∈ s) :
    divided g u v + (u-v)*dividedLeft g u v = deriv g u := by
  have hd := ((hasDerivAt_id u).sub_const v).mul (hasDerivAt_divided_left hs hconv hg hu hv)
  have he : (fun z => g z-g v) =ᶠ[𝓝 u] (fun z => (z-v)*divided g z v) := by
    filter_upwards [hs.mem_nhds hu] with z hz
    exact (divided_fundamental hconv hg hz hv).symm
  have hid := (hd.congr_of_eventuallyEq he).unique ((hg u hu).differentiableAt.hasDerivAt.sub_const (g v))
  simpa using hid

theorem correction_identity {s : Set ℂ} (hs : IsOpen s) (hconv : Convex ℝ s)
    {g : ℂ → ℂ} (hg : AnalyticOnNhd ℂ g s) (c : ℂ) {u v : ℂ}
    (hu : u ∈ s) (hv : v ∈ s) (hv0 : v ≠ 0) (huv : u ≠ v)
    (hH : chordDenominator c g u v ≠ 0) :
    correction c g u v = (c-u^2*deriv g u)/(c+u*(g u-exterior c g v⁻¹)) - (1-u/v)⁻¹ := by
  have hD := divided_fundamental hconv hg hu hv
  have hE := divided_derivative_relation hs hconv hg hu hv
  have huv' : v-u ≠ 0 := sub_ne_zero.mpr (Ne.symm huv)
  have he : c+u*(g u-exterior c g v⁻¹) = chordDenominator c g u v * (v-u)/v := by
    simp only [exterior,inv_inv,chordDenominator]
    field_simp
    linear_combination -u*v*hD
  rw [he,correction]
  have hd0 : chordDenominator c g u v * (v-u) / v ≠ 0 := div_ne_zero (mul_ne_zero hH huv') hv0
  field_simp
  simp only [chordDenominator]
  linear_combination -u^2*hE

end DirectCrouzeix.Faber

end

end

end OAI