lean/OAI/Geometry/Arnold/MidpointGeometry.lean

import OAI.Geometry.Arnold.Midpoint

namespace OAI

noncomputable section
open scoped BigOperators ContDiff
open scoped Topology NNReal
open Set Filter

namespace ArnoldCounterexample.Field
open ArnoldCounterexample.Quadric ArnoldCounterexample.ThreePoints ArnoldCounterexample.Factors
open scoped ContDiff Topology NNReal
open Set Filter

lemma normSquared_add (z w : CV) : normSquared (z+w) =
    normSquared z + 2 * (bilinear (star z) w).re + normSquared w := by
  simp only [normSquared, bilinear, Pi.add_apply, Pi.star_apply, Complex.normSq_add,
    Complex.re_sum, Finset.sum_add_distrib, Finset.mul_sum]
  have h : ∑ i : Fin 5, 2 * (z i * (starRingEnd ℂ) (w i)).re =
      ∑ i : Fin 5, 2 * (star (z i) * w i).re := by
    apply Finset.sum_congr rfl
    intro i _
    simp [Complex.mul_re]
  rw [h]
  ring

lemma real_smul_complex (s : ℝ) (z : CV) : s • z = (s : ℂ) • z := by
  ext i; simp [Complex.real_smul]

lemma normSquared_real_smul (s : ℝ) (z : CV) : normSquared (s • z) = s^2 * normSquared z := by
  rw [real_smul_complex, normSquared_smul, Complex.normSq_ofReal]
  ring

lemma quadricPolynomial_real_smul (s : ℝ) (z : CV) :
    quadricPolynomial (s • z) = (s : ℂ)^2 * quadricPolynomial z := by
  rw [real_smul_complex, quadricPolynomial_smul]

lemma quadratic_skew (z v : CV) (hv : SkewAt z v) (s : ℝ) :
    normSquared (z+s•v) = normSquared z + s^2 * normSquared v ∧
    quadricPolynomial (z+s•v) = quadricPolynomial z + (s:ℂ)^2 * quadricPolynomial v := by
  constructor
  · rw [normSquared_add, normSquared_real_smul, real_smul_complex, bilinear_smul_right]
    simp [Complex.mul_re, hv.1]
  · rw [quadricPolynomial_add, quadricPolynomial_real_smul, real_smul_complex, bilinear_smul_right, hv.2]
    ring

lemma normSquared_le_maxNorm (z : CV) : normSquared z ≤ 5 * ‖z‖^2 := by
  calc
    normSquared z ≤ ∑ i : Fin 5, ‖z‖^2 := by
      apply Finset.sum_le_sum
      intro i _
      rw [Complex.normSq_eq_norm_sq]
      exact pow_le_pow_left₀ (norm_nonneg _) (norm_le_pi_norm z i) 2
    _ = _ := by simp

lemma norm_le_one_of_normSquared {z : CV} (hz : normSquared z ≤ 1) : ‖z‖ ≤ 1 :=
  norm_bound_of_normSquared (by norm_num) (by simpa using hz)

variable {C : ℝ≥0} (hX : LipschitzWith C vectorField) (τ : ℝ → ℝ)
  (hτ : ∀ t, ‖τ t‖ * C < 1)

lemma center_quadratic (p : ℝ × CV) :
    normSquared p.2 = normSquared (Midpoint.resolvent vectorField hX τ hτ p) +
      (τ p.1)^2 * normSquared (vectorField (Midpoint.resolvent vectorField hX τ hτ p)) := by
  let u := Midpoint.resolvent vectorField hX τ hτ p
  have he : p.2 = u + (-τ p.1) • vectorField u := by
    have hh := Midpoint.resolvent_eq vectorField hX τ hτ p
    change u = p.2 + τ p.1 • vectorField u at hh
    rw [neg_smul]
    simpa only [sub_eq_add_neg] using (eq_sub_iff_add_eq.mpr hh.symm)
  rw [he, (quadratic_skew u _ (vectorField_skew u) _).1, neg_sq]

lemma midpoint_preserves (t : ℝ) (z : UnitQuadric) :
    IsUnitQuadric (Midpoint.step vectorField hX τ hτ (t,z)) := by
  let u := Midpoint.resolvent vectorField hX τ hτ (t,z)
  have he : (z : CV) = u + (-τ t) • vectorField u := by
    have hh := Midpoint.resolvent_eq vectorField hX τ hτ (t,z)
    change u = (z:CV) + τ t • vectorField u at hh
    rw [neg_smul]
    simpa only [sub_eq_add_neg] using (eq_sub_iff_add_eq.mpr hh.symm)
  have hp := (quadratic_skew u _ (vectorField_skew u) (-τ t)).2
  rw [← he, z.property.2, Complex.ofReal_neg, neg_sq] at hp
  rw [Midpoint.step_center]
  exact ⟨((quadratic_skew u _ (vectorField_skew u) (τ t)).1).trans
    ((center_quadratic hX τ hτ (t,z)).symm.trans z.property.1),
    ((quadratic_skew u _ (vectorField_skew u) (τ t)).2).trans hp.symm⟩

lemma center_norm_le_one (t : ℝ) (z : UnitQuadric) :
    normSquared (Midpoint.resolvent vectorField hX τ hτ (t,z)) ≤ 1 := by
  have he := center_quadratic hX τ hτ (t,z)
  rw [z.property.1] at he
  nlinarith [mul_nonneg (sq_nonneg (τ t)) (normSquared_nonneg
    (vectorField (Midpoint.resolvent vectorField hX τ hτ (t,z))))]

lemma center_norm_gt_half (hsmall : ∀ t, ‖τ t‖ * C < 1/10) (t : ℝ) (z : UnitQuadric) :
    1/2 < normSquared (Midpoint.resolvent vectorField hX τ hτ (t,z)) := by
  let u := Midpoint.resolvent vectorField hX τ hτ (t,z)
  have hu : ‖u‖ ≤ 1 := norm_le_one_of_normSquared (center_norm_le_one hX τ hτ t z)
  have hXb : ‖vectorField u‖ ≤ C := by
    have h := hX.dist_le_mul u 0
    rw [vectorField_zero, dist_zero_right, dist_zero_right] at h
    exact h.trans (by nlinarith [C.coe_nonneg])
  have hb : normSquared (vectorField u) ≤ 5 * (C:ℝ)^2 :=
    (normSquared_le_maxNorm _).trans (by nlinarith [norm_nonneg (vectorField u)])
  have he := center_quadratic hX τ hτ (t,z)
  change normSquared (z:CV) = normSquared u + (τ t)^2 * normSquared (vectorField u) at he
  rw [z.property.1] at he
  have hs := hsmall t
  have hn : 0 ≤ ‖τ t‖ * (C:ℝ) := mul_nonneg (norm_nonneg _) C.coe_nonneg
  have hsq : (τ t)^2 * (C:ℝ)^2 < 1/100 := by
    simp only [Real.norm_eq_abs] at hs hn
    nlinarith [sq_abs (τ t)]
  have hmul := mul_le_mul_of_nonneg_left hb (sq_nonneg (τ t))
  nlinarith

end ArnoldCounterexample.Field

end

end OAI