Datasets:

roseking
/

openAI-math-dataset

import Mathlib
import OAI.Probability.GaussianInformation.Dyadic.Covers

namespace OAI
open MeasureTheory ProbabilityTheory Set
open scoped BigOperators ENNReal
open MeasureTheory Metric Set
open scoped ENNReal NNReal
namespace RepeatedLocalization

def euclideanGridCell {n : ℕ} (base : Fin n → ℝ) (h : ℝ) (a : Fin n → ℤ) :
    Set (EuclideanSpace ℝ (Fin n)) :=
  {x | ∀ i, base i + h * a i ≤ x i ∧ x i < base i + h * (a i + 1)}

lemma measurableSet_euclideanGridCell {n : ℕ} (base : Fin n → ℝ)
    (h : ℝ) (a : Fin n → ℤ) : MeasurableSet (euclideanGridCell base h a) := by
  unfold euclideanGridCell
  simp only [ofPred_forall]
  apply MeasurableSet.iInter
  intro i
  have hm : Measurable (fun x : EuclideanSpace ℝ (Fin n) => x i) :=
    (measurable_pi_apply i).comp (WithLp.measurable_ofLp 2 (Fin n → ℝ))
  exact (measurableSet_le (measurable_const (a := base i + h * a i)) hm).inter
    (measurableSet_lt hm (measurable_const (a := base i + h * (a i + 1))))

lemma volume_euclideanGridCell {n : ℕ} (base : Fin n → ℝ)
    {h : ℝ} (hh : 0 ≤ h) (a : Fin n → ℤ) :
    volume (euclideanGridCell base h a) = ENNReal.ofReal (h ^ n) := by
  rw [← (PiLp.volume_preserving_toLp (Fin n)).measure_preimage
    (measurableSet_euclideanGridCell base h a).nullMeasurableSet]
  have he : WithLp.toLp 2 ⁻¹' euclideanGridCell base h a =
      Set.pi Set.univ (fun i => Ico (base i + h * a i) (base i + h * (a i + 1))) := by
    ext x
    simp [euclideanGridCell]
  rw [he, Real.volume_pi_Ico]
  have hf : ∀ i, base i + h * (a i + 1) - (base i + h * a i) = h := by
    intro i
    ring
  simp only [hf, Finset.prod_const, Finset.card_univ, Fintype.card_fin,
    ← ENNReal.ofReal_pow hh]

lemma euclideanGridCell_disjoint {n : ℕ} (base : Fin n → ℝ) {h : ℝ} (hh : 0 < h)
    {a b : Fin n → ℤ} (hab : a ≠ b) :
    Disjoint (euclideanGridCell base h a) (euclideanGridCell base h b) := by
  rw [Function.ne_iff] at hab
  obtain ⟨i, hi⟩ := hab
  apply Set.disjoint_left.mpr
  intro x hx hy
  obtain hl | hl := lt_or_gt_of_ne hi
  · have ht : (a i : ℝ) + 1 ≤ b i := by exact_mod_cast hl
    have hs := mul_le_mul_of_nonneg_left ht hh.le
    linarith [(hx i).2, (hy i).1]
  · have ht : (b i : ℝ) + 1 ≤ a i := by exact_mod_cast hl
    have hs := mul_le_mul_of_nonneg_left ht hh.le
    linarith [(hy i).2, (hx i).1]

lemma dist_le_of_mem_euclideanGridCell {n : ℕ} (base : Fin n → ℝ)
    {h : ℝ} (hh : 0 ≤ h) (a : Fin n → ℤ)
    {x y : EuclideanSpace ℝ (Fin n)} (hx : x ∈ euclideanGridCell base h a)
    (hy : y ∈ euclideanGridCell base h a) : dist x y ≤ Real.sqrt n * h := by
  have hsq : ‖x - y‖ ^ 2 ≤ (n : ℝ) * h^2 := by
    rw [EuclideanSpace.real_norm_sq_eq]
    calc
      _ ≤ ∑ i : Fin n, h^2 := Finset.sum_le_sum fun i _ => by
        have hx1 := (hx i).1
        have hx2 := (hx i).2
        have hy1 := (hy i).1
        have hy2 := (hy i).2
        change (x i - y i)^2 ≤ h^2
        have hab : |x i - y i| ≤ h := abs_le.mpr ⟨by nlinarith, by nlinarith⟩
        nlinarith [sq_abs (x i - y i)]
      _ = _ := by simp
  rw [dist_eq_norm]
  have hn : 0 ≤ (n : ℝ) := Nat.cast_nonneg _
  have hs := Real.sq_sqrt hn
  have hnn := Real.sqrt_nonneg (n : ℝ)
  have hm : 0 ≤ Real.sqrt n * h := mul_nonneg hnn hh
  nlinarith [norm_nonneg (x - y)]

lemma euclidean_closedBall_volume_le (n : ℕ) (hn : 0 < n)
    (x : EuclideanSpace ℝ (Fin n)) {r : ℝ} (hr : 0 < r) :
    volume.real (closedBall x r) ≤
      (r * (Real.exp 1 * Real.sqrt Real.pi) / Real.sqrt n)^n := by
  have : Nonempty (Fin n) := Fin.pos_iff_nonempty.mp hn
  rw [measureReal_def, volume.addHaar_closedBall_eq_addHaar_ball]
  exact euclidean_ball_volume_le n hn x hr

lemma grid_ball_count {n : ℕ} (hn : 0 < n) (base : Fin n → ℝ)
    {h R L : ℝ} (hh : 0 < h) (hR : 0 ≤ R) (_ : 0 ≤ L)
    (hscale : R ≤ L * (Real.sqrt n * h)) (c : EuclideanSpace ℝ (Fin n))
    (s : Finset (Fin n → ℤ))
    (hs : ∀ a ∈ s, (euclideanGridCell base h a ∩ closedBall c R).Nonempty) :
    (s.card : ℝ) ≤ ((L + 1) * (Real.exp 1 * Real.sqrt Real.pi))^n := by
  classical
  let A := ⋃ a ∈ s, euclideanGridCell base h a
  have hsub : A ⊆ closedBall c (R + Real.sqrt n * h) := by
    intro x hx
    obtain ⟨a, ha, hxa⟩ := mem_iUnion₂.mp hx
    obtain ⟨y, hya, hy⟩ := hs a ha
    have hdist := dist_le_of_mem_euclideanGridCell base hh.le a hxa hya
    exact (dist_triangle x y c).trans (add_le_add hdist hy) |>.trans_eq (add_comm _ _)
  have hmeasure : volume A = (s.card : ℝ≥0∞) * ENNReal.ofReal (h^n) := by
    rw [measure_biUnion_finset
      (fun a _ b _ hab => euclideanGridCell_disjoint base hh hab)
      (fun a _ => measurableSet_euclideanGridCell base h a)]
    simp [volume_euclideanGridCell base hh.le]
  have hra : 0 < R + Real.sqrt n * h := by positivity
  have hfin : volume (closedBall c (R + Real.sqrt n * h)) ≠ ∞ := measure_closedBall_lt_top.ne
  have hreal : (s.card : ℝ) * h^n ≤ volume.real (closedBall c (R + Real.sqrt n * h)) := by
    have hm := ENNReal.toReal_mono hfin (measure_mono hsub)
    rw [hmeasure] at hm
    simpa [measureReal_def, ENNReal.toReal_ofReal (pow_nonneg hh.le n)] using hm
  have hC : 0 ≤ Real.exp 1 * Real.sqrt Real.pi := by positivity
  have hn' : 0 < Real.sqrt (n : ℝ) := by positivity
  have hscaled : (R + Real.sqrt n * h) * (Real.exp 1 * Real.sqrt Real.pi) / Real.sqrt n ≤
      h * ((L + 1) * (Real.exp 1 * Real.sqrt Real.pi)) := by
    apply (div_le_iff₀ hn').mpr
    calc
      _ ≤ (L * (Real.sqrt n * h) + Real.sqrt n * h) * (Real.exp 1 * Real.sqrt Real.pi) :=
        mul_le_mul_of_nonneg_right (add_le_add hscale le_rfl) hC
      _ = _ := by ring
  have hcard := hreal.trans (euclidean_closedBall_volume_le n hn c hra)
  have hpow := pow_le_pow_left₀ (by positivity : 0 ≤
    (R + Real.sqrt n * h) * (Real.exp 1 * Real.sqrt Real.pi) / Real.sqrt n) hscaled n
  have hfinal := hcard.trans hpow
  rw [mul_pow, mul_comm (h^n)] at hfinal
  exact (mul_le_mul_iff_left₀ (pow_pos hh n)).mp hfinal

lemma exists_mem_euclideanGridCell {n : ℕ} (base : Fin n → ℝ)
    {h : ℝ} (hh : 0 < h) (x : EuclideanSpace ℝ (Fin n)) :
    ∃ a : Fin n → ℤ, x ∈ euclideanGridCell base h a := by
  refine ⟨fun i => ⌊(x i - base i) / h⌋, fun i => ?_⟩
  have hl := (le_div_iff₀ hh).mp (Int.floor_le ((x i - base i) / h))
  have hu := (div_lt_iff₀ hh).mp (Int.lt_floor_add_one ((x i - base i) / h))
  constructor <;> nlinarith

lemma finite_grid_hitting_ball {n : ℕ} (hn : 0 < n) (base : Fin n → ℝ)
    {h R L : ℝ} (hh : 0 < h) (hR : 0 ≤ R) (hL : 0 ≤ L)
    (hscale : R ≤ L * (Real.sqrt n * h)) (c : EuclideanSpace ℝ (Fin n)) :
    {a : Fin n → ℤ | (euclideanGridCell base h a ∩ closedBall c R).Nonempty}.Finite := by
  by_contra hinf
  let C := ((L + 1) * (Real.exp 1 * Real.sqrt Real.pi))^n
  obtain ⟨s, hs, hcard⟩ := Set.Infinite.exists_subset_card_eq hinf (⌈C⌉₊ + 1)
  have hb := grid_ball_count hn base hh hR hL hscale c s (fun a ha => hs ha)
  have hceil := Nat.le_ceil C
  rw [hcard, Nat.cast_add, Nat.cast_one] at hb
  dsimp [C] at hceil
  linarith

lemma measure_ball_le_of_grid {α : Type*} [MeasurableSpace α]
    (μ : Measure α) {n : ℕ} (hn : 0 < n) (ξ : α → EuclideanSpace ℝ (Fin n))
    (base : Fin n → ℝ) {h R L : ℝ} (hh : 0 < h) (hR : 0 ≤ R) (hL : 0 ≤ L)
    (hscale : R ≤ L * (Real.sqrt n * h)) (c : EuclideanSpace ℝ (Fin n))
    (b : ℝ≥0∞) (hgrid : ∀ a, μ (ξ ⁻¹' euclideanGridCell base h a) ≤ b) :
    μ (ξ ⁻¹' closedBall c R) ≤
      ENNReal.ofReal (((L + 1) * (Real.exp 1 * Real.sqrt Real.pi))^n) * b := by
  classical
  let hs := finite_grid_hitting_ball hn base hh hR hL hscale c
  let s := hs.toFinset
  have hsub : ξ ⁻¹' closedBall c R ⊆ ⋃ a ∈ s, ξ ⁻¹' euclideanGridCell base h a := by
    intro z hz
    obtain ⟨a, ha⟩ := exists_mem_euclideanGridCell base hh (ξ z)
    exact mem_iUnion₂.mpr ⟨a, hs.mem_toFinset.mpr ⟨ξ z, ha, hz⟩, ha⟩
  have hcard := grid_ball_count hn base hh hR hL hscale c s
    (fun a ha => hs.mem_toFinset.mp ha)
  calc
    μ (ξ ⁻¹' closedBall c R) ≤ μ (⋃ a ∈ s, ξ ⁻¹' euclideanGridCell base h a) := measure_mono hsub
    _ ≤ ∑ a ∈ s, μ (ξ ⁻¹' euclideanGridCell base h a) := measure_biUnion_finset_le _ _
    _ ≤ ∑ a ∈ s, b := Finset.sum_le_sum fun a _ => hgrid a
    _ = (s.card : ℝ≥0∞) * b := by simp
    _ ≤ ENNReal.ofReal (((L + 1) * (Real.exp 1 * Real.sqrt Real.pi))^n) * b := by
      gcongr
      simpa only [ENNReal.ofReal_natCast] using ENNReal.ofReal_le_ofReal hcard

end RepeatedLocalization

end OAI

	import Mathlib
	import OAI.Probability.GaussianInformation.Dyadic.Covers

	namespace OAI
	open MeasureTheory ProbabilityTheory Set
	open scoped BigOperators ENNReal
	open MeasureTheory Metric Set
	open scoped ENNReal NNReal
	namespace RepeatedLocalization

	def euclideanGridCell {n : ℕ} (base : Fin n → ℝ) (h : ℝ) (a : Fin n → ℤ) :
	Set (EuclideanSpace ℝ (Fin n)) :=
	{x \| ∀ i, base i + h * a i ≤ x i ∧ x i < base i + h * (a i + 1)}

	lemma measurableSet_euclideanGridCell {n : ℕ} (base : Fin n → ℝ)
	(h : ℝ) (a : Fin n → ℤ) : MeasurableSet (euclideanGridCell base h a) := by
	unfold euclideanGridCell
	simp only [ofPred_forall]
	apply MeasurableSet.iInter
	intro i
	have hm : Measurable (fun x : EuclideanSpace ℝ (Fin n) => x i) :=
	(measurable_pi_apply i).comp (WithLp.measurable_ofLp 2 (Fin n → ℝ))
	exact (measurableSet_le (measurable_const (a := base i + h * a i)) hm).inter
	(measurableSet_lt hm (measurable_const (a := base i + h * (a i + 1))))

	lemma volume_euclideanGridCell {n : ℕ} (base : Fin n → ℝ)
	{h : ℝ} (hh : 0 ≤ h) (a : Fin n → ℤ) :
	volume (euclideanGridCell base h a) = ENNReal.ofReal (h ^ n) := by
	rw [← (PiLp.volume_preserving_toLp (Fin n)).measure_preimage
	(measurableSet_euclideanGridCell base h a).nullMeasurableSet]
	have he : WithLp.toLp 2 ⁻¹' euclideanGridCell base h a =
	Set.pi Set.univ (fun i => Ico (base i + h * a i) (base i + h * (a i + 1))) := by
	ext x
	simp [euclideanGridCell]
	rw [he, Real.volume_pi_Ico]
	have hf : ∀ i, base i + h * (a i + 1) - (base i + h * a i) = h := by
	intro i
	ring
	simp only [hf, Finset.prod_const, Finset.card_univ, Fintype.card_fin,
	← ENNReal.ofReal_pow hh]

	lemma euclideanGridCell_disjoint {n : ℕ} (base : Fin n → ℝ) {h : ℝ} (hh : 0 < h)
	{a b : Fin n → ℤ} (hab : a ≠ b) :
	Disjoint (euclideanGridCell base h a) (euclideanGridCell base h b) := by
	rw [Function.ne_iff] at hab
	obtain ⟨i, hi⟩ := hab
	apply Set.disjoint_left.mpr
	intro x hx hy
	obtain hl \| hl := lt_or_gt_of_ne hi
	· have ht : (a i : ℝ) + 1 ≤ b i := by exact_mod_cast hl
	have hs := mul_le_mul_of_nonneg_left ht hh.le
	linarith [(hx i).2, (hy i).1]
	· have ht : (b i : ℝ) + 1 ≤ a i := by exact_mod_cast hl
	have hs := mul_le_mul_of_nonneg_left ht hh.le
	linarith [(hy i).2, (hx i).1]

	lemma dist_le_of_mem_euclideanGridCell {n : ℕ} (base : Fin n → ℝ)
	{h : ℝ} (hh : 0 ≤ h) (a : Fin n → ℤ)
	{x y : EuclideanSpace ℝ (Fin n)} (hx : x ∈ euclideanGridCell base h a)
	(hy : y ∈ euclideanGridCell base h a) : dist x y ≤ Real.sqrt n * h := by
	have hsq : ‖x - y‖ ^ 2 ≤ (n : ℝ) * h^2 := by
	rw [EuclideanSpace.real_norm_sq_eq]
	calc
	_ ≤ ∑ i : Fin n, h^2 := Finset.sum_le_sum fun i _ => by
	have hx1 := (hx i).1
	have hx2 := (hx i).2
	have hy1 := (hy i).1
	have hy2 := (hy i).2
	change (x i - y i)^2 ≤ h^2
	have hab : \|x i - y i\| ≤ h := abs_le.mpr ⟨by nlinarith, by nlinarith⟩
	nlinarith [sq_abs (x i - y i)]
	_ = _ := by simp
	rw [dist_eq_norm]
	have hn : 0 ≤ (n : ℝ) := Nat.cast_nonneg _
	have hs := Real.sq_sqrt hn
	have hnn := Real.sqrt_nonneg (n : ℝ)
	have hm : 0 ≤ Real.sqrt n * h := mul_nonneg hnn hh
	nlinarith [norm_nonneg (x - y)]

	lemma euclidean_closedBall_volume_le (n : ℕ) (hn : 0 < n)
	(x : EuclideanSpace ℝ (Fin n)) {r : ℝ} (hr : 0 < r) :
	volume.real (closedBall x r) ≤
	(r * (Real.exp 1 * Real.sqrt Real.pi) / Real.sqrt n)^n := by
	have : Nonempty (Fin n) := Fin.pos_iff_nonempty.mp hn
	rw [measureReal_def, volume.addHaar_closedBall_eq_addHaar_ball]
	exact euclidean_ball_volume_le n hn x hr

	lemma grid_ball_count {n : ℕ} (hn : 0 < n) (base : Fin n → ℝ)
	{h R L : ℝ} (hh : 0 < h) (hR : 0 ≤ R) (_ : 0 ≤ L)
	(hscale : R ≤ L * (Real.sqrt n * h)) (c : EuclideanSpace ℝ (Fin n))
	(s : Finset (Fin n → ℤ))
	(hs : ∀ a ∈ s, (euclideanGridCell base h a ∩ closedBall c R).Nonempty) :
	(s.card : ℝ) ≤ ((L + 1) * (Real.exp 1 * Real.sqrt Real.pi))^n := by
	classical
	let A := ⋃ a ∈ s, euclideanGridCell base h a
	have hsub : A ⊆ closedBall c (R + Real.sqrt n * h) := by
	intro x hx
	obtain ⟨a, ha, hxa⟩ := mem_iUnion₂.mp hx
	obtain ⟨y, hya, hy⟩ := hs a ha
	have hdist := dist_le_of_mem_euclideanGridCell base hh.le a hxa hya
	exact (dist_triangle x y c).trans (add_le_add hdist hy) \|>.trans_eq (add_comm _ _)
	have hmeasure : volume A = (s.card : ℝ≥0∞) * ENNReal.ofReal (h^n) := by
	rw [measure_biUnion_finset
	(fun a _ b _ hab => euclideanGridCell_disjoint base hh hab)
	(fun a _ => measurableSet_euclideanGridCell base h a)]
	simp [volume_euclideanGridCell base hh.le]
	have hra : 0 < R + Real.sqrt n * h := by positivity
	have hfin : volume (closedBall c (R + Real.sqrt n * h)) ≠ ∞ := measure_closedBall_lt_top.ne
	have hreal : (s.card : ℝ) * h^n ≤ volume.real (closedBall c (R + Real.sqrt n * h)) := by
	have hm := ENNReal.toReal_mono hfin (measure_mono hsub)
	rw [hmeasure] at hm
	simpa [measureReal_def, ENNReal.toReal_ofReal (pow_nonneg hh.le n)] using hm
	have hC : 0 ≤ Real.exp 1 * Real.sqrt Real.pi := by positivity
	have hn' : 0 < Real.sqrt (n : ℝ) := by positivity
	have hscaled : (R + Real.sqrt n * h) * (Real.exp 1 * Real.sqrt Real.pi) / Real.sqrt n ≤
	h * ((L + 1) * (Real.exp 1 * Real.sqrt Real.pi)) := by
	apply (div_le_iff₀ hn').mpr
	calc
	_ ≤ (L * (Real.sqrt n * h) + Real.sqrt n * h) * (Real.exp 1 * Real.sqrt Real.pi) :=
	mul_le_mul_of_nonneg_right (add_le_add hscale le_rfl) hC
	_ = _ := by ring
	have hcard := hreal.trans (euclidean_closedBall_volume_le n hn c hra)
	have hpow := pow_le_pow_left₀ (by positivity : 0 ≤
	(R + Real.sqrt n * h) * (Real.exp 1 * Real.sqrt Real.pi) / Real.sqrt n) hscaled n
	have hfinal := hcard.trans hpow
	rw [mul_pow, mul_comm (h^n)] at hfinal
	exact (mul_le_mul_iff_left₀ (pow_pos hh n)).mp hfinal

	lemma exists_mem_euclideanGridCell {n : ℕ} (base : Fin n → ℝ)
	{h : ℝ} (hh : 0 < h) (x : EuclideanSpace ℝ (Fin n)) :
	∃ a : Fin n → ℤ, x ∈ euclideanGridCell base h a := by
	refine ⟨fun i => ⌊(x i - base i) / h⌋, fun i => ?_⟩
	have hl := (le_div_iff₀ hh).mp (Int.floor_le ((x i - base i) / h))
	have hu := (div_lt_iff₀ hh).mp (Int.lt_floor_add_one ((x i - base i) / h))
	constructor <;> nlinarith

	lemma finite_grid_hitting_ball {n : ℕ} (hn : 0 < n) (base : Fin n → ℝ)
	{h R L : ℝ} (hh : 0 < h) (hR : 0 ≤ R) (hL : 0 ≤ L)
	(hscale : R ≤ L * (Real.sqrt n * h)) (c : EuclideanSpace ℝ (Fin n)) :
	{a : Fin n → ℤ \| (euclideanGridCell base h a ∩ closedBall c R).Nonempty}.Finite := by
	by_contra hinf
	let C := ((L + 1) * (Real.exp 1 * Real.sqrt Real.pi))^n
	obtain ⟨s, hs, hcard⟩ := Set.Infinite.exists_subset_card_eq hinf (⌈C⌉₊ + 1)
	have hb := grid_ball_count hn base hh hR hL hscale c s (fun a ha => hs ha)
	have hceil := Nat.le_ceil C
	rw [hcard, Nat.cast_add, Nat.cast_one] at hb
	dsimp [C] at hceil
	linarith

	lemma measure_ball_le_of_grid {α : Type*} [MeasurableSpace α]
	(μ : Measure α) {n : ℕ} (hn : 0 < n) (ξ : α → EuclideanSpace ℝ (Fin n))
	(base : Fin n → ℝ) {h R L : ℝ} (hh : 0 < h) (hR : 0 ≤ R) (hL : 0 ≤ L)
	(hscale : R ≤ L * (Real.sqrt n * h)) (c : EuclideanSpace ℝ (Fin n))
	(b : ℝ≥0∞) (hgrid : ∀ a, μ (ξ ⁻¹' euclideanGridCell base h a) ≤ b) :
	μ (ξ ⁻¹' closedBall c R) ≤
	ENNReal.ofReal (((L + 1) * (Real.exp 1 * Real.sqrt Real.pi))^n) * b := by
	classical
	let hs := finite_grid_hitting_ball hn base hh hR hL hscale c
	let s := hs.toFinset
	have hsub : ξ ⁻¹' closedBall c R ⊆ ⋃ a ∈ s, ξ ⁻¹' euclideanGridCell base h a := by
	intro z hz
	obtain ⟨a, ha⟩ := exists_mem_euclideanGridCell base hh (ξ z)
	exact mem_iUnion₂.mpr ⟨a, hs.mem_toFinset.mpr ⟨ξ z, ha, hz⟩, ha⟩
	have hcard := grid_ball_count hn base hh hR hL hscale c s
	(fun a ha => hs.mem_toFinset.mp ha)
	calc
	μ (ξ ⁻¹' closedBall c R) ≤ μ (⋃ a ∈ s, ξ ⁻¹' euclideanGridCell base h a) := measure_mono hsub
	_ ≤ ∑ a ∈ s, μ (ξ ⁻¹' euclideanGridCell base h a) := measure_biUnion_finset_le _ _
	_ ≤ ∑ a ∈ s, b := Finset.sum_le_sum fun a _ => hgrid a
	_ = (s.card : ℝ≥0∞) * b := by simp
	_ ≤ ENNReal.ofReal (((L + 1) * (Real.exp 1 * Real.sqrt Real.pi))^n) * b := by
	gcongr
	simpa only [ENNReal.ofReal_natCast] using ENNReal.ofReal_le_ofReal hcard

	end RepeatedLocalization

	end OAI