| import OAI.Probability.Renewal.Centered |
| import OAI.Probability.Renewal.GaussianCore |
|
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| namespace OAI |
|
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| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology ENNReal |
| namespace HoneycombLocal |
|
|
| def localIntegrand (p : PMF ℕ) (k m : ℕ) (σ t : ℝ) : ℂ := |
| charFun (centeredMeasure p) (t / σ) ^ k * |
| Complex.exp (-(((m : ℝ) - k * natMean p) / σ * t : ℝ) * Complex.I) |
|
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| lemma localIntegrand_eq (p : PMF ℕ) (k m : ℕ) (σ t : ℝ) : |
| localIntegrand p k m σ t = latticeChar p (t / σ) ^ k * |
| Complex.exp (-((m : ℝ) * (t / σ) : ℝ) * Complex.I) := by |
| rw [localIntegrand, centered_char, mul_pow, ← Complex.exp_nat_mul] |
| have hid : ((k : ℂ) * (-((natMean p * (t / σ) : ℝ) : ℂ) * Complex.I)) + |
| (-(((m : ℝ) - k * natMean p) / σ * t : ℝ) * Complex.I) = |
| -((m : ℝ) * (t / σ) : ℝ) * Complex.I := by |
| push_cast |
| ring |
| rw [mul_comm (Complex.exp _), mul_assoc, ← Complex.exp_add, hid] |
|
|
| lemma continuous_localIntegrand (p : PMF ℕ) (k m : ℕ) (σ : ℝ) : |
| Continuous (localIntegrand p k m σ) := by |
| unfold localIntegrand |
| fun_prop |
|
|
| lemma norm_localIntegrand (p : PMF ℕ) (k m : ℕ) (σ t : ℝ) : |
| ‖localIntegrand p k m σ t‖ = ‖latticeChar p (t / σ)‖ ^ k := by |
| rw [localIntegrand_eq, norm_mul, norm_pow] |
| simp only [← Complex.ofReal_neg, Complex.norm_exp_ofReal_mul_I, mul_one] |
|
|
| lemma scaled_inversion (p : PMF ℕ) (k m : ℕ) {σ : ℝ} (hσ : 0 < σ) : |
| (∫ t : ℝ in Icc (-Real.pi * σ) (Real.pi * σ), localIntegrand p k m σ t) = |
| (2 * Real.pi * σ : ℂ) * ((sumLaw p k m).toReal : ℂ) := by |
| rw [integral_Icc_eq_integral_Ioc, ← intervalIntegral.integral_of_le (by nlinarith [Real.pi_pos] : -Real.pi * σ ≤ Real.pi * σ)] |
| simp_rw [localIntegrand_eq] |
| rw [intervalIntegral.integral_comp_div (fun t => latticeChar p t ^ k * |
| Complex.exp (-((m : ℝ) * t : ℝ) * Complex.I)) hσ.ne'] |
| simp only [mul_div_cancel_right₀ _ hσ.ne', sumLaw_inversion, Complex.real_smul] |
| ring |
|
|
| def gaussianIntegrand (b t : ℝ) : ℂ := |
| Complex.exp (-((t : ℂ) ^ 2 / 2)) * Complex.exp (-((b * t : ℝ) : ℂ) * Complex.I) |
|
|
| lemma continuous_gaussianIntegrand (b : ℝ) : Continuous (gaussianIntegrand b) := by |
| unfold gaussianIntegrand |
| fun_prop |
|
|
| lemma gaussianIntegrand_re (b t : ℝ) : |
| (gaussianIntegrand b t).re = Real.exp (-t ^ 2 / 2) * Real.cos (b * t) := by |
| have hid : -((t : ℂ) ^ 2 / 2) = ((-t ^ 2 / 2 : ℝ) : ℂ) := by push_cast; ring |
| rw [gaussianIntegrand, hid, ← Complex.ofReal_exp] |
| rw [Complex.mul_re] |
| simp only [Complex.ofReal_re, Complex.ofReal_im, zero_mul, sub_zero] |
| simp [Complex.exp_re, Real.cos_neg] |
|
|
| lemma gaussian_core_lower {ω b : ℝ} (hω : 1 ≤ ω) (hb : |b| * ω ≤ 1) : |
| Real.exp (-1) ≤ (∫ t : ℝ in Icc (-ω) ω, gaussianIntegrand b t).re := by |
| have hω0 : 0 < ω := lt_of_lt_of_le zero_lt_one hω |
| have hcos {t : ℝ} (ht : t ∈ Icc (-ω) ω) : (1 : ℝ) / 2 ≤ Real.cos (b * t) := by |
| have habs : |b * t| ≤ 1 := by |
| rw [abs_mul] |
| exact (mul_le_mul_of_nonneg_left (abs_le.mpr ht) (abs_nonneg b)).trans hb |
| have hsq : (b * t) ^ 2 ≤ 1 := by nlinarith [sq_abs (b * t), abs_nonneg (b * t)] |
| nlinarith [Real.one_sub_sq_div_two_le_cos (x := b * t)] |
| have hcont : Continuous (fun t => (gaussianIntegrand b t).re) := Complex.continuous_re.comp (continuous_gaussianIntegrand b) |
| change Real.exp (-1) ≤ RCLike.re (∫ t : ℝ in Icc (-ω) ω, gaussianIntegrand b t) |
| rw [← integral_re (continuous_gaussianIntegrand b).integrableOn_Icc] |
| calc |
| Real.exp (-1) = ∫ _ : ℝ in Icc (-1) 1, Real.exp (-1) / 2 := by norm_num; ring |
| _ ≤ ∫ t : ℝ in Icc (-1) 1, (gaussianIntegrand b t).re := by |
| apply setIntegral_mono_on (integrable_const _) hcont.integrableOn_Icc measurableSet_Icc |
| intro t ht |
| have htω : t ∈ Icc (-ω) ω := ⟨by linarith [ht.1], ht.2.trans hω⟩ |
| rw [gaussianIntegrand_re] |
| have ht2 : t ^ 2 ≤ 1 := by nlinarith [ht.1, ht.2] |
| have hexp : Real.exp (-1) ≤ Real.exp (-t ^ 2 / 2) := Real.exp_le_exp.mpr (by linarith) |
| have hh := mul_le_mul hexp (hcos htω) (by norm_num : (0 : ℝ) ≤ 1 / 2) (Real.exp_pos _).le |
| nlinarith |
| _ ≤ ∫ t : ℝ in Icc (-ω) ω, (gaussianIntegrand b t).re := by |
| apply setIntegral_mono_set hcont.integrableOn_Icc |
| · filter_upwards [ae_restrict_mem measurableSet_Icc] with t ht |
| rw [gaussianIntegrand_re] |
| exact mul_nonneg (Real.exp_pos _).le (by linarith [hcos ht]) |
| · exact ae_of_all _ (fun t ht => ⟨by linarith [ht.1], ht.2.trans hω⟩) |
|
|
| lemma local_core_error (p : PMF ℕ) |
| (h3 : Integrable (fun j : ℕ => (j : ℝ) ^ 3) p.toMeasure) |
| {k : ℕ} (hk : 0 < k) (m : ℕ) {σ ω : ℝ} (hσ : 0 < σ) (hω : 0 ≤ ω) |
| (hvar : σ ^ 2 = k * natVariance p) (hsmall : ω ^ 2 ≤ 2 * k) : |
| ‖(∫ t : ℝ in Icc (-ω) ω, localIntegrand p k m σ t) - |
| (∫ t : ℝ in Icc (-ω) ω, gaussianIntegrand (((m : ℝ) - k * natMean p) / σ) t)‖ ≤ |
| 2 * ω * ((k * natThird p / σ ^ 3) * ω ^ 3 / 2 + ω ^ 4 / (4 * k)) := by |
| have h1 : Integrable (fun j : ℕ => (j : ℝ)) p.toMeasure := by |
| have hh := (nat_memLp p (by norm_num : (3 : ℕ) ≠ 0) h3).mono_exponent (by norm_num : (1 : ENNReal) ≤ 3) |
| simpa using hh.integrable |
| rw [← integral_sub (continuous_localIntegrand p k m σ).integrableOn_Icc |
| (continuous_gaussianIntegrand _).integrableOn_Icc] |
| have hh := norm_setIntegral_le_of_norm_le_const (μ := volume) (s := Icc (-ω) ω) |
| (f := fun t => localIntegrand p k m σ t - gaussianIntegrand (((m : ℝ) - k * natMean p) / σ) t) |
| (C := (k * natThird p / σ ^ 3) * ω ^ 3 / 2 + ω ^ 4 / (4 * k)) (by simp) (fun t ht => ?_) |
| · rw [Real.volume_real_Icc] at hh |
| simp only [sub_neg_eq_add, max_eq_left (by linarith : 0 ≤ ω + ω)] at hh |
| exact hh.trans_eq (by ring) |
| · have ht2 : t ^ 2 ≤ ω ^ 2 := by |
| nlinarith [mul_nonneg (sub_nonneg.mpr ht.2) (by linarith [ht.1] : 0 ≤ ω + t)] |
| have h := normalized_gaussian_power_error (centered_memLp p (by norm_num) h3) |
| (centered_mean p h1) hk hσ (by simpa only [centered_variance] using hvar) t (ht2.trans hsmall) |
| rw [centered_third] at h |
| dsimp only [localIntegrand, gaussianIntegrand] |
| rw [← sub_mul, norm_mul, ← Complex.ofReal_neg, Complex.norm_exp_ofReal_mul_I, mul_one] |
| refine h.trans ?_ |
| apply add_le_add |
| · apply div_le_div_of_nonneg_right _ (by norm_num) |
| exact mul_le_mul_of_nonneg_left (pow_le_pow_left₀ (abs_nonneg t) (abs_le.mpr ht) 3) |
| (by positivity [natThird_nonneg p]) |
| · apply div_le_div_of_nonneg_right _ (by positivity) |
| nlinarith [mul_nonneg (sub_nonneg.mpr ht2) (by positivity : 0 ≤ ω ^ 2 + t ^ 2)] |
|
|
| lemma fourier_tail_bound {f : ℝ → ℂ} (hf : Continuous f) {g : ℝ → ℝ} |
| (hg : Integrable g) (hg0 : ∀ t, 0 ≤ g t) {R ω η : ℝ} (hR : 0 ≤ R) (hη : 0 ≤ η) |
| (hbound : ∀ t ∈ Icc (-R) R, ‖f t‖ ≤ g t + η) : |
| ‖∫ t : ℝ in Icc (-R) R \ Icc (-ω) ω, f t‖ ≤ |
| (∫ t : ℝ in (Icc (-ω) ω)ᶜ, g t) + 2 * R * η := by |
| let s := Icc (-R) R \ Icc (-ω) ω |
| have hfi : IntegrableOn f s := hf.integrableOn_Icc.mono_set sdiff_subset |
| have hci : IntegrableOn (fun _ : ℝ => η) s := |
| (integrableOn_const (C := η) (s := Icc (-R) R) (by simp)).mono_set sdiff_subset |
| calc |
| ‖∫ t in s, f t‖ ≤ ∫ t in s, ‖f t‖ := norm_integral_le_integral_norm _ |
| _ ≤ ∫ t in s, g t + η := by |
| apply setIntegral_mono_on hfi.norm (hg.integrableOn.add hci) (measurableSet_Icc.diff measurableSet_Icc) |
| exact fun t ht => hbound t ht.1 |
| _ = (∫ t in s, g t) + ∫ _ in s, η := integral_add hg.integrableOn hci |
| _ ≤ (∫ t in (Icc (-ω) ω)ᶜ, g t) + ∫ _ : ℝ in Icc (-R) R, η := by |
| apply add_le_add |
| · apply setIntegral_mono_set hg.integrableOn (ae_of_all _ hg0) |
| exact ae_of_all _ (fun _ ht => ht.2) |
| · apply setIntegral_mono_set (integrable_const _) (ae_of_all _ (fun _ => hη)) |
| exact ae_of_all _ (fun _ ht => ht.1) |
| _ = _ := by |
| simp only [integral_const, smul_eq_mul, measureReal_restrict_apply_univ, Real.volume_real_Icc] |
| rw [max_eq_left (by linarith : 0 ≤ R - -R)] |
| ring |
|
|
| lemma local_lower_of_envelope (p : PMF ℕ) |
| (h3 : Integrable (fun j : ℕ => (j : ℝ) ^ 3) p.toMeasure) |
| {k : ℕ} (hk : 0 < k) (m : ℕ) {σ ω η : ℝ} (hσ : 0 < σ) (hω : 1 ≤ ω) |
| (hvar : σ ^ 2 = k * natVariance p) (hsmall : ω ^ 2 ≤ 2 * k) |
| (hwidth : ω ≤ Real.pi * σ) |
| (hcenter : |((m : ℝ) - k * natMean p) / σ| * ω ≤ 1) |
| {g : ℝ → ℝ} (hg : Integrable g) (hg0 : ∀ t, 0 ≤ g t) (hη : 0 ≤ η) |
| (hbound : ∀ t ∈ Icc (-Real.pi * σ) (Real.pi * σ), |
| ‖latticeChar p (t / σ)‖ ^ k ≤ g t + η) |
| (htail : (∫ t : ℝ in (Icc (-ω) ω)ᶜ, g t) ≤ Real.exp (-1) / 4) |
| (hhigh : 2 * (Real.pi * σ) * η ≤ Real.exp (-1) / 4) |
| (herror : 2 * ω * ((k * natThird p / σ ^ 3) * ω ^ 3 / 2 + ω ^ 4 / (4 * k)) ≤ |
| Real.exp (-1) / 4) : |
| Real.exp (-1) / (8 * Real.pi * σ) ≤ (sumLaw p k m).toReal := by |
| have hω0 : 0 ≤ ω := zero_le_one.trans hω |
| have hR : 0 ≤ Real.pi * σ := by positivity |
| have hsub : Icc (-ω) ω ⊆ Icc (-Real.pi * σ) (Real.pi * σ) := |
| fun _ ht => ⟨by linarith [ht.1], ht.2.trans hwidth⟩ |
| have he := (local_core_error p h3 hk m hσ hω0 hvar hsmall).trans herror |
| have hc := gaussian_core_lower hω hcenter |
| have ht := fourier_tail_bound (ω := ω) (continuous_localIntegrand p k m σ) hg hg0 hR hη (by |
| intro t ht |
| rw [norm_localIntegrand] |
| exact hbound t (by simpa [neg_mul] using ht)) |
| have htail' : ‖∫ t : ℝ in Icc (-Real.pi * σ) (Real.pi * σ) \ Icc (-ω) ω, |
| localIntegrand p k m σ t‖ ≤ Real.exp (-1) / 2 := by |
| have hh := ht.trans (add_le_add htail hhigh) |
| simp only [← neg_mul] at hh |
| linarith |
| have hdiff := Complex.re_le_norm ((∫ t : ℝ in Icc (-ω) ω, gaussianIntegrand |
| (((m : ℝ) - k * natMean p) / σ) t) - ∫ t : ℝ in Icc (-ω) ω, localIntegrand p k m σ t) |
| rw [norm_sub_rev, Complex.sub_re] at hdiff |
| have hl := (abs_le.mp (Complex.abs_re_le_norm (∫ t : ℝ in Icc (-Real.pi * σ) (Real.pi * σ) \ Icc (-ω) ω, |
| localIntegrand p k m σ t))).1 |
| have hlower : -Real.exp (-1) / 2 ≤ |
| (∫ t : ℝ in Icc (-Real.pi * σ) (Real.pi * σ) \ Icc (-ω) ω, |
| localIntegrand p k m σ t).re := by linarith |
| rw [setIntegral_sdiff measurableSet_Icc (continuous_localIntegrand p k m σ).integrableOn_Icc hsub, |
| Complex.sub_re, scaled_inversion p k m hσ] at hlower |
| norm_num [Complex.mul_re] at hlower |
| apply (div_le_iff₀ (by positivity : 0 < 8 * Real.pi * σ)).mpr |
| nlinarith |
|
|
| end HoneycombLocal |
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| end |
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| end OAI |
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