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67aee1d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 | import OAI.Analysis.FiniteFactor.Sampling.SecondDensity
namespace OAI
namespace FiniteFactor
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal
noncomputable section
/-- Exact x- and y-tests in the second diagonal orientation. -/
def samplingTestedSecondWeight (h α : ℝ) (E G : Set ℝ) (s t : ℝ) : ℝ := by
classical
exact if Int.fract (s-t) ∈ E ∧ s ∈ G then samplingSecondWeight h α s t else 0
def samplingTestedSecondDensity (h α : ℝ) (E G : Set ℝ) (s : ℝ) : ℝ :=
∫ t, samplingTestedSecondWeight h α E G s t ∂unitIntervalMeasure
theorem samplingTestedSecondWeight_measurable (h α : ℝ) (E G : Set ℝ)
(hE : MeasurableSet E) (hG : MeasurableSet G) :
Measurable (fun p : ℝ × ℝ => samplingTestedSecondWeight h α E G p.1 p.2) := by
unfold samplingTestedSecondWeight
exact Measurable.ite ((hE.preimage (sourceFract_measurable.comp
(measurable_fst.sub measurable_snd))).inter (hG.preimage measurable_fst))
(samplingSecondWeight_measurable h α) measurable_const
theorem samplingTestedSecondWeight_bounds (h α : ℝ) (E G : Set ℝ) (s t : ℝ)
(hh : 0 < h) (hα : 0 < α) :
0 ≤ samplingTestedSecondWeight h α E G s t ∧
samplingTestedSecondWeight h α E G s t ≤ samplingSecondWeight h α s t := by
have hn : 0 ≤ samplingSecondWeight h α s t := (samplingWeight_bounds false h α _ t hh hα).1
unfold samplingTestedSecondWeight
split_ifs <;> simp [hn]
theorem samplingTestedSecondDensity_measurable (h α : ℝ) (E G : Set ℝ)
(hE : MeasurableSet E) (hG : MeasurableSet G) :
Measurable (samplingTestedSecondDensity h α E G) :=
(samplingTestedSecondWeight_measurable h α E G hE hG).stronglyMeasurable.integral_prod_right.measurable
theorem samplingTestedSecondDensity_bounds (h α : ℝ) (E G : Set ℝ)
(hh : 0 < h) (hα : 0 < α) (hE : MeasurableSet E) (hG : MeasurableSet G) (s : ℝ) :
0 ≤ samplingTestedSecondDensity h α E G s ∧
samplingTestedSecondDensity h α E G s ≤ samplingSecondDensity h α s := by
have hm : Measurable (samplingTestedSecondWeight h α E G s) :=
(samplingTestedSecondWeight_measurable h α E G hE hG).comp (measurable_const.prodMk measurable_id)
have hi := (samplingSecondWeight_integrable h α s hh hα).mono' hm.aestronglyMeasurable
(Eventually.of_forall (fun t => by
rw [Real.norm_eq_abs,abs_of_nonneg (samplingTestedSecondWeight_bounds h α E G s t hh hα).1]
exact (samplingTestedSecondWeight_bounds h α E G s t hh hα).2))
exact ⟨integral_nonneg (fun t => (samplingTestedSecondWeight_bounds h α E G s t hh hα).1),
integral_mono_ae hi (samplingSecondWeight_integrable h α s hh hα)
(Eventually.of_forall (fun t => (samplingTestedSecondWeight_bounds h α E G s t hh hα).2))⟩
theorem samplingTestedSecondDensity_square (h α : ℝ) (E G : Set ℝ)
(hh : 0 < h) (hh1 : h < 1) (hα : 0 < α) (hE : MeasurableSet E) (hG : MeasurableSet G) :
(∀ᵐ s ∂unitIntervalMeasure, 0 ≤ samplingTestedSecondDensity h α E G s ∧
samplingTestedSecondDensity h α E G s ≤ 1/(1-h)) ∧
(∫ s, samplingTestedSecondDensity h α E G s^2 ∂unitIntervalMeasure) ≤ 1+h^2/(1-h)^2 := by
let f := samplingTestedSecondDensity h α E G
let q := samplingIdealSecond h
let C := 1/(1-h)
have hC : 0 ≤ C := by dsimp [C]; positivity
have hf := samplingTestedSecondDensity_measurable h α E G hE hG
have hq := samplingIdealSecond_measurable h hh
have hnon (s : ℝ) : 0 ≤ f s := (samplingTestedSecondDensity_bounds h α E G hh hα hE hG s).1
have hle : ∀ᵐ s ∂unitIntervalMeasure, f s ≤ q s :=
(samplingSecondDensity_le_ideal h α hh hh1 hα).mono (fun s hs =>
(samplingTestedSecondDensity_bounds h α E G hh hα hE hG s).2.trans hs)
have hqb (s : ℝ) : 0 ≤ q s ∧ q s ≤ C := (samplingIdealSecond_bounds h s hh hh1).imp_right And.left
have hq2 (s : ℝ) : q s^2 ≤ C^2 := by nlinarith [(hqb s).1,(hqb s).2]
have hf2 : ∀ᵐ s ∂unitIntervalMeasure, f s^2 ≤ q s^2 := hle.mono (fun s hs => by nlinarith [hnon s,(hqb s).1])
have hfi : Integrable (fun s => f s^2) unitIntervalMeasure :=
(integrable_const (C^2)).mono' (hf.pow_const 2).aestronglyMeasurable (hf2.mono (fun s hs => by
rw [Real.norm_eq_abs,abs_of_nonneg (sq_nonneg _)]
exact hs.trans (hq2 s)))
have hqi : Integrable (fun s => q s^2) unitIntervalMeasure :=
(integrable_const (C^2)).mono' (hq.pow_const 2).aestronglyMeasurable (Eventually.of_forall (fun s => by
rw [Real.norm_eq_abs,abs_of_nonneg (sq_nonneg _)]
exact hq2 s))
exact ⟨hle.mono (fun s hs => ⟨hnon s,hs.trans (hqb s).2⟩),
(integral_mono_ae hfi hqi hf2).trans (samplingIdealSecond_square h hh hh1).2⟩
end
end FiniteFactor
end OAI
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