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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 87 88 89 90 91 92 93 94 95 | import OAI.Analysis.FiniteFactor.Sampling.FirstDiscrepancy
import OAI.Analysis.FiniteFactor.Sampling.OrientedMoments
namespace OAI
namespace FiniteFactor
open MeasureTheory Filter
open scoped Classical
noncomputable section
/-- The triangular second orientation has one reflected interval of phases. -/
theorem samplingSecondTriangular_interval (h α : ℝ) (G : Set ℝ) (a c s t : ℝ)
(hs : s ∈ Set.Ico (0 : ℝ) 1) (ht : t ∈ Set.Ico (0 : ℝ) 1) :
samplingOrientedWeight true true h α (Set.Icc a c) G s t =
(Set.Ioc 0 s ∩ Set.Icc (s-c) (s-a)).indicator (fun _ => if s ∈ G then (1 : ℝ) else 0) t := by
by_cases hts : t ≤ s
· have hf : Int.fract (s-t) = s-t := Int.fract_eq_self.mpr ⟨by linarith,by linarith [hs.2,ht.1]⟩
simp only [samplingOrientedWeight,ite_true,samplingWeight,hf,
Set.indicator,Set.mem_inter_iff,Set.mem_Ioc,Set.mem_Icc]
split_ifs <;> simp_all <;> grind
· have hst : s-t < 0 := by linarith
have hfloor : ⌊s-t⌋ = (-1 : ℤ) := Int.floor_eq_iff.mpr
⟨by norm_num; linarith [hs.1,ht.2],by norm_num; linarith⟩
have hf : Int.fract (s-t) = s-t+1 := by rw [Int.fract,hfloor]; norm_num
simp only [samplingOrientedWeight,ite_true,samplingWeight,hf,
Set.indicator,Set.mem_inter_iff,Set.mem_Ioc,Set.mem_Icc]
split_ifs <;> simp_all
grind
/-- Reflection identifies the actual triangular second weight with a first
weight. This equality concerns values; it introduces no new random law. -/
theorem samplingSecondTriangular_reflect (h α : ℝ) (G : Set ℝ) (a c s t : ℝ)
(hh : 0 < h) (hh1 : h < 1) (hα : 0 < α)
(hs : s ∈ Set.Ioo (0 : ℝ) 1) (ht : t ∈ Set.Ico (0 : ℝ) 1) (hts : t ≠ s) :
samplingOrientedWeight true true h α (Set.Icc a c) G s t =
samplingTestedFirstWeight true h α (if s ∈ G then Set.univ else ∅)
(Set.Icc (1-c) (1-a)) (1-s) t := by
rw [samplingSecondTriangular_interval h α G a c s t ⟨hs.1.le,hs.2⟩ ht,
samplingTestedFirstWeight_interval true h α (if s ∈ G then Set.univ else ∅)
(1-c) (1-a) (1-s) t hh hh1 hα ⟨by linarith [hs.2],by linarith [hs.1]⟩]
have h1 : 1-(1-s) = s := by ring
have h2 : (1-c)-(1-s) = s-c := by ring
have h3 : (1-a)-(1-s) = s-a := by ring
simp only [samplingFirstLength,samplingFirstHeight,ite_true,h1,h2,h3]
simp only [Set.indicator,Set.mem_inter_iff,Set.mem_Ioc,Set.mem_Ioo,Set.mem_Icc]
by_cases hG : s ∈ G <;> simp only [hG,ite_true,ite_false,Set.mem_univ,Set.notMem_empty]
all_goals split_ifs <;> simp_all <;> grind
/-- Rectangle discrepancy controls the remaining triangular second case. -/
theorem samplingSecondTriangular_pair_bound {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (X Y : Ω → ℝ)
(hX : Measurable X) (hY : Measurable Y)
[NullSingletonClass (μ.map X)] [NullSingletonClass (μ.map Y)]
(hXs : ∀ᵐ x ∂μ, X x ∈ Set.Ico (0 : ℝ) 1)
(hYs : ∀ᵐ x ∂μ, Y x ∈ Set.Ico (0 : ℝ) 1)
(δ : ℝ) (hδ : 0 ≤ δ) (hrect : ∀ a b c d : ℝ,
|(μ {x | X x ∈ Set.Icc a b ∧ Y x ∈ Set.Icc c d}).toReal -
(unitIntervalMeasure (Set.Icc a b)).toReal*(unitIntervalMeasure (Set.Icc c d)).toReal| ≤ δ)
(h α : ℝ) (G : Set ℝ) (a c s : ℝ)
(hh : 0 < h) (hh1 : h < 1) (hα : 0 < α) (hs : s ∈ Set.Ioo (0 : ℝ) 1) :
|(∫ x, samplingOrientedWeight true true h α (Set.Icc a c) G s (X x)*
samplingOrientedWeight true true h α (Set.Icc a c) G s (Y x) ∂μ)-
samplingOrientedDensity true true h α (Set.Icc a c) G s^2| ≤ δ := by
have he := samplingSecondTriangular_reflect h α G a c s
have hp : (∫ x, samplingOrientedWeight true true h α (Set.Icc a c) G s (X x)*
samplingOrientedWeight true true h α (Set.Icc a c) G s (Y x) ∂μ) =
∫ x, samplingTestedFirstWeight true h α (if s ∈ G then Set.univ else ∅)
(Set.Icc (1-c) (1-a)) (1-s) (X x)*
samplingTestedFirstWeight true h α (if s ∈ G then Set.univ else ∅)
(Set.Icc (1-c) (1-a)) (1-s) (Y x) ∂μ := by
apply integral_congr_ae
have hxne : ∀ᵐ x ∂μ, X x ≠ s := ae_of_ae_map hX.aemeasurable
(ae_iff.mpr (show (μ.map X) {t | ¬t ≠ s} = 0 by simp))
have hyne : ∀ᵐ x ∂μ, Y x ≠ s := ae_of_ae_map hY.aemeasurable
(ae_iff.mpr (show (μ.map Y) {t | ¬t ≠ s} = 0 by simp))
filter_upwards [hXs,hYs,hxne,hyne] with x hx hy hxn hyn
rw [he (X x) hh hh1 hα hs hx hxn,he (Y x) hh hh1 hα hs hy hyn]
have hm : samplingOrientedDensity true true h α (Set.Icc a c) G s =
samplingTestedFirstDensity true h α (if s ∈ G then Set.univ else ∅)
(Set.Icc (1-c) (1-a)) (1-s) := by
apply integral_congr_ae
have : NullSingletonClass unitIntervalMeasure := by unfold unitIntervalMeasure; infer_instance
filter_upwards [self_mem_ae_restrict measurableSet_Ico,
ae_iff.mpr (show unitIntervalMeasure {t | ¬t ≠ s} = 0 by simp)] with t ht hts
exact he t hh hh1 hα hs ht hts
rw [hp,hm]
apply (samplingTestedFirst_pair_bound μ X Y hX hY δ hrect true h α _
(1-c) (1-a) (1-s) hh hh1 hα ⟨by linarith [hs.2],by linarith [hs.1]⟩).trans
by_cases hG : s ∈ G <;> simp [hG,samplingFirstHeight,hδ]
end
end FiniteFactor
end OAI
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