| import OAI.Analysis.FiniteFactor.Sampling.FirstDiscrepancy |
| import OAI.Analysis.FiniteFactor.Sampling.OrientedMoments |
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| namespace OAI |
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| namespace FiniteFactor |
| open MeasureTheory Filter |
| open scoped Classical |
| noncomputable section |
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| / |
| 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 |
|
|
| / |
| 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 |
|
|
| / |
| 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 |
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| end OAI |
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