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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 96 97 98 99 100 101 102 103 | import OAI.Analysis.FiniteFactor.Sampling.FirstInterval
import OAI.Analysis.FiniteFactor.OrbitSampling.Weights
namespace OAI
namespace FiniteFactor
open MeasureTheory Filter
open scoped Classical ENNReal
noncomputable section
/-- Rectangle discrepancy passes exactly to the first-oriented tested weights. -/
theorem samplingTestedFirst_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)]
(δ : ℝ) (hrect : ∀ a b c d : ℝ,
|(μ {ω | X ω ∈ Set.Icc a b ∧ Y ω ∈ Set.Icc c d}).toReal -
(unitIntervalMeasure (Set.Icc a b)).toReal*(unitIntervalMeasure (Set.Icc c d)).toReal| ≤ δ)
(b : Bool) (h α : ℝ) (E : Set ℝ) (a c s : ℝ)
(hh : 0 < h) (hh1 : h < 1) (hα : 0 < α) (hs : s ∈ Set.Ico (0 : ℝ) 1) :
|(∫ ω, samplingTestedFirstWeight b h α E (Set.Icc a c) s (X ω) *
samplingTestedFirstWeight b h α E (Set.Icc a c) s (Y ω) ∂μ) -
samplingTestedFirstDensity b h α E (Set.Icc a c) s ^ 2| ≤
δ*(if s ∈ E then samplingFirstHeight b h α s else 0)^2 := by
have : NullSingletonClass unitIntervalMeasure := by unfold unitIntervalMeasure; infer_instance
let A := max 0 (a-s)
let B := min (samplingFirstLength b h s) (c-s)
let C := if s ∈ E then samplingFirstHeight b h α s else 0
let D := Set.Icc A B
have hrep (ν : Measure ℝ) [NullSingletonClass ν] :
samplingTestedFirstWeight b h α E (Set.Icc a c) s =ᵐ[ν] D.indicator (fun _ => C) :=
samplingTestedFirstWeight_closed_interval_ae ν b h α E a c s hh hh1 hα hs
have hprod : (∫ ω, samplingTestedFirstWeight b h α E (Set.Icc a c) s (X ω) *
samplingTestedFirstWeight b h α E (Set.Icc a c) s (Y ω) ∂μ) =
C^2*(μ {ω | X ω ∈ D ∧ Y ω ∈ D}).toReal := by
calc
_ = ∫ ω, ({ω | X ω ∈ D ∧ Y ω ∈ D}).indicator (fun _ => C^2) ω ∂μ := by
apply integral_congr_ae
filter_upwards [ae_of_ae_map hX.aemeasurable (hrep (μ.map X)),
ae_of_ae_map hY.aemeasurable (hrep (μ.map Y))] with ω hx hy
rw [hx,hy]
simp only [Set.indicator,Set.mem_ofPred_eq]
split_ifs <;> simp_all [pow_two]
_ = _ := by
have hD : MeasurableSet {ω | X ω ∈ D ∧ Y ω ∈ D} :=
(measurableSet_Icc.preimage hX).inter (measurableSet_Icc.preimage hY)
rw [integral_indicator hD]
simp [integral_const,measureReal_def,smul_eq_mul,mul_comm,D]
have hmean : samplingTestedFirstDensity b h α E (Set.Icc a c) s =
C*(unitIntervalMeasure D).toReal := by
unfold samplingTestedFirstDensity
rw [integral_congr_ae (hrep unitIntervalMeasure),integral_indicator measurableSet_Icc]
simp [integral_const,measureReal_def,smul_eq_mul,mul_comm,D]
rw [hprod,hmean]
calc
_ = C^2*|(μ {ω | X ω ∈ D ∧ Y ω ∈ D}).toReal -
(unitIntervalMeasure D).toReal*(unitIntervalMeasure D).toReal| := by
calc
_ = |C^2*((μ {ω | X ω ∈ D ∧ Y ω ∈ D}).toReal -
(unitIntervalMeasure D).toReal*(unitIntervalMeasure D).toReal)| := by congr 1; ring
_ = _ := by rw [abs_mul,abs_of_nonneg (sq_nonneg C)]
_ ≤ C^2*δ := mul_le_mul_of_nonneg_left (hrect A B A B) (sq_nonneg C)
_ = _ := by ring
/-- The actual selected phase laws are atomless, despite possible dependence
between two phases. Only their already proved marginal domination is used. -/
theorem sourceSampling_phase_null_singletons (stage : ℕ) (hs : 200000 ≤ sourceBlockSize stage)
(i : SourceCoordinate) (hi : i.1 ≤ stage) (n : ℤ) (hn : n ∈ sourceSamplingPowers stage) :
NullSingletonClass (sourceAngleLaw.map (fun θ => Int.fract ((n : ℝ)*θ i))) := by
have : NullSingletonClass unitIntervalMeasure := by unfold unitIntervalMeasure; infer_instance
constructor
intro x
have hdom := sourceSampling_phase_domination stage hs i hi n hn
exact le_antisymm ((hdom {x}).trans_eq (by simp)) bot_le
/-- Actual first-tested pair discrepancy, with all probabilistic assumptions
discharged by the source's deterministic separated powers. -/
theorem sourceSampling_first_pair_bound (stage : ℕ) (hs : 200000 ≤ sourceBlockSize stage)
(i : SourceCoordinate) (hi : i.1 ≤ stage) (n n' : ℤ)
(hn : n ∈ sourceSamplingPowers stage) (hn' : n' ∈ sourceSamplingPowers stage) (hne : n ≠ n')
(b : Bool) (E : Set ℝ) (a c s : ℝ) (hsx : s ∈ Set.Ico (0 : ℝ) 1) :
|(∫ θ, samplingTestedFirstWeight b (sourceSamplingWidth stage) (sourceSamplingCutoff stage)
E (Set.Icc a c) s (Int.fract ((n : ℝ)*θ i)) *
samplingTestedFirstWeight b (sourceSamplingWidth stage) (sourceSamplingCutoff stage)
E (Set.Icc a c) s (Int.fract ((n' : ℝ)*θ i)) ∂sourceAngleLaw) -
samplingTestedFirstDensity b (sourceSamplingWidth stage) (sourceSamplingCutoff stage) E (Set.Icc a c) s ^ 2| ≤
(26*(sourceBlockSize stage : ℝ)^(-80 : ℝ))*
(if s ∈ E then samplingFirstHeight b (sourceSamplingWidth stage) (sourceSamplingCutoff stage) s else 0)^2 := by
have := sourceSampling_phase_null_singletons stage hs i hi n hn
have := sourceSampling_phase_null_singletons stage hs i hi n' hn'
have hp := sourceSampling_parameters stage
exact samplingTestedFirst_pair_bound sourceAngleLaw
(fun θ => Int.fract ((n : ℝ)*θ i)) (fun θ => Int.fract ((n' : ℝ)*θ i))
(sourceFract_measurable.comp (measurable_const.mul (measurable_pi_apply i)))
(sourceFract_measurable.comp (measurable_const.mul (measurable_pi_apply i))) _
(sourceSampling_rectangle_bound stage hs i hi n n' hn hn' hne)
b _ _ E a c s hp.1 hp.2.1 hp.2.2 hsx
end
end FiniteFactor
end OAI
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