| 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 |
|
|
