| import OAI.Analysis.FiniteFactor.OrbitSampling.CutoffMass |
| import OAI.Analysis.FiniteFactor.OrbitSampling.RectangleApproximation |
|
|
| namespace OAI |
|
|
| namespace FiniteFactor |
| open MeasureTheory Filter Finset |
| open scoped Classical ENNReal Topology |
| noncomputable section |
|
|
| /-- Explicit finite-coordinate rectangle mass of the triangular/diagonal |
| model. The triangular term retains the actual source weight integral. -/ |
| def sourceSamplingLimitRectangleMass (I : Finset SourceCoordinate) (T : SourceSamplingRationalTest) : ℝ := |
| ∏ i ∈ I ∪ T.1, |
| if i ∈ I then |
| ∫ s, samplingTestedFirstDensity true (sourceSamplingWidth 0) (sourceSamplingCutoff 0) |
| (sourceSamplingRationalX T i) (sourceSamplingRationalY T i) s ∂unitIntervalMeasure |
| else (unitIntervalMeasure (sourceSamplingRationalX T i ∩ sourceSamplingRationalY T i)).toReal |
|
|
| /-- Exact finite-coordinate factorization of the first ideal integral. -/ |
| theorem sourceSamplingIdealDensity_integral (I : Finset SourceCoordinate) (stage : ℕ) |
| (E G : SourceCoordinate → Set ℝ) (hE : ∀ i, MeasurableSet (E i)) (hG : ∀ i, MeasurableSet (G i)) : |
| (∫ x, sourceSamplingIdealDensity stage false (fun i => decide (i ∈ I)) E G x ∂sourceSamplingHaar) = |
| ∏ i ∈ sourceSamplingPrefix stage, ∫ s, |
| samplingTestedFirstDensity (decide (i ∈ I)) (sourceSamplingWidth stage) (sourceSamplingCutoff stage) |
| (E i) (G i) s ∂unitIntervalMeasure := by |
| exact integral_infinitePi_finset_prod _ _ _ (fun i => |
| samplingTestedFirstDensity_measurable (decide (i ∈ I)) _ _ (E i) (G i) (hE i) (hG i)) |
|
|
| /-- Tests and triangular coordinates occupy a fixed finite set. Every other |
| prefix coordinate contributes its exact retained mass, 1-2α. -/ |
| theorem sourceSamplingIdeal_rectangle_split (I : Finset SourceCoordinate) (T : SourceSamplingRationalTest) |
| (stage : ℕ) (hK : I ∪ T.1 ⊆ sourceSamplingPrefix stage) : |
| (∫ x, sourceSamplingIdealDensity stage false (fun i => decide (i ∈ I)) |
| (sourceSamplingRationalX T) (sourceSamplingRationalY T) x ∂sourceSamplingHaar) = |
| (∏ i ∈ I ∪ T.1, ∫ s, |
| samplingTestedFirstDensity (decide (i ∈ I)) (sourceSamplingWidth stage) (sourceSamplingCutoff stage) |
| (sourceSamplingRationalX T i) (sourceSamplingRationalY T i) s ∂unitIntervalMeasure) * |
| (1-2*sourceSamplingCutoff stage)^((sourceSamplingPrefix stage \ (I ∪ T.1)).card) := by |
| rw [sourceSamplingIdealDensity_integral I stage (sourceSamplingRationalX T) (sourceSamplingRationalY T) (fun _ => measurableSet_Icc) (fun _ => measurableSet_Icc)] |
| let F := fun i => ∫ s, samplingTestedFirstDensity (decide (i ∈ I)) (sourceSamplingWidth stage) |
| (sourceSamplingCutoff stage) (sourceSamplingRationalX T i) (sourceSamplingRationalY T i) s ∂unitIntervalMeasure |
| have hf (i : SourceCoordinate) (hi : i ∈ sourceSamplingPrefix stage \ (I ∪ T.1)) : |
| F i = 1-2*sourceSamplingCutoff stage := by |
| have hnot := (Finset.mem_sdiff.mp hi).2 |
| have hI : i ∉ I := fun h => hnot (Finset.mem_union_left _ h) |
| have hT : i ∉ T.1 := fun h => hnot (Finset.mem_union_right _ h) |
| simp only [F,decide_eq_false hI,sourceSamplingRationalX,sourceSamplingRationalY, |
| sourceSamplingRationalEndpoints,dite_eq_right hT,Rat.cast_zero,Rat.cast_one] |
| have hp := sourceSampling_parameters stage |
| exact sampling_diagonal_whole_mass _ _ hp.1 hp.2.1 hp.2.2 (sourceSamplingCutoff_le_half stage) |
| have hp := Finset.prod_sdiff hK (f := F) |
| rw [Finset.prod_congr rfl hf,Finset.prod_const] at hp |
| simpa only [mul_comm] using hp.symm |
|
|
| /-- The deterministic ideal product limit includes all growing-prefix cutoff |
| losses and has exactly the tested diagonal-coordinate limits. -/ |
| theorem sourceSamplingIdeal_rectangle_tendsto (I : Finset SourceCoordinate) (T : SourceSamplingRationalTest) : |
| Tendsto (fun stage => ∫ x, sourceSamplingIdealDensity stage false (fun i => decide (i ∈ I)) |
| (sourceSamplingRationalX T) (sourceSamplingRationalY T) x ∂sourceSamplingHaar) |
| atTop (𝓝 (sourceSamplingLimitRectangleMass I T)) := by |
| let F := fun stage i => ∫ s, samplingTestedFirstDensity (decide (i ∈ I)) (sourceSamplingWidth stage) |
| (sourceSamplingCutoff stage) (sourceSamplingRationalX T i) (sourceSamplingRationalY T i) s ∂unitIntervalMeasure |
| have hcoord (i : SourceCoordinate) : Tendsto (fun stage => F stage i) atTop |
| (𝓝 (if i ∈ I then ∫ s, samplingTestedFirstDensity true (sourceSamplingWidth 0) (sourceSamplingCutoff 0) |
| (sourceSamplingRationalX T i) (sourceSamplingRationalY T i) s ∂unitIntervalMeasure |
| else (unitIntervalMeasure (sourceSamplingRationalX T i ∩ sourceSamplingRationalY T i)).toReal)) := by |
| by_cases hi : i ∈ I |
| · simp only [ite_eq_left hi,F,decide_eq_true hi] |
| have he (stage : ℕ) : (∫ s, samplingTestedFirstDensity true (sourceSamplingWidth stage) (sourceSamplingCutoff stage) |
| (sourceSamplingRationalX T i) (sourceSamplingRationalY T i) s ∂unitIntervalMeasure) = |
| ∫ s, samplingTestedFirstDensity true (sourceSamplingWidth 0) (sourceSamplingCutoff 0) |
| (sourceSamplingRationalX T i) (sourceSamplingRationalY T i) s ∂unitIntervalMeasure := by |
| apply integral_congr_ae |
| exact Eventually.of_forall (fun s => samplingTestedFirstDensity_triangular_parameters _ _ _ _ _ _ _) |
| simp_rw [he] |
| exact tendsto_const_nhds |
| · simp only [ite_eq_right hi,F,decide_eq_false hi] |
| exact sourceSampling_diagonal_integral_tendsto _ measurableSet_Icc _ _ |
| have hfixed := tendsto_finsetProd (I ∪ T.1) (fun i _ => hcoord i) |
| have htail := sourceSampling_untested_mass_tendsto |
| (fun stage => (sourceSamplingPrefix stage \ (I ∪ T.1)).card) |
| (fun stage => Finset.card_le_card Finset.sdiff_subset) |
| have hall := hfixed.mul htail |
| simp only [mul_one] at hall |
| apply hall.congr' |
| filter_upwards [sourceSamplingPrefix_eventually_covers (I ∪ T.1)] with stage hstage |
| exact (sourceSamplingIdeal_rectangle_split I T stage hstage).symm |
| |
| /-- Actual untrimmed rational rectangle masses converge to the explicit |
| triangular/diagonal product value on one conull event for all I and tests. -/ |
| theorem sourceSampling_rectangle_limit_event : |
| ∀ᵐ θ ∂sourceAngleLaw, ∀ (I : Finset SourceCoordinate) (T : SourceSamplingRationalTest), |
| Tendsto (fun stage => ((sourceSamplingUntrimmed I stage θ) |
| (sourceSamplingRectangle (sourceSamplingRationalX T) (sourceSamplingRationalY T))).toReal) |
| atTop (𝓝 (sourceSamplingLimitRectangleMass I T)) := by |
| filter_upwards [sourceSampling_rectangle_approximation_event] with θ hθ |
| intro I T |
| have hh := (hθ I T).add (sourceSamplingIdeal_rectangle_tendsto I T) |
| simpa only [sub_add_cancel,zero_add] using hh |
| |
| end |
| end FiniteFactor |
| |
| end OAI |
| |
