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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 99 100 101 102 103 104 105 106 | 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
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