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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 | import OAI.Analysis.FiniteFactor.OrbitSampling.SecondMarginal
namespace OAI
namespace FiniteFactor
open MeasureTheory Filter Finset
open scoped Classical ENNReal
noncomputable section
def sourceSamplingRectangle (E G : SourceCoordinate → Set ℝ) :
Set ((SourceCoordinate → ℝ) × (SourceCoordinate → ℝ)) :=
{p | ∀ i, p.1 i ∈ E i ∧ p.2 i ∈ G i}
theorem sourceSamplingRectangle_measurable (E G : SourceCoordinate → Set ℝ)
(hE : ∀ i, MeasurableSet (E i)) (hG : ∀ i, MeasurableSet (G i)) :
MeasurableSet (sourceSamplingRectangle E G) := by
simp only [sourceSamplingRectangle,Set.ofPred_forall]
apply MeasurableSet.iInter
intro i
exact ((hE i).preimage ((measurable_pi_apply i).comp measurable_fst)).inter
((hG i).preimage ((measurable_pi_apply i).comp measurable_snd))
theorem sourceSamplingProduct_first_indicator (I : Finset SourceCoordinate) (stage : ℕ)
(E G : SourceCoordinate → Set ℝ)
(houtside : ∀ i ∉ sourceSamplingPrefix stage,
Set.Ico (0 : ℝ) 1 ⊆ E i ∧ Set.Ico (0 : ℝ) 1 ⊆ G i)
(θ x : SourceCoordinate → ℝ) (n : ℤ)
(hx : ∀ i, x i ∈ Set.Ico (0 : ℝ) 1) :
sourceSamplingProduct stage false (fun i => decide (i ∈ I)) E G n θ x =
if (x,coordinateRotate θ n x) ∈ sourceSamplingRectangle E G
then sourceSamplingWeight I stage θ n x else 0 := by
have hy (i) : coordinateRotate θ n x i ∈ Set.Ico (0 : ℝ) 1 :=
⟨Int.fract_nonneg _,Int.fract_lt_one _⟩
have he : (∀ i ∈ sourceSamplingPrefix stage, x i ∈ E i ∧ coordinateRotate θ n x i ∈ G i) ↔
(x,coordinateRotate θ n x) ∈ sourceSamplingRectangle E G := by
constructor
· intro h i
by_cases hi : i ∈ sourceSamplingPrefix stage
· exact h i hi
· exact ⟨(houtside i hi).1 (hx i),(houtside i hi).2 (hy i)⟩
· intro h i _
exact h i
unfold sourceSamplingProduct samplingOrientedWeight samplingTestedFirstWeight
simp only [Bool.false_eq_true,ite_false,← coordinateRotate_phase]
simp only [Finset.prod_ite_zero,he]
rfl
/-- Inverse rotation exchanges the two actual tested product orientations. -/
theorem sourceSamplingProduct_second_inverse (I : Finset SourceCoordinate) (stage : ℕ)
(E G : SourceCoordinate → Set ℝ) (θ y : SourceCoordinate → ℝ) (n : ℤ)
(hy : ∀ i, y i ∈ Set.Ico (0 : ℝ) 1) :
sourceSamplingProduct stage false (fun i => decide (i ∈ I)) E G n θ
(coordinateRotate θ (-n) y) =
sourceSamplingProduct stage true (fun i => decide (i ∈ I)) E G n θ y := by
have hinv : coordinateRotate θ n (coordinateRotate θ (-n) y) = y := by
simpa only [neg_neg] using coordinateRotate_neg_cube θ y (-n) hy
apply Finset.prod_congr rfl
intro i _
have hi := congrFun hinv i
rw [coordinateRotate_phase] at hi
simp only [samplingOrientedWeight,Bool.false_eq_true,ite_false,ite_true,samplingTestedFirstWeight,hi]
rw [coordinateRotate_neg_phase]
/-- The restricted orbit-pair summand is exactly the pushforward of its
first tested product, with no assumed marginal law. -/
theorem sourceSamplingSummand_restrict_rectangle (I : Finset SourceCoordinate) (stage : ℕ)
(E G : SourceCoordinate → Set ℝ)
(hE : ∀ i, MeasurableSet (E i)) (hG : ∀ i, MeasurableSet (G i))
(houtside : ∀ i ∉ sourceSamplingPrefix stage,
Set.Ico (0 : ℝ) 1 ⊆ E i ∧ Set.Ico (0 : ℝ) 1 ⊆ G i)
(θ : SourceCoordinate → ℝ) (n : ℤ) :
(sourceSamplingSummand I stage θ n).restrict (sourceSamplingRectangle E G) =
(sourceSamplingHaar.withDensity (fun x => ENNReal.ofReal
(sourceSamplingProduct stage false (fun i => decide (i ∈ I)) E G n θ x))).map
(fun x => (x,coordinateRotate θ n x)) := by
have hp : Measurable (fun x : SourceCoordinate → ℝ => (x,coordinateRotate θ n x)) :=
measurable_id.prodMk (coordinateRotate_preserving θ n).measurable
have hR := sourceSamplingRectangle_measurable E G hE hG
unfold sourceSamplingSummand sourceSamplingWeightedMeasure
rw [Measure.restrict_map hp hR,restrict_withDensity (hp hR),← withDensity_indicator (hp hR)]
congr 1
apply withDensity_congr_ae
filter_upwards [sourceSamplingHaar_cube] with x hx
rw [sourceSamplingProduct_first_indicator I stage E G houtside θ x n hx]
change (if (x,coordinateRotate θ n x) ∈ sourceSamplingRectangle E G
then ENNReal.ofReal (sourceSamplingWeight I stage θ n x) else 0) = _
split_ifs <;> simp
end
end FiniteFactor
end OAI
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