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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 | import OAI.Analysis.FiniteFactor.OrbitSampling.Cube
import OAI.Analysis.FiniteFactor.OrbitSampling.WeightedTail
namespace OAI
namespace FiniteFactor
open MeasureTheory Filter Finset
open scoped Classical ENNReal Topology
noncomputable section
def sourceSamplingOrdered : Set SourceSamplingPair := {p | ∀ i, p.1 i ≤ p.2 i}
theorem sourceSamplingOrdered_measurable : MeasurableSet sourceSamplingOrdered := by
simp only [sourceSamplingOrdered,Set.ofPred_forall]
exact MeasurableSet.iInter (fun i => measurableSet_le
((measurable_pi_apply i).comp measurable_fst) ((measurable_pi_apply i).comp measurable_snd))
/-- The actual ordered restriction of the already constructed average. -/
def sourceSamplingTrimmed (I : Finset SourceCoordinate) (s : ℕ) (θ : SourceCoordinate → ℝ) :
Measure SourceSamplingPair := (sourceSamplingUntrimmed I s θ).restrict sourceSamplingOrdered
instance sourceSamplingTrimmed_finite (I : Finset SourceCoordinate) (s : ℕ)
(θ : SourceCoordinate → ℝ) : IsFiniteMeasure (sourceSamplingTrimmed I s θ) := by
unfold sourceSamplingTrimmed
infer_instance
theorem sourceSamplingTrimmed_ordered (I : Finset SourceCoordinate) (s : ℕ)
(θ : SourceCoordinate → ℝ) : ∀ᵐ p ∂sourceSamplingTrimmed I s θ, ∀ i, p.1 i ≤ p.2 i :=
ae_restrict_mem sourceSamplingOrdered_measurable
theorem sourceSamplingTrimmed_le (I : Finset SourceCoordinate) (s : ℕ)
(θ : SourceCoordinate → ℝ) : sourceSamplingTrimmed I s θ ≤ sourceSamplingUntrimmed I s θ :=
Measure.restrict_le_self
theorem sourceSamplingPowers_positive (s : ℕ) (n : ℤ) (hn : n ∈ sourceSamplingPowers s) :
0 < n ∧ (n : ℝ) ≤ samplingQ (sourceBlockSize s) := by
by_cases hs : 200000 ≤ sourceBlockSize s
· exact ⟨((sourceSamplingPowers_spec s hs).1 n hn).2.2,
((sourceSamplingPowers_spec s hs).1 n hn).2.1⟩
· simp [sourceSamplingPowers,hs] at hn
/-- Outside the finite prefix, every failure of global order is an actual
member of the source's countable tail-failure event. -/
theorem sourceSampling_order_failure_tail (s n : ℕ) (θ x : SourceCoordinate → ℝ)
(hprefix : ∀ i ∈ sourceSamplingPrefix s, x i < coordinateRotate θ (n : ℤ) x i)
(hbad : (x,coordinateRotate θ (n : ℤ) x) ∉ sourceSamplingOrdered) :
x ∈ sourceTailOrderFailure θ (s+1) n := by
have hb : ∃ i, coordinateRotate θ (n : ℤ) x i < x i := by
simpa only [sourceSamplingOrdered,Set.mem_ofPred_eq,not_forall,not_le] using hbad
obtain ⟨i,hi⟩ := hb
have hout : i ∉ sourceSamplingPrefix s := fun hh => (hprefix i hh).asymm hi
have hk : s+1 ≤ i.1 := by
have hh := (CouplingParameters.mem_blockPrefix sourceBlockSize (s+1) i)
change i ∉ CouplingParameters.blockPrefix sourceBlockSize (s+1) at hout
have hnot : ¬ i.1 < s+1 := fun hi => hout (hh.mpr hi)
omega
rcases i with ⟨k,j⟩
obtain ⟨r,rfl⟩ := Nat.exists_eq_add_of_le hk
apply Set.mem_iUnion.mpr ⟨r,?_⟩
apply Set.mem_iUnion.mpr ⟨j,?_⟩
simpa only [coordinateRotate,Int.cast_natCast,Set.mem_ofPred_eq] using hi
/-- Discarded mass of each real summand obeys the preserved relative
weighted-tail estimate. This includes summands of mass zero. -/
theorem sourceSamplingSummand_discarded_le (I : Finset SourceCoordinate) (s : ℕ)
(θ : SourceCoordinate → ℝ) (hθ : SourceAngles θ) (n : ℤ)
(hn : n ∈ sourceSamplingPowers s) :
sourceSamplingSummand I s θ n sourceSamplingOrderedᶜ ≤
sourceSamplingSummand I s θ n Set.univ *
ENNReal.ofReal (4*(sourceBlockSize s : ℝ)^(-(sourceBlockSize s : ℝ)/20)) := by
have hp := sourceSamplingPowers_positive s n hn
have he : (n.toNat : ℤ) = n := Int.toNat_of_nonneg hp.1.le
have hm : Measurable (fun x => (x,coordinateRotate θ n x)) :=
measurable_id.prodMk (coordinateRotate_preserving θ n).measurable
rw [sourceSamplingSummand,Measure.map_apply hm sourceSamplingOrdered_measurable.compl,
Measure.map_apply hm MeasurableSet.univ,Set.preimage_univ]
have hc : (n.toNat : ℝ) = (n : ℝ) := by exact_mod_cast he
have hb := sourceSampling_weighted_tail_small I s θ hθ n.toNat (by rw [hc]; exact hp.2)
rw [he] at hb
apply le_trans _ hb
apply measure_mono_ae
filter_upwards [sourceSamplingWeightedMeasure_orders I s θ n] with x hx
intro hb
change (x,coordinateRotate θ n x) ∉ sourceSamplingOrdered at hb
exact sourceSampling_order_failure_tail s n.toNat θ x (by simpa [he] using hx) (by simpa [he] using hb)
/-- The average loses at most the same relative error as each summand. -/
theorem sourceSamplingUntrimmed_discarded_le (I : Finset SourceCoordinate) (s : ℕ)
(θ : SourceCoordinate → ℝ) (hθ : SourceAngles θ) :
sourceSamplingUntrimmed I s θ sourceSamplingOrderedᶜ ≤
sourceSamplingUntrimmed I s θ Set.univ *
ENNReal.ofReal (4*(sourceBlockSize s : ℝ)^(-(sourceBlockSize s : ℝ)/20)) := by
simp only [sourceSamplingUntrimmed,Measure.smul_apply,smul_eq_mul,Measure.finsetSum_apply]
rw [mul_assoc,Finset.sum_mul]
exact mul_le_mul' le_rfl (Finset.sum_le_sum (fun n hn => sourceSamplingSummand_discarded_le I s θ hθ n hn))
end
end FiniteFactor
end OAI
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