| 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)) |
|
|
| / |
| 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 |
| |
| / |
| 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 |
| |
| / |
| 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) |
| |
| / |
| 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 |
| |
