openAI-math-dataset / lean /OAI /Probability /Ballisticity /Geometry /WeightedConditionedTubeTransfer.lean
| import Mathlib | |
| import OAI.Probability.Ballisticity.Estimates.RecordPrefixControl | |
| namespace OAI | |
| section | |
| section | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Pointwise Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Pointwise Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical BoundedContinuousFunction | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical BoundedContinuousFunction | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| namespace DirectionalTransience | |
| theorem weightedConditioned_tube_transfer {d : ℕ} (ν : Measure (Row d)) [IsProbabilityMeasure ν] | |
| (ℓ : Vector d) (f : Direction d) (htrans : DirectionallyTransient ν ℓ) | |
| (b : ℕ → ℝ) (H : ℕ → ℕ) (z : ℕ → ℝ) (hz : Tendsto z atTop atTop) | |
| (hsmall : ∀ᶠ i in atTop, (conditionedLaw ν ℓ).real | |
| (MedianTubeFailure ℓ f b (H i) (z i/10)) ≤ 3/10) | |
| (J : ℝ) (hJ : 0 ≤ J) (g : Environment d → ℝ≥0∞) | |
| (hg : @Measurable _ _ (rowSigma {x | dot (realPosition x) ℓ ≤ J}) _ g) | |
| (hgle : ∀ ω, g ω ≤ 1) {ε : ℝ} (hε : 0 < ε) : | |
| ∀ᶠ i in atTop, (weightedConditioned ν ℓ g).real (MedianTubeFailure ℓ f b (H i) (z i)) ≤ | |
| (weightedConditioned ν ℓ g).real Set.univ * | |
| (conditionedLaw ν ℓ).real (MedianTubeFailure ℓ f b (H i) (z i/10))+ε := by | |
| let μ := conditionedLaw ν ℓ | |
| let : IsProbabilityMeasure μ := conditionedLaw_probability ν ℓ | |
| (ne_of_gt (noDrop_positive_of_directionallyTransient ν ℓ htrans)) | |
| let η := weightedConditioned ν ℓ g | |
| let : IsFiniteMeasure η := weightedConditioned_finite ν ℓ g (hg.mono (rowSigma_le _) le_rfl) hgle | |
| have hac : η ≪ μ := Measure.absolutelyContinuous_of_le | |
| (weightedConditioned_le ν ℓ g (hg.mono (rowSigma_le _) le_rfl) hgle) | |
| have hcnt : Measurable (fun X => (highRecordCount ℓ J X:ℝ)) := | |
| (measurable_of_countable (fun k : ℕ => (k:ℝ))).comp (measurable_highRecordCount ℓ J hJ) | |
| have hc := fixed_real_tail_small η _ hcnt (fun n : ℕ => (n:ℝ)) tendsto_natCast_atTop_atTop | |
| obtain ⟨R,hR⟩ := (hc.eventually (gt_mem_nhds (show (0:ℝ) < ε/2 by positivity))).exists | |
| have hR' : η.real {X | R < highRecordCount ℓ J X} < ε/2 := by | |
| apply lt_of_le_of_lt (measureReal_mono (μ := η) ?_) hR | |
| intro X hX | |
| change (R:ℝ) ≤ |(highRecordCount ℓ J X:ℝ)| | |
| rw [abs_of_nonneg (Nat.cast_nonneg _)] | |
| exact_mod_cast hX.le | |
| have hp := fixed_real_tail_small η _ (measurable_recordPrefixControl ℓ f b J hJ) | |
| (fun i => z i/10) (hz.atTop_div_const (by norm_num)) | |
| have hpe := hp.eventually (gt_mem_nhds (show (0:ℝ) < ε/2 by positivity)) | |
| have hmed := median_shift_eventually ν ℓ f htrans b H z hz hsmall R | |
| filter_upwards [hpe,hmed,hz.eventually_gt_atTop 0] with i hi hmi hzi | |
| let A := MedianTubeFailure ℓ f b (H i) (z i/10) | |
| let B := highSuffix ℓ J ⁻¹' A | |
| let C := {X | R < highRecordCount ℓ J X} | |
| let E := {X | z i/10 ≤ |recordPrefixControl ℓ f b J X|} | |
| have hs := hac.ae_le (medianTube_suffix_subset_ae ν ℓ f htrans b J hJ (H i) R hzi hmi) | |
| have hb : η.real (MedianTubeFailure ℓ f b (H i) (z i)) ≤ η.real (B ∪ C ∪ E) := | |
| ENNReal.toReal_mono (measure_ne_top _ _) (measure_mono_ae hs) | |
| have hmap := weightedConditioned_highSuffix ν ℓ htrans J hJ g hg hgle | |
| have hB : η.real B = η.real Set.univ * μ.real A := by | |
| have hh := congrArg (fun ρ : Measure (Path d) => ρ A) hmap | |
| rw [Measure.map_apply (measurable_highSuffix ℓ J hJ) | |
| (measurableSet_medianTubeFailure ℓ f b (H i) (z i/10)),Measure.smul_apply] at hh | |
| have hhr := congrArg ENNReal.toReal hh | |
| simpa only [smul_eq_mul,ENNReal.toReal_mul,Measure.real,η,μ,B] using hhr | |
| have hu := measureReal_union_le (μ := η) (B ∪ C) E | |
| have hu' := measureReal_union_le (μ := η) B C | |
| change η.real E < ε/2 at hi | |
| change η.real C < ε/2 at hR' | |
| change η.real (MedianTubeFailure ℓ f b (H i) (z i)) ≤ η.real Set.univ*μ.real A+ε | |
| linarith | |
| end DirectionalTransience | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| end | |
| end | |
| end OAI | |