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