| import Mathlib |
| import OAI.Probability.Ballisticity.Geometry.WindowWeak |
| import OAI.Probability.Ballisticity.Estimates.BadOccupationLimit |
| import OAI.Probability.Ballisticity.Stationary.BadArraySampling |
|
|
| namespace OAI |
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| section |
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|
| open MeasureTheory ProbabilityTheory InformationTheory Filter |
| open scoped ENNReal NNReal Classical Topology |
| namespace DirectionalTransience |
| namespace OperationalConstants |
| variable {d : ℕ} {ν : Measure (Row d)} |
| {e f : Direction d} {D : ℝ} |
|
|
| lemma paddedTimes_stopping [IsProbabilityMeasure ν] |
| (C : OperationalConstants ν e f D) (hef : e.1≠f.1) |
| (N : ℕ) (i : ℤ) (n : ℕ) : |
| MeasurableSet[rowSigma (BelowHeight (realPosition (step e)) n)] |
| {ω | C.paddedTimes hef N ω i=n} := |
| episodeBoundary_height_event (k:=C.k) e f hef (episodeScaleRadius ν e f) |
| C.fexp C.g C.χ C.b C.sfloor C.radius_nonneg N i.toNat n |
|
|
| lemma paddedTimes_mono [IsProbabilityMeasure ν] |
| (C : OperationalConstants ν e f D) (hef : e.1≠f.1) |
| (N : ℕ) (ω : Environment d) : Monotone (fun i : ℕ => C.paddedTimes hef N ω i) := by |
| simpa only [paddedTimes,EpisodeChainLedger.arrayTime_nat] using |
| episodeBoundary_mono (k:=C.k) e f hef (episodeScaleRadius ν e f) |
| C.fexp C.g C.χ C.b C.sfloor C.radius_nonneg N ω |
|
|
| lemma paddedTimes_active [IsProbabilityMeasure ν] |
| (C : OperationalConstants ν e f D) (hef : e.1≠f.1) |
| (N i : ℕ) (ω : Environment d) : |
| C.paddedTimes hef N ω i<N ↔ i<C.activeCount hef N ω := by |
| simpa only [paddedTimes,EpisodeChainLedger.arrayTime_nat,activeCount] using |
| EpisodeChainLedger.active_iff_lt_number (k:=C.k) e f hef (episodeScaleRadius ν e f) |
| C.fexp C.g C.χ C.b C.sfloor C.radius_nonneg N ω i |
|
|
| noncomputable def windowEntropyConstant (C : OperationalConstants ν e f D) : ℝ := |
| (8/3:ℝ)*D*C.injectionCost |
|
|
| lemma windowEntropyConstant_nonneg [IsProbabilityMeasure ν] |
| (C : OperationalConstants ν e f D) (hD : 0≤D) : |
| 0≤C.windowEntropyConstant := mul_nonneg (mul_nonneg (by norm_num) hD) C.cost_nonneg |
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| variable [IsProbabilityMeasure ν] |
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| lemma bad_window_kl_le (C : OperationalConstants ν e f D) (hef : e.1≠f.1) (hD : 0≤D) |
| (N m : ℕ) (hN : C.sfloor ≤ (N:ℝ)) |
| (hlarge : 32*C.b ≤ (1/2:ℝ)*Real.log (N:ℝ)) |
| (hmass : (N:ℝ)^(-D) ≤ (environmentLaw ν).real (badCrossingEvent e N (1/2))) : |
| let W := actualCurrentLaw e m |
| (C.occupation hef N ((environmentLaw ν)[|badCrossingEvent e N (1/2)])) |
| klDiv (W : Measure (CurrentWindow e)) (currentReferenceLaw e ν W : Measure (CurrentWindow e)) ≤ |
| ENNReal.ofReal ((m:ℝ)*C.windowEntropyConstant) := by |
| let Q := (environmentLaw ν)[|badCrossingEvent e N (1/2)] |
| have hN0 : 0<N := by exact_mod_cast lt_of_lt_of_le zero_lt_one (C.hs.trans hN) |
| have : IsProbabilityMeasure Q := |
| cond_isProbabilityMeasure (badCrossingEvent_pos e N (1/2) D (environmentLaw ν) hN0 hmass) |
| have hkl := badCrossing_cond_entropy e N (1/2) D (environmentLaw ν) hN0 hmass |
| have hfin : klDiv Q (environmentLaw ν)≠∞ := ne_top_of_le_ne_top ENNReal.ofReal_ne_top hkl |
| have hpos := C.bad_raw_mass_pos hef N hN hmass |
| have hb := actualOccupation_window_kl_le e ν Q hfin |
| (C.paddedTimes hef N) (C.paddedStages hef N) |
| (C.paddedTimes_measurable hef N) (C.paddedStages_measurable hef N) |
| (C.paddedTimes_stopping hef N) (C.paddedTimes_mono hef N) N m (C.activeCount hef N) |
| (C.paddedTimes_active hef N) hpos |
| have he := EpisodeChainLedger.operational_raw_mass (k:=C.k) e f hef (episodeScaleRadius ν e f) |
| C.fexp C.g C.χ C.b C.sfloor C.radius_nonneg N ν Q |
| change (actualOccupationRaw e ν Q (C.paddedTimes hef N) (C.paddedStages hef N) |
| (C.paddedTimes_measurable hef N) N (C.activeCount hef N)).real Set.univ = _ at he |
| rw [he] at hb hpos |
| apply hb.trans |
| apply ENNReal.ofReal_le_ofReal |
| have hlog : 0≤Real.log (N:ℝ) := Real.log_nonneg (by exact_mod_cast Nat.one_le_iff_ne_zero.mpr hN0.ne') |
| have hkreal : (klDiv Q (environmentLaw ν)).toReal ≤ D*Real.log (N:ℝ) := by |
| have hh := ENNReal.toReal_mono ENNReal.ofReal_ne_top hkl |
| rwa [ENNReal.toReal_ofReal (mul_nonneg hD hlog)] at hh |
| obtain ⟨L,hL,hcharge⟩ := C.finite_charge |
| have hl := (C.conditional_episode_drops hef hD N hN hlarge hmass L hL hcharge).1 |
| have hd := mul_le_mul_of_nonneg_left hl hD |
| have hbudget : (klDiv Q (environmentLaw ν)).toReal ≤ |
| C.windowEntropyConstant*(∫ ω, (C.activeCount hef N ω:ℝ) ∂Q) := by |
| unfold windowEntropyConstant |
| dsimp only [Q] |
| nlinarith only [hkreal,hd] |
| apply (div_le_iff₀ hpos).mpr |
| have hh := mul_le_mul_of_nonneg_left hbudget (Nat.cast_nonneg m) |
| simpa only [activeCount,mul_assoc] using hh |
|
|
| lemma bad_occupation_currentOffsets (C : OperationalConstants ν e f D) (hef : e.1≠f.1) |
| (N : ℕ) (hN : C.sfloor≤(N:ℝ)) |
| (hmass : (N:ℝ)^(-D) ≤ (environmentLaw ν).real (badCrossingEvent e N (1/2))) : |
| ∀ᵐ Y ∂(C.occupation hef N ((environmentLaw ν)[|badCrossingEvent e N (1/2)]) : Measure (ActualEpisodeArray e)), |
| ∀ j, ReferenceClasses.Consistent (currentOffsets e j Y) := |
| actualOccupation_currentOffsets e ν _ (C.paddedTimes hef N) (C.paddedStages hef N) |
| (C.paddedTimes_measurable hef N) (C.paddedStages_measurable hef N) N (C.activeCount hef N) |
| (C.bad_raw_mass_pos hef N hN hmass) |
|
|
| theorem bad_limit_window_entropy (C : OperationalConstants ν e f D) (hef : e.1≠f.1) (hD : 0≤D) |
| (Ns : ℕ → ℕ) |
| (hN : ∀ n, C.sfloor ≤ (Ns n:ℝ)) |
| (hlarge : ∀ n, 32*C.b ≤ (1/2:ℝ)*Real.log (Ns n:ℝ)) |
| (hmass : ∀ n, (Ns n:ℝ)^(-D) ≤ (environmentLaw ν).real (badCrossingEvent e (Ns n) (1/2))) |
| (ρ : ProbabilityMeasure (ActualEpisodeArray e)) |
| (hlim : Tendsto (fun n => C.occupation hef (Ns n) |
| ((environmentLaw ν)[|badCrossingEvent e (Ns n) (1/2)])) atTop (𝓝 ρ)) (m : ℕ) : |
| klDiv (actualCurrentLaw e m ρ : Measure (CurrentWindow e)) |
| (currentReferenceLaw e ν (actualCurrentLaw e m ρ) : Measure (CurrentWindow e)) ≤ |
| ENNReal.ofReal ((m:ℝ)*C.windowEntropyConstant) := |
| actualCurrentLaw_kl_weak_limit e ν m _ ρ hlim |
| (fun n => C.bad_occupation_currentOffsets hef (Ns n) (hN n) (hmass n)) |
| ((m:ℝ)*C.windowEntropyConstant) (mul_nonneg (Nat.cast_nonneg m) (C.windowEntropyConstant_nonneg hD)) |
| (fun n => C.bad_window_kl_le hef hD (Ns n) m (hN n) (hlarge n) (hmass n)) |
|
|
| end OperationalConstants |
| end DirectionalTransience |
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| end |
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| end OAI |
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