| import Mathlib |
| import OAI.Probability.Ballisticity.Geometry.OccupationWindow |
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|
| namespace OAI |
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| section |
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| open MeasureTheory ProbabilityTheory InformationTheory |
| open scoped ENNReal Classical |
| namespace DirectionalTransience |
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|
| noncomputable def actualCurrentLaw {d : ℕ} (e : Direction d) (m : ℕ) |
| (L : ProbabilityMeasure (ActualEpisodeArray e)) : ProbabilityMeasure (CurrentWindow e) := |
| L.map (typedCurrentArrayWindow e 0 m) |
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|
| lemma finiteMeasure_ne_zero_of_real_mass_pos {X : Type*} [MeasurableSpace X] |
| (μ : FiniteMeasure X) (h : 0<(μ : Measure X).real Set.univ) : μ≠0 := by |
| intro hz |
| rw [hz] at h |
| simp at h |
|
|
| theorem actualOccupation_window_kl_le {d : ℕ} (e : Direction d) |
| (ν : Measure (Row d)) [IsProbabilityMeasure ν] |
| (Q : Measure (Environment d)) [IsProbabilityMeasure Q] |
| (hfin : klDiv Q (environmentLaw ν)≠∞) |
| (t J : Environment d → ℤ → ℕ) |
| (ht : ∀ i, Measurable fun ω => t ω i) (hJ : ∀ i, Measurable fun ω => J ω i) |
| (hst : ∀ (i : ℤ) (n : ℕ), MeasurableSet[rowSigma (BelowHeight (realPosition (step e)) n)] {ω | t ω i=n}) |
| (hmono : ∀ ω, Monotone (fun i : ℕ => t ω i)) |
| (N m : ℕ) (M : Environment d → ℕ) |
| (hact : ∀ (i : ℕ) ω, t ω i<N ↔ i<M ω) |
| (hpos : 0<(actualOccupationRaw e ν Q t J ht N M).real Set.univ) : |
| let W := (actualCurrentLaw e m (actualOccupation e ν Q t J ht N M) : Measure (CurrentWindow e)) |
| klDiv W (W.fst.compProd (currentWindowReference e ν)) ≤ |
| ENNReal.ofReal ((m:ℝ)*(klDiv Q (environmentLaw ν)).toReal / |
| (actualOccupationRaw e ν Q t J ht N M).real Set.univ) := by |
| have : Nonempty (Row d) := nonempty_of_isProbabilityMeasure ν |
| have : Nonempty (ReferenceClasses.Data (HorizontalSpace e)) := ⟨ReferenceClasses.anchorOffsets (fun _ => 0)⟩ |
| let filler : Environment d := fun _ => Classical.choice ‹Nonempty (Row d)› |
| let μ : FiniteMeasure (RawCurrentData e × LocalUpperField e) := |
| ⟨rawCurrentOccupation e ν Q (fun i ω => t ω i) (fun i => ht i) filler N m,inferInstance⟩ |
| let ρ : FiniteMeasure (ActualEpisodeArray e) := ⟨actualOccupationRaw e ν Q t J ht N M,inferInstance⟩ |
| have he := rawCurrentOccupation_actual e ν Q t J ht hJ hst filler N m M hact |
| have hmass : (μ : Measure (RawCurrentData e × LocalUpperField e)).real Set.univ= |
| (ρ : Measure (ActualEpisodeArray e)).real Set.univ := |
| Entropy.real_mass_eq_of_maps_equal _ _ _ _ (rawCurrentProjection_measurable e) |
| (typedCurrentArrayWindow_measurable e 0 m) he |
| have hμpos : 0<(μ : Measure (RawCurrentData e × LocalUpperField e)).real Set.univ := by rw [hmass]; exact hpos |
| have hn := Entropy.normalize_maps_equal μ ρ (rawCurrentProjection e) (typedCurrentArrayWindow e 0 m) |
| (rawCurrentProjection_measurable e) (typedCurrentArrayWindow_measurable e 0 m) |
| (finiteMeasure_ne_zero_of_real_mass_pos μ hμpos) (finiteMeasure_ne_zero_of_real_mass_pos ρ hpos) he |
| have hb := currentOccupation_kl_le e ν Q hfin (fun i ω => t ω i) |
| (fun i => ht i) (fun i => hst i) hmono filler N m hμpos |
| change klDiv ((μ.normalize : Measure _).map (rawCurrentProjection e)) |
| (((μ.normalize : Measure _).map (rawCurrentProjection e)).fst.compProd (currentWindowReference e ν)) ≤ _ at hb |
| rw [hn] at hb |
| change klDiv ((ρ.normalize : Measure _).map (typedCurrentArrayWindow e 0 m)) |
| (((ρ.normalize : Measure _).map (typedCurrentArrayWindow e 0 m)).fst.compProd (currentWindowReference e ν)) ≤ _ |
| convert hb using 1 |
| exact congrArg (fun x : ℝ => ENNReal.ofReal ((m:ℝ)*(klDiv Q (environmentLaw ν)).toReal/x)) hmass.symm |
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| end DirectionalTransience |
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| end |
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| end OAI |
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