| import Mathlib |
| import OAI.Probability.Ballisticity.Estimates.AnchorFrameReference |
| import OAI.Probability.Ballisticity.Geometry.RawWindow |
| import OAI.Probability.Ballisticity.Estimates.InputMarginal |
|
|
| namespace OAI |
|
|
| section |
|
|
| open MeasureTheory ProbabilityTheory InformationTheory |
| open scoped ENNReal Classical |
| namespace DirectionalTransience |
|
|
| abbrev CurrentProfiles {d : ℕ} (e : Direction d) := |
| (ℕ × HorizontalSpace e) → Set.Icc (0 : ℝ) 1 |
|
|
| abbrev CurrentWindow {d : ℕ} (e : Direction d) := |
| (ReferenceClasses.Data (HorizontalSpace e) × CurrentProfiles e) × |
| ReferenceClasses.AllFields (HorizontalSpace e) (Row d) |
|
|
| noncomputable def rawCurrentProfiles {d : ℕ} (e : Direction d) |
| (z : RawCurrentData e) : CurrentProfiles e := |
| fun p => globalHorizontalCoordinate e z.1 z.2.1 (z.2.2 p.1 + p.2) |
|
|
| lemma rawCurrentProfileCoordinate_measurable {d : ℕ} (e : Direction d) |
| (a : ℕ) (z : HorizontalSpace e) : |
| Measurable (fun p : RawCurrentData e => globalHorizontalCoordinate e p.1 p.2.1 (p.2.2 a+z)) := by |
| have hw : Measurable (fun p : RawCurrentData e => p.2.1) := measurable_fst.comp measurable_snd |
| have hn : Measurable (fun p : RawCurrentData e => p.1) := measurable_fst |
| have hu : Measurable (fun p : RawCurrentData e => p.2.2 a) := |
| (measurable_pi_apply a).comp (measurable_snd.comp measurable_snd) |
| have ha : Measurable (fun p : RawCurrentData e => ((p.2.1,p.1),p.2.2 a+z)) := |
| (hw.prodMk hn).prodMk (hu.add_const z) |
| have hm := (globalHorizontalCoordinate_joint_measurable e).comp ha |
| simpa only [Function.comp_def] using hm |
|
|
| lemma rawCurrentProfiles_measurable {d : ℕ} (e : Direction d) : |
| Measurable (rawCurrentProfiles e) := |
| Measurable.of_eval fun p => rawCurrentProfileCoordinate_measurable e p.1 p.2 |
|
|
| noncomputable def rawCurrentProjection {d : ℕ} (e : Direction d) |
| (p : RawCurrentData e × LocalUpperField e) : CurrentWindow e := |
| ((ReferenceClasses.anchorOffsets p.1.2.2,rawCurrentProfiles e p.1), |
| ReferenceClasses.anchorFields p.1.2.2 p.2) |
|
|
| lemma rawCurrentProjection_measurable {d : ℕ} (e : Direction d) : |
| Measurable (rawCurrentProjection e) := by |
| have hU : Measurable (fun p : RawCurrentData e × LocalUpperField e => p.1.2.2) := |
| measurable_snd.comp (measurable_snd.comp measurable_fst) |
| exact ((ReferenceClasses.anchorOffsets_measurable.comp hU).prodMk |
| ((rawCurrentProfiles_measurable e).comp measurable_fst)).prodMk |
| (ReferenceClasses.anchorFields_measurable.comp (hU.prodMk measurable_snd)) |
|
|
| noncomputable def currentWindowReference {d : ℕ} (e : Direction d) (ν : Measure (Row d)) |
| [IsProbabilityMeasure ν] : |
| Kernel (ReferenceClasses.Data (HorizontalSpace e) × CurrentProfiles e) |
| (ReferenceClasses.AllFields (HorizontalSpace e) (Row d)) := |
| (ReferenceClasses.reference ν).comap Prod.fst measurable_fst |
|
|
| instance currentWindowReference_markov {d : ℕ} (e : Direction d) (ν : Measure (Row d)) |
| [IsProbabilityMeasure ν] : IsMarkovKernel (currentWindowReference e ν) := by |
| unfold currentWindowReference |
| infer_instance |
|
|
| lemma rawCurrentProjection_kl_le {d : ℕ} (e : Direction d) |
| (ν : Measure (Row d)) [IsProbabilityMeasure ν] |
| (μ : Measure (RawCurrentData e × LocalUpperField e)) [IsFiniteMeasure μ] : |
| let out := μ.map (rawCurrentProjection e) |
| klDiv out (out.fst.compProd (currentWindowReference e ν)) ≤ |
| klDiv μ (μ.fst.prod (Measure.infinitePi (fun _ : ℕ × HorizontalSpace e => ν))) := by |
| exact ReferenceClasses.anchorFrame_kl_le μ ν (fun z => z.2.2) |
| (measurable_snd.comp measurable_snd) (rawCurrentProfiles e) (rawCurrentProfiles_measurable e) |
|
|
| noncomputable def actualCurrentWindow {d : ℕ} (e : Direction d) |
| (t : Environment d → ℤ → ℕ) (i : ℤ) (m : ℕ) (X : EpisodeInput e) : CurrentWindow e := |
| ((ReferenceClasses.anchorOffsets (fun a => X.1.2 (i,a)), |
| fun p => globalHorizontalCoordinate e (t X.1.1 i) X.1.1 (X.1.2 (i,p.1)+p.2)), |
| fun p => episodeSplicedRow e t X i m p.1 p.2.1 p.2.2) |
|
|
| lemma actualCurrentWindow_measurable {d : ℕ} (e : Direction d) |
| (t : Environment d → ℤ → ℕ) (ht : ∀ i, Measurable fun ω => t ω i) |
| (i : ℤ) (m : ℕ) : Measurable (actualCurrentWindow e t i m) := by |
| have hu : Measurable (fun X : EpisodeInput e => fun a => X.1.2 (i,a)) := |
| Measurable.of_eval fun a => (measurable_pi_apply (i,a)).comp |
| (measurable_snd.comp measurable_fst) |
| have ho := ReferenceClasses.anchorOffsets_measurable.comp hu |
| have hp : Measurable (fun X : EpisodeInput e => fun p : ℕ × HorizontalSpace e => |
| globalHorizontalCoordinate e (t X.1.1 i) X.1.1 (X.1.2 (i,p.1)+p.2)) := |
| Measurable.of_eval fun p => actualArrayCoordinate_measurable e t ht i (i,p.1) p.2 |
| have hf : Measurable (fun X : EpisodeInput e => fun p : ReferenceClasses.Site (HorizontalSpace e) => |
| episodeSplicedRow e t X i m p.1 p.2.1 p.2.2) := |
| Measurable.of_eval fun p => episodeSplicedRow_measurable e t ht i m p.1 p.2.1 p.2.2 |
| exact (ho.prodMk hp).prodMk hf |
|
|
| lemma globalHorizontalCoordinate_stoppedObservation {d : ℕ} (e : Direction d) |
| (τ : Environment d → ℕ) |
| (hτ : ∀ n : ℕ, MeasurableSet[rowSigma (BelowHeight (realPosition (step e)) n)] {ω | τ ω=n}) |
| (filler ω : Environment d) (z : HorizontalSpace e) : |
| globalHorizontalCoordinate e (τ ω) |
| (stoppedObservation (fun n => BelowHeight (realPosition (step e)) n) τ filler ω) z = |
| globalHorizontalCoordinate e (τ ω) ω z := by |
| have h := currentStoppedProfile_observation e τ hτ filler ω |
| have hn : τ (stoppedObservation (fun n => BelowHeight (realPosition (step e)) n) τ filler ω) = τ ω := |
| stoppedRowGraft_stop (fun n => BelowHeight (realPosition (step e)) n) τ hτ ω filler |
| dsimp only [currentStoppedProfile] at h |
| rw [hn] at h |
| apply Subtype.ext |
| exact congrArg (fun μ : Measure (HorizontalSpace e) => (μ {z}).toReal) h |
|
|
| lemma actualCurrentWindow_raw {d : ℕ} (e : Direction d) |
| (t : Environment d → ℤ → ℕ) (i : ℤ) (m : ℕ) |
| (ht : ∀ n : ℕ, MeasurableSet[rowSigma (BelowHeight (realPosition (step e)) n)] |
| {ω | t ω i=n}) (filler : Environment d) (X : EpisodeInput e) : |
| rawCurrentProjection e ((t X.1.1 i, |
| (stoppedObservation (fun n => BelowHeight (realPosition (step e)) n) |
| (fun ω => t ω i) filler X.1.1,fun a => X.1.2 (i,a))), |
| localStoppedField e (fun ω => t ω i) (fun ω => t ω (i+m)) (X.1.1,X.2 (i,m))) = |
| actualCurrentWindow e t i m X := by |
| dsimp only [rawCurrentProjection,actualCurrentWindow,rawCurrentProfiles] |
| refine Prod.ext (Prod.ext rfl ?_) ?_ |
| · funext p |
| exact globalHorizontalCoordinate_stoppedObservation e (fun ω => t ω i) ht filler X.1.1 _ |
| · funext p |
| dsimp only [ReferenceClasses.anchorFields,localStoppedField,episodeSplicedRow] |
|
|
| theorem actualCurrentWindow_law {d : ℕ} (e : Direction d) |
| (ν : Measure (Row d)) [IsProbabilityMeasure ν] |
| (Q : Measure (Environment d)) [IsFiniteMeasure Q] |
| (t : Environment d → ℤ → ℕ) (ht : ∀ i, Measurable fun ω => t ω i) |
| (i : ℤ) (m : ℕ) |
| (hi : ∀ n : ℕ, MeasurableSet[rowSigma (BelowHeight (realPosition (step e)) n)] {ω | t ω i=n}) |
| (him : ∀ n : ℕ, MeasurableSet[rowSigma (BelowHeight (realPosition (step e)) n)] {ω | t ω (i+m)=n}) |
| (filler : Environment d) : |
| (rawCurrentWindow e ν Q (fun ω => t ω i) (fun ω => t ω (i+m)) (ht i) filler).map |
| (rawCurrentProjection e) = |
| (episodeInputLaw e ν Q t ht).map (actualCurrentWindow e t i m) := by |
| unfold rawCurrentWindow |
| rw [← episodeInputLaw_current e ν Q t ht i m] |
| rw [Measure.map_map (rawCurrentWindow_measurable e _ _ hi him filler) |
| (currentInput_measurable e i m)] |
| rw [Measure.map_map (rawCurrentProjection_measurable e)] |
| · congr 1 |
| funext X |
| exact actualCurrentWindow_raw e t i m hi filler X |
| · exact (rawCurrentWindow_measurable e _ _ hi him filler).comp (currentInput_measurable e i m) |
|
|
| end DirectionalTransience |
|
|
| end |
|
|
| end OAI |
|
|
