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import OAI.Probability.Ballisticity.Estimates.LocalStoppedSplice
import OAI.Probability.Ballisticity.Entropy.WindowProjectionEntropy
import OAI.Probability.Ballisticity.Entropy.AnchorEntropy
import OAI.Probability.Ballisticity.Estimates.SamplingRearrangement
import OAI.Probability.Ballisticity.Entropy.StoppedEntropySum
namespace OAI
section
open MeasureTheory ProbabilityTheory InformationTheory
open scoped ENNReal Classical
namespace DirectionalTransience
abbrev RawCurrentData {d : ℕ} (e : Direction d) :=
ℕ × (Environment d × (ℕ → HorizontalSpace e))
noncomputable def rawCurrentWindow {d : ℕ} (e : Direction d)
(ν : Measure (Row d)) (Q : Measure (Environment d))
(τ σ : Environment d → ℕ) (hτ : Measurable τ) (filler : Environment d) :
Measure (RawCurrentData e × LocalUpperField e) :=
(((Q.compProd (currentAnchorKernel e τ hτ)).prod
(Measure.infinitePi (fun _ : ℕ × HorizontalSpace e => ν))).map
(fun p => ((τ p.1.1,
(stoppedObservation (fun n => BelowHeight (realPosition (step e)) n) τ filler p.1.1,p.1.2)),
localStoppedField e τ σ (p.1.1,p.2))))
lemma sampled_stoppedWindow_law {d : ℕ} (e : Direction d)
(ν : Measure (Row d)) [IsProbabilityMeasure ν]
(Q : Measure (Environment d)) [IsFiniteMeasure Q]
(τ σ : Environment d → ℕ) (hτσ : ∀ ω, τ ω ≤ σ ω)
(hτ : ∀ n : ℕ, MeasurableSet[rowSigma (BelowHeight (realPosition (step e)) n)] {ω | τ ω=n})
(hσ : ∀ n : ℕ, MeasurableSet[rowSigma (BelowHeight (realPosition (step e)) n)] {ω | σ ω=n})
(filler : Environment d) :
let G := stoppedObservation (fun n => BelowHeight (realPosition (step e)) n) τ filler
let K := currentAnchorKernel e τ (measurable_of_stop_events _ τ hτ)
(((Q.compProd K).prod (Measure.infinitePi (fun _ : ℕ × HorizontalSpace e => ν))).map
(fun p => ((G p.1.1,p.1.2),localStoppedField e τ σ (p.1.1,p.2)))) =
(((Kernel.id.prod K) ∥ₖ Kernel.id) ∘ₘ stoppedWindowLaw e ν Q τ σ filler) := by
let S : ℕ → Set (Lattice d) := fun n => BelowHeight (realPosition (step e)) n
let G := stoppedObservation S τ filler
let K := currentAnchorKernel e τ (measurable_of_stop_events S τ hτ)
have hG := stoppedObservation_measurable S τ hτ filler
have hK : K.comap G hG = K := by
ext ω : 1
exact currentAnchorKernel_observation e τ hτ filler ω
have h := StoppedWindow.sample_lower_map Q
(Measure.infinitePi (fun _ : ℕ × HorizontalSpace e => ν)) G hG
(localStoppedField e τ σ)
(localStoppedField_measurable e τ σ (measurable_of_stop_events S τ hτ)
(measurable_of_stop_events S σ hσ)) K
rw [hK, localStoppedField_law e ν Q τ σ hτσ hτ hσ filler] at h
exact h
lemma rawCurrentWindow_measurable {d : ℕ} (e : Direction d)
(τ σ : Environment d → ℕ)
(hτ : ∀ n : ℕ, MeasurableSet[rowSigma (BelowHeight (realPosition (step e)) n)] {ω | τ ω=n})
(hσ : ∀ n : ℕ, MeasurableSet[rowSigma (BelowHeight (realPosition (step e)) n)] {ω | σ ω=n})
(filler : Environment d) :
Measurable (fun p : (Environment d × (ℕ → HorizontalSpace e)) × LocalUpperField e =>
((τ p.1.1,(stoppedObservation (fun n => BelowHeight (realPosition (step e)) n)
τ filler p.1.1,p.1.2)),localStoppedField e τ σ (p.1.1,p.2))) := by
let S : ℕ → Set (Lattice d) := fun n => BelowHeight (realPosition (step e)) n
have hw : Measurable (fun p : (Environment d × (ℕ → HorizontalSpace e)) × LocalUpperField e => p.1.1) :=
measurable_fst.comp measurable_fst
exact ((measurable_of_stop_events S τ hτ).comp hw).prodMk
(((stoppedObservation_measurable S τ hτ filler).comp hw).prodMk
(measurable_snd.comp measurable_fst)) |>.prodMk
((localStoppedField_measurable e τ σ (measurable_of_stop_events S τ hτ)
(measurable_of_stop_events S σ hσ)).comp (hw.prodMk measurable_snd))
theorem rawCurrentWindow_kl_le {d : ℕ} (e : Direction d)
(ν : Measure (Row d)) [IsProbabilityMeasure ν]
(Q : Measure (Environment d)) [IsProbabilityMeasure Q]
(τ σ : Environment d → ℕ) (hτσ : ∀ ω, τ ω ≤ σ ω)
(hτ : ∀ n : ℕ, MeasurableSet[rowSigma (BelowHeight (realPosition (step e)) n)] {ω | τ ω=n})
(hσ : ∀ n : ℕ, MeasurableSet[rowSigma (BelowHeight (realPosition (step e)) n)] {ω | σ ω=n})
(filler : Environment d) :
let μ := rawCurrentWindow e ν Q τ σ (measurable_of_stop_events _ τ hτ) filler
klDiv μ (μ.fst.prod (Measure.infinitePi (fun _ : ℕ × HorizontalSpace e => ν))) ≤
ownedWindowEntropy e ν Q τ σ filler := by
let S : ℕ → Set (Lattice d) := fun n => BelowHeight (realPosition (step e)) n
let G := stoppedObservation S τ filler
let K := currentAnchorKernel e τ (measurable_of_stop_events S τ hτ)
let π := Measure.infinitePi (fun _ : ℕ × HorizontalSpace e => ν)
let μ := stoppedWindowLaw e ν Q τ σ filler
let μ' : Measure ((Environment d × (ℕ → HorizontalSpace e)) × LocalUpperField e) :=
(((Kernel.id.prod K) ∥ₖ (Kernel.id : Kernel (LocalUpperField e) (LocalUpperField e))) ∘ₘ μ)
have : IsFiniteMeasure μ := by unfold μ stoppedWindowLaw; infer_instance
have : IsFiniteMeasure μ' := by unfold μ'; infer_instance
let dat : Environment d × (ℕ → HorizontalSpace e) → RawCurrentData e := fun p => (τ p.1,p)
have hdat : Measurable dat := ((measurable_of_stop_events S τ hτ).comp measurable_fst).prodMk measurable_id
have hproj := klDiv_map_le μ' (μ'.fst.prod π) (hdat.prodMap measurable_id)
have hfst := StoppedWindow.map_data_field_fst μ' dat hdat
(fun (_ : Environment d × (ℕ → HorizontalSpace e)) (ξ : LocalUpperField e) => ξ) measurable_snd
change (μ'.map (Prod.map dat id)).fst = μ'.fst.map dat at hfst
have hout : μ'.map (Prod.map dat id) =
rawCurrentWindow e ν Q τ σ (measurable_of_stop_events S τ hτ) filler := by
dsimp only [μ', μ, K]
rw [← sampled_stoppedWindow_law e ν Q τ σ hτσ hτ hσ filler]
rw [Measure.map_map (hdat.prodMap measurable_id)]
· congr 1
funext p
have ht : τ (G p.1.1) = τ p.1.1 := stoppedRowGraft_stop S τ hτ p.1.1 filler
exact Prod.ext (Prod.ext ht rfl) rfl
· exact (((stoppedObservation_measurable S τ hτ filler).comp
(measurable_fst.comp measurable_fst)).prodMk (measurable_snd.comp measurable_fst)).prodMk
((localStoppedField_measurable e τ σ (measurable_of_stop_events S τ hτ)
(measurable_of_stop_events S σ hσ)).comp
((measurable_fst.comp measurable_fst).prodMk measurable_snd))
have href : (μ'.fst.prod π).map (Prod.map dat id) =
(μ'.map (Prod.map dat id)).fst.prod π := by
rw [← Measure.map_prod_map _ _ hdat measurable_id, Measure.map_id]
rw [hfst]
rw [href, hout] at hproj
refine hproj.trans ?_
have h := StoppedWindow.parallel_lower_kl_le μ π (Kernel.id.prod K)
rw [stoppedWindowLaw_fst e ν Q τ σ hτ hσ filler] at h
exact h
end DirectionalTransience
end
end OAI
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