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9b6e014 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 | import Mathlib
import OAI.Probability.Ballisticity.Geometry.RestrictedWindow
import OAI.Probability.Ballisticity.Stationary.ArrayClosed
namespace OAI
section
open MeasureTheory ProbabilityTheory InformationTheory
open scoped ENNReal Classical BigOperators
namespace DirectionalTransience
lemma typedCurrentArrayWindow_iterate_actual {d : ℕ} (e : Direction d)
(t J : Environment d → ℤ → ℕ) (j : ℤ) (m i : ℕ) (X : EpisodeInput e) :
typedCurrentArrayWindow e j m (StationaryCompact.shift^[i] (actualArrayMap e t J X)) =
actualCurrentWindow e t (j+i) m X := by
have hc : ReferenceClasses.Consistent
(currentOffsets e j (StationaryCompact.shift^[i] (actualArrayMap e t J X))) := by
have h := congrArg (fun W : AmbientCurrentWindow e => W.1.1)
(currentArrayWindow_iterate e j m i (actualArrayMap e t J X))
change currentOffsets e j (StationaryCompact.shift^[i] (actualArrayMap e t J X)) =
currentOffsets e (j+i) (actualArrayMap e t J X) at h
rw [h]
exact actual_currentOffsets e t J X (j+i)
apply currentWindowInclude_injective e
rw [currentWindowInclude_typed e j m _ hc,currentArrayWindow_iterate,currentArrayWindow_actual]
lemma rawCurrentOccupation_actual {d : ℕ} (e : Direction d)
(ν : Measure (Row d)) [IsProbabilityMeasure ν]
(Q : Measure (Environment d)) [IsFiniteMeasure Q]
(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})
(filler : Environment d) (N m : ℕ) (M : Environment d → ℕ)
(hact : ∀ (i : ℕ) ω, t ω i<N ↔ i<M ω) :
(rawCurrentOccupation e ν Q (fun i ω => t ω i) (fun i => ht i) filler N m).map
(rawCurrentProjection e) =
(actualOccupationRaw e ν Q t J ht N M).map (typedCurrentArrayWindow e 0 m) := by
rw [rawCurrentOccupation_projected e ν Q t ht hst filler N m]
unfold actualOccupationRaw StationaryCompact.episodeOccupationRaw
rw [Measure.map_finset_sum (typedCurrentArrayWindow_measurable e 0 m).aemeasurable]
apply Finset.sum_congr rfl
intro i _
have hA : {X : EpisodeInput e | t X.1.1 i<N}={X | i<M X.1.1} := by
ext X
exact hact i X.1.1
have hm : Measurable (fun X : EpisodeInput e => StationaryCompact.shift^[i] (actualArrayMap e t J X)) :=
(StationaryCompact.shift_continuous.measurable.iterate i).comp (actualArrayMap_measurable e t J ht hJ)
rw [hA,Measure.map_map (typedCurrentArrayWindow_measurable e 0 m) hm]
congr 1
funext X
exact ((typedCurrentArrayWindow_iterate_actual e t J 0 m i X).trans (by simp)).symm
namespace Entropy
lemma normalize_map_measure {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y]
[Nonempty X] [Nonempty Y] (μ : FiniteMeasure X) (f : X → Y) (hf : Measurable f)
(hμ : μ≠0) :
let ν : FiniteMeasure Y := ⟨(μ : Measure X).map f,inferInstance⟩
(ν.normalize : Measure Y) = (μ.normalize : Measure X).map f := by
let ν : FiniteMeasure Y := ⟨(μ : Measure X).map f,inferInstance⟩
have hm : ν.mass=μ.mass := by
change ((μ : Measure X).map f Set.univ).toNNReal = _
rw [Measure.map_apply hf MeasurableSet.univ,Set.preimage_univ]
rfl
have hν : ν≠0 := by
rw [← ν.mass_nonzero_iff,hm]
exact μ.mass_nonzero_iff.mpr hμ
change (ν.normalize : Measure Y)=_
rw [ν.toMeasure_normalize_eq_of_nonzero hν,μ.toMeasure_normalize_eq_of_nonzero hμ,hm,
Measure.map_smul _ hf.aemeasurable]
rfl
lemma normalize_maps_equal {X Y Z : Type*} [MeasurableSpace X] [MeasurableSpace Y] [MeasurableSpace Z]
[Nonempty X] [Nonempty Y] [Nonempty Z]
(μ : FiniteMeasure X) (ν : FiniteMeasure Y) (f : X → Z) (g : Y → Z)
(hf : Measurable f) (hg : Measurable g) (hμ : μ≠0) (hν : ν≠0)
(h : (μ : Measure X).map f = (ν : Measure Y).map g) :
(μ.normalize : Measure X).map f = (ν.normalize : Measure Y).map g := by
rw [←normalize_map_measure μ f hf hμ,←normalize_map_measure ν g hg hν]
have he : (⟨(μ : Measure X).map f,inferInstance⟩ : FiniteMeasure Z) =
⟨(ν : Measure Y).map g,inferInstance⟩ := Subtype.ext h
rw [he]
lemma real_mass_eq_of_maps_equal {X Y Z : Type*}
[MeasurableSpace X] [MeasurableSpace Y] [MeasurableSpace Z]
(μ : Measure X) (ν : Measure Y) (f : X → Z) (g : Y → Z)
(hf : Measurable f) (hg : Measurable g) (h : μ.map f=ν.map g) :
μ.real Set.univ=ν.real Set.univ := by
have hh := congrArg (fun ρ : Measure Z => ρ.real Set.univ) h
simpa only [Measure.real,Measure.map_apply hf MeasurableSet.univ,
Measure.map_apply hg MeasurableSet.univ,Set.preimage_univ] using hh
end Entropy
end DirectionalTransience
end
end OAI
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