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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 | import Mathlib
import OAI.Probability.Ballisticity.Estimates.BufferStagePacket
import OAI.Probability.Ballisticity.Walk.BufferNormalizedRows
namespace OAI
section
section
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal BigOperators Topology Classical
namespace DirectionalTransience
lemma bufferStageRetainedLaw_height_unconditional {d : ℕ} (e f : Direction d)
(a z₀ r ε α g : ℝ) (π : Environment d → SupportedPairMeasures (PairAtHeight (realPosition (step e)) a))
{H : ℕ} (hH : 0 < H) (ω : Environment d) (κ : ℝ≥0) :
∀ᵐ y ∂bufferStageRetainedLaw e f a z₀ r ε α g π H ω κ,
y ∈ PairAtHeight (realPosition (step e))
(a+bufferFirstFailure (realPosition (step e)) f a (z₀+(1-ε)*r) g π H ω) := by
let ℓ := realPosition (step e)
let z := z₀+(1-ε)*r
let j := bufferFirstFailure ℓ f a z g π H ω
have hπ : ∀ᵐ x ∂(π ω).val, x ∈ PairAtHeight ℓ a := ae_iff.mpr (π ω).property
have hj : 1 ≤ j := (bufferFirstFailure_bounds ℓ f a z g π hH ω).1
unfold bufferStageRetainedLaw
dsimp only
split
· have he : j-1+1=j := by omega
simpa only [he] using upwardRetainedLaw_height e f (j-1) z ω (π ω).val κ a hπ
· rename_i hp
have hjH : j=H := bufferFirstFailure_eq_top_of_pass ℓ f a z g π hH ω (le_of_not_gt hp)
change ∀ᵐ y ∂(if _ then _ else _), y ∈ PairAtHeight ℓ (a+j)
rw [hjH]
split <;> exact retainedPairLaw_height e f H _ ω (π ω).val a hπ
noncomputable def bufferStageStoppedProbability {d : ℕ} (e f : Direction d)
(a z₀ r ε α g : ℝ) (π : Environment d → SupportedPairMeasures (PairAtHeight (realPosition (step e)) a))
(H : ℕ) (hH : 0 < H) (κ : ℝ≥0) (k : ℕ) (ω : Environment d) :
SupportedPairMeasures (PairAtHeight (realPosition (step e)) (a+k)) :=
⟨if bufferFirstFailure (realPosition (step e)) f a (z₀+(1-ε)*r) g π H ω = k
then normalizedMeasure (bufferStageRetainedLaw e f a z₀ r ε α g π H ω κ) else 0, by
split
· rename_i hk
apply ae_iff.mp
apply Measure.ae_smul_measure
simpa only [hk] using bufferStageRetainedLaw_height_unconditional e f a z₀ r ε α g π hH ω κ
· simp⟩
lemma bufferStageStoppedProbability_rows {d : ℕ} (e f : Direction d)
(a z₀ r ε α g : ℝ) (π : Environment d → SupportedPairMeasures (PairAtHeight (realPosition (step e)) a))
(hπ : @Measurable _ _ (rowSigma (BelowHeight (realPosition (step e)) a)) _ π)
(H : ℕ) (hH : 0 < H) (κ : ℝ≥0) (k : ℕ) :
@Measurable _ _ (rowSigma (BelowHeight (realPosition (step e)) (a+k))) _
(bufferStageStoppedProbability e f a z₀ r ε α g π H hH κ k) := by
exact (bufferStage_normalized_stopped_rows e f a z₀ r ε α g π hπ hH κ k).subtype_mk
lemma bufferStageStoppedProbability_gap {d : ℕ} (e f : Direction d)
(a z₀ r ε α g : ℝ) (π : Environment d → SupportedPairMeasures (PairAtHeight (realPosition (step e)) a))
(H : ℕ) (hH : 0 < H) (κ : ℝ≥0) (k : ℕ) (ω : Environment d) :
∀ᵐ y ∂(bufferStageStoppedProbability e f a z₀ r ε α g π H hH κ k ω).val,
z₀+(if BufferStageEvent (realPosition (step e)) f H z₀ r ε α g (π ω).val ω
then (1+α)*r else (1-ε)*r) ≤ signedCoordinate f (y.2-y.1) := by
by_cases hk : bufferFirstFailure (realPosition (step e)) f a (z₀+(1-ε)*r) g π H ω = k
· simp only [bufferStageStoppedProbability,ite_eq_left hk]
apply Measure.ae_smul_measure
by_cases hE : BufferStageEvent (realPosition (step e)) f H z₀ r ε α g (π ω).val ω
· rw [ite_eq_left hE,bufferStageRetainedLaw_success e f a z₀ r ε α g π hH ω κ hE]
exact retainedPairLaw_support _ _ _ _ _ _
· rw [ite_eq_right hE]
exact bufferStageRetainedLaw_failure_support e f a z₀ r ε α g π hH ω κ hE
· simp [bufferStageStoppedProbability,ite_eq_right hk]
end DirectionalTransience
end
end
end OAI
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