| import Mathlib |
| import OAI.Probability.Ballisticity.Walk.RetentionRowsAllHeights |
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| namespace OAI |
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| section |
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| section |
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| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| namespace DirectionalTransience |
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| lemma normalizedMeasure_zero {α : Type*} [MeasurableSpace α] : |
| normalizedMeasure (0 : Measure α) = 0 := by simp [normalizedMeasure] |
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| lemma bufferStage_normalized_stopped_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))) _ |
| (fun ω => 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 |
| have hm := measurable_normalizedMeasure.comp |
| (bufferStageRetainedLaw_stopped_rows e f a z₀ r ε α g π hπ hH κ k) |
| convert hm using 1 |
| funext ω |
| dsimp only [Function.comp_def] |
| split <;> simp_all only [normalizedMeasure_zero] |
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| lemma bufferStage_success_stopped_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) (k : ℕ) : |
| MeasurableSet[rowSigma (BelowHeight (realPosition (step e)) (a+k))] |
| {ω | bufferFirstFailure (realPosition (step e)) f a (z₀+(1-ε)*r) g π H ω = k ∧ |
| BufferStageEvent (realPosition (step e)) f H z₀ r ε α g (π ω).val ω} := by |
| let ℓ := realPosition (step e) |
| by_cases h : k=H |
| · subst k |
| have heq := bufferFirstFailure_eq_measurable ℓ f a (z₀+(1-ε)*r) g π hπ H H |
| have hp : @Measurable _ _ (rowSigma (BelowHeight ℓ (a+H))) _ π := |
| hπ.mono (rowSigma_below_height_mono ℓ a H) le_rfl |
| have hm := (measurableSet_bufferStageEvent_rows ℓ f hH z₀ r ε α g (PairAtHeight ℓ a) |
| (BelowHeight ℓ (a+H)) (fun x hx => strip_subset_below_height ℓ a x hx le_rfl)).preimage |
| (hp.prodMk measurable_id) |
| exact heq.inter hm |
| · have he : {ω | bufferFirstFailure ℓ f a (z₀+(1-ε)*r) g π H ω = k ∧ |
| BufferStageEvent ℓ f H z₀ r ε α g (π ω).val ω} = ∅ := by |
| apply Set.eq_empty_iff_forall_notMem.mpr |
| rintro ω ⟨hk,hE⟩ |
| have hlast := bufferFirstFailure_eq_top_of_pass ℓ f a (z₀+(1-ε)*r) g π hH ω |
| ((bufferStageEvent_iff_last_tests ℓ f a z₀ r ε α g π hH ω).mp hE).1 |
| exact h (hk.symm.trans hlast) |
| change MeasurableSet[rowSigma (BelowHeight ℓ (a+k))] _ |
| rw [he] |
| exact @MeasurableSet.empty _ (rowSigma (BelowHeight ℓ (a+k))) |
| end DirectionalTransience |
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| end |
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| end |
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| end OAI |
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