| import Mathlib |
| import OAI.Probability.Ballisticity.Coupling.FreshBuffer |
| import OAI.Probability.Ballisticity.Estimates.BufferFirstFailure |
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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 Function |
| namespace DirectionalTransience |
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| lemma weighted_countable_partition_lower {Ω I : Type*} [MeasurableSpace Ω] [Countable I] |
| (μ : Measure Ω) (A B : I → Set Ω) (hA : ∀ i, MeasurableSet (A i)) |
| (hB : ∀ i, MeasurableSet (B i)) (hdis : Pairwise (Disjoint on A)) |
| (hsub : ∀ i, B i ⊆ A i) (w : Ω → ℝ≥0∞) (c : ℝ≥0∞) |
| (hbound : ∀ i, c*(∫⁻ ω in A i, w ω ∂μ) ≤ ∫⁻ ω in B i, w ω ∂μ) : |
| c*(∫⁻ ω in ⋃ i, A i, w ω ∂μ) ≤ ∫⁻ ω in ⋃ i, B i, w ω ∂μ := by |
| rw [lintegral_iUnion hA hdis,lintegral_iUnion hB |
| (fun i j hij => (hdis hij).mono (hsub i) (hsub j)),←ENNReal.tsum_mul_left] |
| exact ENNReal.tsum_le_tsum hbound |
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| lemma actualFreshBuffer_countable_branches {d : ℕ} (ν : Measure (Row d)) |
| [IsProbabilityMeasure ν] (hue : UniformElliptic ν) (e f : Direction d) (hef : e.1 ≠ f.1) |
| (htrans : DirectionallyTransient ν (realPosition (step e))) : |
| ∃ α c : ℝ, 0 < α ∧ 0 < c ∧ ∀ ε : ℝ, 0 < ε → |
| ∃ g R : ℝ, 0 < g ∧ 0 < R ∧ ∀ {I : Type} [Countable I] |
| (a r : I → ℝ) (_ : ∀ i, R ≤ r i) (z₀ : ℝ) |
| (π : (i : I) → Environment d → SupportedPairMeasures (PairAtHeight (realPosition (step e)) (a i))) |
| (_ : ∀ i, @Measurable _ _ (rowSigma (BelowHeight (realPosition (step e)) (a i))) _ (π i)) |
| (A : I → Set (Environment d)) |
| (_ : ∀ i, MeasurableSet[rowSigma (BelowHeight (realPosition (step e)) (a i))] (A i)) |
| (_ : Pairwise (Disjoint on A)) |
| (_ : ∀ i η, η ∈ A i → IsProbabilityMeasure (π i η).val ∧ |
| ∀ᵐ x ∂(π i η).val, z₀+r i ≤ signedCoordinate f (x.2-x.1)) |
| (w : Environment d → ℝ≥0∞) |
| (_ : ∀ i, @Measurable _ _ (rowSigma (BelowHeight (realPosition (step e)) (a i))) _ |
| ((A i).indicator w)), |
| let H := fun i => ⌊fluctuationScale (independentConditionedPairLaw ν (realPosition (step e))) |
| (commonIncrementProcess (realPosition (step e)) f 0) (r i)⌋₊ |
| ENNReal.ofReal c*(∫⁻ ω in ⋃ i, A i, w ω ∂environmentLaw ν) ≤ |
| ∫⁻ ω in ⋃ i, {ω | ω ∈ A i ∧ BufferStageEvent (realPosition (step e)) f (H i) |
| z₀ (r i) ε α g (π i ω).val ω}, w ω ∂environmentLaw ν := by |
| obtain ⟨α,c,hα,hc,hchance⟩ := buffer_stage_fresh_success_weights ν hue e f hef htrans |
| refine ⟨α,c,hα,hc,fun ε hε => ?_⟩ |
| obtain ⟨g,R,hg,hR,hgood⟩ := hchance ε hε |
| refine ⟨g,R,hg,hR,?_⟩ |
| intro I _ a r hr z₀ π hπ A hA hdis hvalid w hw |
| let ℓ := realPosition (step e) |
| let H := fun i => ⌊fluctuationScale (independentConditionedPairLaw ν ℓ) |
| (commonIncrementProcess ℓ f 0) (r i)⌋₊ |
| have hH (i : I) : 0 < H i := (hgood (r i) (hr i)).1 |
| have hS (i : I) : Disjoint (BelowHeight ℓ (a i)) (BelowHeight ℓ (a i))ᶜ := disjoint_compl_right |
| have hC (i : I) (x : Lattice d × Lattice d) (hx : x ∈ PairAtHeight ℓ (a i)) : |
| Strip ℓ x.1 (H i) ⊆ (BelowHeight ℓ (a i))ᶜ ∧ Strip ℓ x.2 (H i) ⊆ (BelowHeight ℓ (a i))ᶜ := by |
| constructor |
| · intro y hy |
| change ¬ dot (realPosition y) ℓ < a i |
| have hh := hy.1 |
| rw [hx.1] at hh |
| exact not_lt_of_ge hh |
| · intro y hy |
| change ¬ dot (realPosition y) ℓ < a i |
| have hh := hy.1 |
| rw [hx.2] at hh |
| exact not_lt_of_ge hh |
| let B := fun i => {ω | ω ∈ A i ∧ BufferStageEvent ℓ f (H i) z₀ (r i) ε α g (π i ω).val ω} |
| have hAm (i : I) : MeasurableSet (A i) := (rowSigma_le _) _ (hA i) |
| have hBm (i : I) : MeasurableSet (B i) := by |
| apply (hAm i).inter |
| have hp := (hπ i).mono (rowSigma_le _) le_rfl |
| have hm := (measurableSet_bufferStageEvent_rows ℓ f (hH i) z₀ (r i) ε α g |
| (PairAtHeight ℓ (a i)) Set.univ (fun x hx => ⟨Set.subset_univ _,Set.subset_univ _⟩)).preimage |
| (hp.prodMk (measurable_id.mono le_rfl (rowSigma_le Set.univ))) |
| exact hm |
| apply weighted_countable_partition_lower (environmentLaw ν) A B hAm hBm hdis |
| (fun i _ h => h.1) w (ENNReal.ofReal c) |
| intro i |
| have hh := (hgood (r i) (hr i)).2 z₀ (PairAtHeight ℓ (a i)) (BelowHeight ℓ (a i)) |
| (BelowHeight ℓ (a i))ᶜ (hS i) (hC i) (π i) (hπ i) (A i) (hA i) (hvalid i) |
| ((A i).indicator w) (hw i) |
| rw [lintegral_congr_ae (ae_restrict_mem (hAm i) |>.mono (fun ω hω => Set.indicator_of_mem hω _))] at hh |
| have he : (∫⁻ ω in B i, (A i).indicator w ω ∂environmentLaw ν) = |
| ∫⁻ ω in B i, w ω ∂environmentLaw ν := by |
| apply lintegral_congr_ae |
| filter_upwards [ae_restrict_mem (hBm i)] with ω hω |
| exact Set.indicator_of_mem hω.1 _ |
| exact he ▸ hh |
| end DirectionalTransience |
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| end |
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| end |
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| end OAI |
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