| import Mathlib |
| import OAI.Probability.Ballisticity.Walk.FreshRowWeights |
|
|
| namespace OAI |
|
|
| section |
|
|
| section |
|
|
| open MeasureTheory ProbabilityTheory Filter Function |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| namespace DirectionalTransience |
|
|
| def pairLawValidSet {d : ℕ} (f : Direction d) (z : ℝ) {C : Set (Lattice d × Lattice d)} |
| (π : Environment d → SupportedPairMeasures C) : Set (Environment d) := |
| {ω | (π ω).val Set.univ=1 ∧ (π ω).val {x | ¬ z ≤ signedCoordinate f (x.2-x.1)}=0} |
|
|
| lemma pairLawValidSet_iff {d : ℕ} (f : Direction d) (z : ℝ) {C : Set (Lattice d × Lattice d)} |
| (π : Environment d → SupportedPairMeasures C) (ω : Environment d) : |
| ω ∈ pairLawValidSet f z π ↔ IsProbabilityMeasure (π ω).val ∧ |
| ∀ᵐ x ∂(π ω).val, z ≤ signedCoordinate f (x.2-x.1) := by |
| constructor |
| · rintro ⟨hp,hg⟩ |
| exact ⟨⟨hp⟩,ae_iff.mpr hg⟩ |
| · rintro ⟨hp,hg⟩ |
| let := hp |
| exact ⟨measure_univ,ae_iff.mp hg⟩ |
|
|
| lemma pairLawValidSet_measurable {d : ℕ} (f : Direction d) (z : ℝ) {C : Set (Lattice d × Lattice d)} |
| (π : Environment d → SupportedPairMeasures C) {m : MeasurableSpace (Environment d)} (hπ : @Measurable _ _ m _ π) : |
| MeasurableSet[m] (pairLawValidSet f z π) := by |
| have hm := hπ.subtype_val |
| exact (measurableSet_eq_fun ((Measure.measurable_coe MeasurableSet.univ).comp hm) measurable_const).inter |
| (measurableSet_eq_fun ((Measure.measurable_coe (Set.to_countable _).measurableSet).comp hm) measurable_const) |
|
|
| lemma weighted_test_ae_valid {Ω : Type*} [mΩ : MeasurableSpace Ω] {m : MeasurableSpace Ω} |
| (hm : m ≤ mΩ) (μ : @Measure Ω mΩ) (A V E : Set Ω) (hA : MeasurableSet[m] A) |
| (hV : MeasurableSet[m] V) (w : Ω → ℝ≥0∞) (c : ℝ≥0∞) |
| (hvalid : ∀ᵐ ω ∂μ, ω ∈ A → ω ∈ V) |
| (hbound : ∀ A', MeasurableSet[m] A' → A' ⊆ A → A' ⊆ V → |
| c*(∫⁻ ω in A', w ω ∂μ) ≤ ∫⁻ ω in A' ∩ E, w ω ∂μ) : |
| c*(∫⁻ ω in A, w ω ∂μ) ≤ ∫⁻ ω in A ∩ E, w ω ∂μ := by |
| have hv : (fun ω => ω ∈ A ∩ V) =ᵐ[μ] (fun ω => ω ∈ A) := hvalid.mono fun ω h => by |
| simp only [Set.mem_inter_iff] |
| exact propext ⟨And.left,fun hA => ⟨hA,h hA⟩⟩ |
| have hvE : (fun ω => ω ∈ (A ∩ V) ∩ E) =ᵐ[μ] (fun ω => ω ∈ A ∩ E) := hv.mono fun ω h => by |
| change (ω ∈ A ∩ V ∧ ω ∈ E) = (ω ∈ A ∧ ω ∈ E) |
| change (ω ∈ A ∩ V) = (ω ∈ A) at h |
| rw [h] |
| have hh := hbound (A ∩ V) (hA.inter hV) Set.inter_subset_left Set.inter_subset_right |
| have hr : μ.restrict (A ∩ V)=μ.restrict A := by |
| rw [Set.inter_comm, ← Measure.restrict_restrict (hm V hV)] |
| exact Measure.restrict_eq_self_of_ae_mem ((ae_restrict_iff' (hm A hA)).mpr hvalid) |
| have hrE : μ.restrict ((A ∩ V) ∩ E)=μ.restrict (A ∩ E) := Measure.restrict_congr_set hvE |
| rwa [hr,hrE] at hh |
|
|
| lemma fresh_buffer_weight_ae_valid {d : ℕ} (ν : Measure (Row d)) [IsProbabilityMeasure ν] |
| (ℓ : Vector d) (f : Direction d) {H : ℕ} (hH : 0 < H) (z₀ r ε α g c : ℝ) |
| (C : Set (Lattice d × Lattice d)) (D S : Set (Lattice d)) |
| (hdis : Disjoint D S) (hC : ∀ x ∈ C, Strip ℓ x.1 H ⊆ S ∧ Strip ℓ x.2 H ⊆ S) |
| (hchance : ∀ π : SupportedPairMeasures C, IsProbabilityMeasure π.val → |
| (∀ᵐ x ∂π.val, z₀+r ≤ signedCoordinate f (x.2-x.1)) → |
| ENNReal.ofReal c ≤ environmentLaw ν {ω | BufferStageEvent ℓ f H z₀ r ε α g π.val ω}) |
| (π : Environment d → SupportedPairMeasures C) |
| (hπ : @Measurable _ _ (rowSigma D) _ π) (A : Set (Environment d)) |
| (hA : MeasurableSet[rowSigma D] A) |
| (hvalid : ∀ᵐ ω ∂environmentLaw ν, ω ∈ A → IsProbabilityMeasure (π ω).val ∧ |
| ∀ᵐ x ∂(π ω).val, z₀+r ≤ signedCoordinate f (x.2-x.1)) |
| (w : Environment d → ℝ≥0∞) (hw : @Measurable _ _ (rowSigma D) _ w) : |
| ENNReal.ofReal c*(∫⁻ ω in A, w ω ∂environmentLaw ν) ≤ |
| ∫⁻ ω in {ω | ω ∈ A ∧ BufferStageEvent ℓ f H z₀ r ε α g (π ω).val ω}, w ω ∂environmentLaw ν := by |
| apply weighted_test_ae_valid (mΩ := (inferInstance : MeasurableSpace (Environment d))) (m := rowSigma D) (rowSigma_le D) (environmentLaw ν) A (pairLawValidSet f (z₀+r) π) |
| {ω | BufferStageEvent ℓ f H z₀ r ε α g (π ω).val ω} hA |
| (pairLawValidSet_measurable f (z₀+r) π hπ) w (ENNReal.ofReal c) |
| (hvalid.mono fun ω h hω => (pairLawValidSet_iff f (z₀+r) π ω).mpr (h hω)) |
| intro A' hA' hsub hV |
| apply fresh_buffer_weighted_test ν ℓ f hH z₀ r ε α g C hdis hC π hπ A' hA' w hw |
| intro ω hω |
| obtain ⟨hp,hgap⟩ := (pairLawValidSet_iff f (z₀+r) π ω).mp (hV hω) |
| exact hchance (π ω) hp hgap |
| end DirectionalTransience |
|
|
| end |
|
|
| end |
|
|
| end OAI |
|
|
