| import OAI.Probability.DirectionalWalk.WordPosterior |
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|
| namespace OAI |
|
|
| open MeasureTheory ProbabilityTheory Filter Preorder |
| open scoped ENNReal BigOperators Topology |
|
|
| namespace DirectionalZeroOne |
|
|
| open scoped Classical |
|
|
| noncomputable def freshPathKernel {d : ℕ} (x : Site d) : Kernel (Environment d) (Path d) := |
| (quenchedKernel d).comap (fun ω => (ω,x)) (measurable_id.prodMk measurable_const) |
| |
| instance {d : ℕ} (x : Site d) : IsMarkovKernel (freshPathKernel x) := by |
| unfold freshPathKernel |
| infer_instance |
| |
| noncomputable def freshJoint {d : ℕ} (μ : Measure (Row d)) [IsProbabilityMeasure μ] |
| (x : Site d) : Measure (Environment d × Path d) := |
| environmentLaw μ ⊗ₘ freshPathKernel x |
| |
| instance {d : ℕ} (μ : Measure (Row d)) [IsProbabilityMeasure μ] (x : Site d) : |
| IsProbabilityMeasure (freshJoint μ x) := by |
| unfold freshJoint |
| infer_instance |
| |
| lemma freshJoint_apply {d : ℕ} (μ : Measure (Row d)) [IsProbabilityMeasure μ] (x : Site d) |
| {A : Set (Environment d × Path d)} (hA : MeasurableSet A) : |
| freshJoint μ x A = ∫⁻ ω, quenchedKernel d (ω,x) (Prod.mk ω ⁻¹' A) ∂environmentLaw μ := by |
| rw [freshJoint,Measure.compProd_apply hA] |
| rfl |
| |
| lemma freshJoint_path_marginal {d : ℕ} (μ : Measure (Row d)) [IsProbabilityMeasure μ] (x : Site d) : |
| MeasurePreserving Prod.snd (freshJoint μ x) (annealed μ x) := by |
| refine ⟨measurable_snd,?_⟩ |
| ext A hA |
| rw [Measure.map_apply measurable_snd hA,freshJoint_apply μ x (hA.preimage measurable_snd),annealed_apply μ x hA] |
| rfl |
| |
| lemma freshJoint_lintegral {d : ℕ} (μ : Measure (Row d)) [IsProbabilityMeasure μ] (x : Site d) |
| (f : Environment d × Path d → ℝ≥0∞) (hf : Measurable f) : |
| (∫⁻ ξ, f ξ ∂freshJoint μ x) = |
| ∫⁻ ω, ∫⁻ X, f (ω,X) ∂quenchedKernel d (ω,x) ∂environmentLaw μ := by |
| rw [freshJoint,Measure.lintegral_compProd hf] |
| rfl |
| |
| lemma measurable_random_visitAfter {d : ℕ} (R K : Set (Site d)) (A : Environment d → Set (Site d)) |
| (hA : ∀ y, MeasurableSet {ω | y ∈ A ω}) : |
| MeasurableSet {ξ : Environment d × Path d | ξ.2 ∈ visitAfter R (A ξ.1) K} := by |
| have he : {ξ : Environment d × Path d | ξ.2 ∈ visitAfter R (A ξ.1) K} = |
| ⋃ y : Site d, {ξ | y ∈ A ξ.1} ∩ {ξ | ξ.2 ∈ visitAfter R {y} K} := by |
| ext ⟨ω,X⟩ |
| simp only [Set.mem_iUnion,Set.mem_inter_iff,Set.mem_ofPred_eq] |
| constructor |
| · rintro ⟨n,hAn,hR,hK⟩ |
| exact ⟨X n,hAn,n,rfl,hR,hK⟩ |
| · rintro ⟨y,hy,n,hAn,hR,hK⟩ |
| exact ⟨n,hAn ▸ hy,hR,hK⟩ |
| rw [he] |
| exact MeasurableSet.iUnion (fun y => ((hA y).preimage measurable_fst).inter |
| ((measurableSet_visitAfter R {y} K).preimage measurable_snd)) |
| |
| noncomputable def quenchedReturn {d : ℕ} (e : Step d) (k N : ℤ) |
| (ω : Environment d) (y : Site d) : ℝ≥0∞ := |
| quenchedKernel d (ω,y) (hitBefore (axisLower e k) (axisUpper e N)) |
| |
| lemma measurable_quenchedReturn {d : ℕ} (e : Step d) (k N : ℤ) (y : Site d) : |
| Measurable (fun ω => quenchedReturn e k N ω y) := |
| ((quenchedKernel d).measurable_coe (measurableSet_hitBefore _ _)).comp |
| (measurable_id.prodMk measurable_const) |
| |
| def returnAtLeast {d : ℕ} (e : Step d) (k N : ℤ) (a : ℝ≥0∞) |
| (ω : Environment d) : Set (Site d) := |
| {y | k ≤ axisHeight e y ∧ axisHeight e y < N ∧ a ≤ quenchedReturn e k N ω y} |
| |
| def returnBin {d : ℕ} (e : Step d) (k N : ℤ) (a : ℝ≥0∞) |
| (ω : Environment d) : Set (Site d) := |
| {y | y ∈ returnAtLeast e k N a ω ∧ quenchedReturn e k N ω y < 2*a} |
| |
| lemma measurableSet_returnAtLeast {d : ℕ} (e : Step d) (k N : ℤ) (a : ℝ≥0∞) (y : Site d) : |
| MeasurableSet {ω | y ∈ returnAtLeast e k N a ω} := |
| (MeasurableSet.const _).inter ((MeasurableSet.const _).inter |
| (measurableSet_le measurable_const (measurable_quenchedReturn e k N y))) |
| |
| lemma measurableSet_returnBin {d : ℕ} (e : Step d) (k N : ℤ) (a : ℝ≥0∞) (y : Site d) : |
| MeasurableSet {ω | y ∈ returnBin e k N a ω} := |
| (measurableSet_returnAtLeast e k N a y).inter |
| (measurableSet_lt (measurable_quenchedReturn e k N y) measurable_const) |
| |
| lemma freshJoint_return_visit_bound {d : ℕ} (μ : Measure (Row d)) [IsProbabilityMeasure μ] |
| (hell : StrictEllipticity μ) (e : Step d) {r N : ℕ} (hrN : r ≤ N) (a : ℝ≥0∞) : |
| a * freshJoint μ 0 {ξ | ξ.2 ∈ visitAfter (axisUpper e r) (returnAtLeast e 0 N a ξ.1) |
| (axisLower e 0 ∪ axisUpper e N)} ≤ axisReachProb μ e r - axisReachProb μ e N := by |
| rw [freshJoint_apply μ 0 (measurable_random_visitAfter _ _ _ (measurableSet_returnAtLeast e 0 N a))] |
| exact annealed_axis_visitAfter_bound μ hell e hrN a _ (by |
| filter_upwards [] with ω |
| exact fun y hy => hy.2.2) |
| |
| lemma weighted_disjoint_measure_bound_ae {Ω ι : Type*} [MeasurableSpace Ω] [Countable ι] |
| (μ : Measure Ω) (s : ι → Set Ω) (hs : ∀ i, MeasurableSet (s i)) |
| (hd : Pairwise (fun i j => Disjoint (s i) (s j))) (w : ι → ℝ≥0∞) (F : Ω → ℝ≥0∞) |
| (E : Set Ω) (hsub : ∀ i, s i ⊆ E) (hbound : ∀ i, ∀ᵐ ω ∂μ, ω ∈ s i → w i ≤ F ω) : |
| ∑' i, w i * μ (s i) ≤ ∫⁻ ω in E, F ω ∂μ := by |
| calc |
| ∑' i, w i * μ (s i) ≤ ∑' i, ∫⁻ ω in s i, F ω ∂μ := by |
| apply ENNReal.tsum_le_tsum |
| intro i |
| rw [← setLIntegral_const] |
| exact lintegral_mono_ae ((ae_restrict_iff' (hs i)).mpr (hbound i)) |
| _ = ∫⁻ ω in ⋃ i, s i, F ω ∂μ := (lintegral_iUnion hs hd F).symm |
| _ ≤ _ := lintegral_mono_set (Set.iUnion_subset hsub) |
| |
| lemma weighted_contact_bin_bound_ae {d : ℕ} (P Q : Measure (Path d)) [IsProbabilityMeasure P] |
| [IsProbabilityMeasure Q] (A : Set (Word d)) (D V : Set (Path d)) (K : Set (Site d)) |
| (hD : MeasurableSet D) (w : Word d → ℝ≥0∞) (M : Path d → ℝ≥0∞) |
| (hM : Measurable M) (hA : ∀ a ∈ A, wordEnd a ∈ K) |
| (hAV : ∀ a ∈ A, wordCylinder a ⊆ V) |
| (hw : ∀ a ∈ A, ∀ᵐ X ∂P, X ∈ wordCylinder a → w a ≤ M X) (c : ℝ≥0∞) |
| (hc : Q (D ∩ {B | ∃ n, B n ∈ K}) ≤ c) : |
| ∑' a : A, w a * (P (wordCylinder a) * Q (D ∩ contactAvoid a)) ≤ |
| c * ∫⁻ X in V, M X ∂P := by |
| let R : Set (Path d) := D ∩ {B | ∃ n, B n ∈ K} |
| have hR : MeasurableSet R := hD.inter (by |
| simp only [Set.ofPred_exists] |
| exact MeasurableSet.iUnion (fun n => (Set.to_countable K).measurableSet.preimage (measurable_pi_apply n))) |
| have hs (a : A) : MeasurableSet (wordCylinder a.val ×ˢ (D ∩ contactAvoid a.val)) := |
| (measurableSet_pathCylinder _ _).prod (hD.inter (measurableSet_contactAvoid _)) |
| have hd : Pairwise (fun a b : A => Disjoint |
| (wordCylinder a.val ×ˢ (D ∩ contactAvoid a.val)) |
| (wordCylinder b.val ×ˢ (D ∩ contactAvoid b.val))) := by |
| intro a b hab |
| apply (contactRectangles_disjoint (i := a.val) (j := b.val) (fun h => hab (Subtype.ext h))).mono |
| · exact fun _ h => ⟨h.1,h.2.2⟩ |
| · exact fun _ h => ⟨h.1,h.2.2⟩ |
| have hsub (a : A) : wordCylinder a.val ×ˢ (D ∩ contactAvoid a.val) ⊆ V ×ˢ R := by |
| rintro ⟨X,B⟩ ⟨hX,hB⟩ |
| obtain ⟨n,hn⟩ := hB.2.1 |
| exact ⟨hAV _ a.property hX,hB.1,n,hn ▸ hA _ a.property⟩ |
| have hb (a : A) : ∀ᵐ p ∂P.prod Q, p ∈ wordCylinder a.val ×ˢ (D ∩ contactAvoid a.val) → w a.val ≤ M p.1 := by |
| have hfst : ∀ᵐ p ∂P.prod Q, p.1 ∈ wordCylinder a.val → w a.val ≤ M p.1 := |
| (measurePreserving_fst (μ := P) (ν := Q)).quasiMeasurePreserving.ae (hw a a.property) |
| filter_upwards [hfst] with p hp |
| exact fun h => hp h.1 |
| have h := weighted_disjoint_measure_bound_ae (P.prod Q) |
| (fun a : A => wordCylinder a.val ×ˢ (D ∩ contactAvoid a.val)) hs hd |
| (fun a => w a) (fun p => M p.1) (V ×ˢ R) hsub hb |
| simp only [Measure.prod_prod] at h |
| apply h.trans |
| rw [setLIntegral_prod (fun p : Path d × Path d => M p.1) (hM.comp measurable_fst).aemeasurable.restrict] |
| simp only [setLIntegral_const] |
| rw [lintegral_mul_const _ hM,mul_comm] |
| exact mul_le_mul_left hc _ |
| |
| end DirectionalZeroOne |
| |
| end OAI |
| |
