| import Mathlib |
| import OAI.Probability.Ballisticity.Coupling.CountableFreshBuffer |
| import OAI.Probability.Ballisticity.Estimates.BufferTree |
| import OAI.Probability.Ballisticity.Coupling.FreshBufferAE |
|
|
| namespace OAI |
|
|
| section |
|
|
| section |
|
|
| open MeasureTheory ProbabilityTheory Filter Function |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| namespace DirectionalTransience |
|
|
| noncomputable def positiveBufferPlan {d : ℕ} (ν : Measure (Row d)) (e f : Direction d) (r : ℝ) : ℕ := |
| max 1 ⌊fluctuationScale (independentConditionedPairLaw ν (realPosition (step e))) |
| (commonIncrementProcess (realPosition (step e)) f 0) r⌋₊ |
|
|
| lemma positiveBufferPlan_pos {d : ℕ} (ν : Measure (Row d)) (e f : Direction d) (r : ℝ) : |
| 0 < positiveBufferPlan ν e f r := lt_of_lt_of_le Nat.zero_lt_one (le_max_left _ _) |
|
|
| def BufferChanceData {d : ℕ} (ν : Measure (Row d)) (e f : Direction d) |
| (α c ε g R : ℝ) : Prop := |
| ∀ r : ℝ, R ≤ r → |
| positiveBufferPlan ν e f r = ⌊fluctuationScale (independentConditionedPairLaw ν (realPosition (step e))) |
| (commonIncrementProcess (realPosition (step e)) f 0) r⌋₊ ∧ |
| ∀ z₀ : ℝ, ∀ π : Measure (Lattice d × Lattice d), IsProbabilityMeasure π → |
| (∀ᵐ x ∂π, z₀+r ≤ signedCoordinate f (x.2-x.1)) → |
| ENNReal.ofReal c ≤ environmentLaw ν {ω | BufferStageEvent (realPosition (step e)) f |
| (positiveBufferPlan ν e f r) z₀ r ε α g π ω} |
|
|
| lemma bufferStageEvent_threshold_mono {d : ℕ} (ℓ : Vector d) (f : Direction d) |
| (H : ℕ) (z₀ r ε α : ℝ) {g g' : ℝ} (hgg : g' ≤ g) |
| (π : Measure (Lattice d × Lattice d)) (ω : Environment d) : |
| BufferStageEvent ℓ f H z₀ r ε α g π ω → BufferStageEvent ℓ f H z₀ r ε α g' π ω := by |
| rintro ⟨ht,hf⟩ |
| exact ⟨fun j hj hjH => (ENNReal.ofReal_le_ofReal hgg).trans (ht j hj hjH), |
| (ENNReal.ofReal_le_ofReal hgg).trans hf⟩ |
|
|
| theorem actual_buffer_chance_data {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 ∧ g ≤ 1 ∧ 0 < R ∧ BufferChanceData ν e f α c ε g R := by |
| obtain ⟨α,c,hα,hc,hchance⟩ := buffer_stage_success_probability ν hue e f hef htrans |
| refine ⟨α,c,hα,hc,fun ε hε => ?_⟩ |
| obtain ⟨g,R,hg,hR,hgood⟩ := hchance ε hε |
| refine ⟨min g 1,R,lt_min hg zero_lt_one,min_le_right _ _,hR,?_⟩ |
| intro r hr |
| obtain ⟨hH,hπ⟩ := hgood r hr |
| have hplan : positiveBufferPlan ν e f r = ⌊fluctuationScale (independentConditionedPairLaw ν (realPosition (step e))) |
| (commonIncrementProcess (realPosition (step e)) f 0) r⌋₊ := max_eq_right hH |
| refine ⟨hplan,fun z₀ π hp hgap => ?_⟩ |
| have hh := hπ z₀ π hp hgap |
| have hb : ENNReal.ofReal c ≤ environmentLaw ν {ω | BufferStageEvent (realPosition (step e)) f |
| (positiveBufferPlan ν e f r) z₀ r ε α g π ω} := by |
| rw [hplan] |
| have he : {ω | BufferStageEvent (realPosition (step e)) f |
| ⌊fluctuationScale (independentConditionedPairLaw ν (realPosition (step e))) |
| (commonIncrementProcess (realPosition (step e)) f 0) r⌋₊ z₀ r ε α g π ω} = |
| {ω | BufferStageSuccess (realPosition (step e)) f |
| ⌊fluctuationScale (independentConditionedPairLaw ν (realPosition (step e))) |
| (commonIncrementProcess (realPosition (step e)) f 0) r⌋₊ z₀ r ε α g π ω} := by |
| ext ω |
| exact bufferStageEvent_iff e f hH z₀ r ε α hg.le π ω |
| rw [he] |
| exact (ENNReal.ofReal_le_iff_le_toReal (measure_ne_top _ _)).mpr hh |
| exact hb.trans (measure_mono fun ω hω => bufferStageEvent_threshold_mono _ _ _ _ _ _ _ |
| (min_le_left g 1) π ω hω) |
| end DirectionalTransience |
|
|
| end |
|
|
| section |
|
|
| open MeasureTheory ProbabilityTheory Filter Function |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| namespace DirectionalTransience |
|
|
| lemma bufferTree_weighted_success {d : ℕ} (ν : Measure (Row d)) [IsProbabilityMeasure ν] |
| (e f : Direction d) (hef : e.1 ≠ f.1) {R₀ : ℝ} (hR₀ : 0 < R₀) |
| (z₀ ε α g c : ℝ) {κ : ℝ≥0} (hκpos : 0 < κ) (hκ1 : κ ≤ 1) |
| (hκae : ∀ᵐ ω ∂environmentLaw ν, ∀ y u, κ ≤ (ω y).1 u) |
| (hg : 0 < g) (hg1 : g ≤ 1) (hε : 0 ≤ ε) |
| (hdata : BufferChanceData ν e f α c ε g R₀) (root : AdaptedBufferNode e) |
| (hr : R₀ ≤ root.radius) (hroot : BufferNodeValid e f z₀ κ root) |
| (hactive : root.active=Set.univ) (s : ℝ) (n : ℕ) : |
| let B := bufferTreeSuccess e f hef R₀ z₀ ε α g κ (positiveBufferPlan ν e f) |
| (positiveBufferPlan_pos ν e f) root |
| ENNReal.ofReal c*(∫⁻ ω, ENNReal.ofReal (failureWeight B s n ω) ∂environmentLaw ν) ≤ |
| ∫⁻ ω in B n, ENNReal.ofReal (failureWeight B s n ω) ∂environmentLaw ν := by |
| let H := positiveBufferPlan ν e f |
| let hH := positiveBufferPlan_pos ν e f |
| let nodes := fun l : BufferWords n => bufferNodeAt e f hef R₀ z₀ ε α g κ H hH root l.val |
| let A := fun l : BufferWords n => (nodes l).active |
| let E := fun l : BufferWords n => {ω | BufferStageEvent (realPosition (step e)) f (H (nodes l).radius) |
| z₀ (nodes l).radius ε α g ((nodes l).law ω).val ω} |
| let B := bufferTreeSuccess e f hef R₀ z₀ ε α g κ H hH root |
| let w := fun ω => ENNReal.ofReal (failureWeight B s n ω) |
| have hAm (l) : MeasurableSet (A l) := rowSigma_le _ _ (nodes l).active_rows |
| have hEm (l) : MeasurableSet (E l) := by |
| have hp := (nodes l).law_rows.mono (rowSigma_le _) le_rfl |
| exact (measurableSet_bufferStageEvent_rows (realPosition (step e)) f (hH _) z₀ (nodes l).radius ε α g |
| (PairAtHeight (realPosition (step e)) (nodes l).level) Set.univ |
| (fun x hx => ⟨Set.subset_univ _,Set.subset_univ _⟩)).preimage |
| (hp.prodMk (measurable_id.mono le_rfl (rowSigma_le Set.univ))) |
| have hpart := bufferWords_partition e f hef R₀ z₀ ε α g κ H hH root n |
| have hbound (l : BufferWords n) : ENNReal.ofReal c*(∫⁻ ω in A l, w ω ∂environmentLaw ν) ≤ |
| ∫⁻ ω in A l ∩ E l, w ω ∂environmentLaw ν := by |
| let N := nodes l |
| let D := BelowHeight (realPosition (step e)) N.level |
| have hC (x : Lattice d × Lattice d) (hx : x ∈ PairAtHeight (realPosition (step e)) N.level) : |
| Strip (realPosition (step e)) x.1 (H N.radius) ⊆ Dᶜ ∧ |
| Strip (realPosition (step e)) x.2 (H N.radius) ⊆ Dᶜ := by |
| constructor |
| · intro y hy |
| change ¬ dot (realPosition y) (realPosition (step e)) < N.level |
| have hh := hy.1 |
| rw [hx.1] at hh |
| exact not_lt_of_ge hh |
| · intro y hy |
| change ¬ dot (realPosition y) (realPosition (step e)) < N.level |
| have hh := hy.1 |
| rw [hx.2] at hh |
| exact not_lt_of_ge hh |
| have hNr : R₀ ≤ N.radius := bufferNodeAt_radius e f hef R₀ z₀ ε α g κ H hH root hr l.val |
| have hNv := bufferNodeAt_valid e f hef hR₀.le z₀ ε α g hκpos hκ1 hg hg1 hε H hH root hr hroot l.val |
| have hvalid : ∀ᵐ ω ∂environmentLaw ν, ω ∈ N.active → IsProbabilityMeasure (N.law ω).val ∧ |
| ∀ᵐ x ∂(N.law ω).val, z₀+N.radius ≤ signedCoordinate f (x.2-x.1) := |
| hκae.mono fun ω hω hA => hNv ω hω hA |
| have hw := buffer_failureWeight_on_branch e f hef R₀ z₀ ε α g κ H hH root s n l |
| have hh := fresh_buffer_weight_ae_valid ν (realPosition (step e)) f (hH N.radius) |
| z₀ N.radius ε α g c (PairAtHeight (realPosition (step e)) N.level) D Dᶜ |
| disjoint_compl_right hC (fun π hp hgap => (hdata N.radius hNr).2 z₀ π.val hp hgap) |
| N.law N.law_rows N.active N.active_rows hvalid (N.active.indicator w) hw |
| have hleft : (∫⁻ ω in N.active, N.active.indicator w ω ∂environmentLaw ν) = |
| ∫⁻ ω in N.active, w ω ∂environmentLaw ν := by |
| apply lintegral_congr_ae |
| filter_upwards [ae_restrict_mem (hAm l)] with ω hω |
| exact Set.indicator_of_mem hω _ |
| have hright : (∫⁻ ω in N.active ∩ E l, N.active.indicator w ω ∂environmentLaw ν) = |
| ∫⁻ ω in N.active ∩ E l, w ω ∂environmentLaw ν := by |
| apply lintegral_congr_ae |
| filter_upwards [ae_restrict_mem ((hAm l).inter (hEm l))] with ω hω |
| exact Set.indicator_of_mem hω.1 _ |
| change ENNReal.ofReal c*(∫⁻ ω in N.active, N.active.indicator w ω ∂environmentLaw ν) ≤ |
| ∫⁻ ω in N.active ∩ E l, N.active.indicator w ω ∂environmentLaw ν at hh |
| rw [hleft,hright] at hh |
| exact hh |
| have hh := weighted_countable_partition_lower (environmentLaw ν) A (fun l => A l ∩ E l) hAm |
| (fun l => (hAm l).inter (hEm l)) hpart.2 (fun _ => Set.inter_subset_left) w (ENNReal.ofReal c) hbound |
| have ha : (⋃ l, A l)=Set.univ := hpart.1.trans hactive |
| have hb : (⋃ l, A l ∩ E l)=B n := by |
| unfold B bufferTreeSuccess |
| congr 1 |
| funext l |
| ext ω |
| simp only [A,E,nodes,bufferNodeDecision,Set.mem_inter_iff,Set.mem_ofPred,decide_eq_true_eq] |
| rw [ha,Measure.restrict_univ,hb] at hh |
| exact hh |
|
|
| lemma bufferTree_real_success {d : ℕ} (ν : Measure (Row d)) [IsProbabilityMeasure ν] |
| (e f : Direction d) (hef : e.1 ≠ f.1) {R₀ : ℝ} (hR₀ : 0 < R₀) |
| (z₀ ε α g c : ℝ) (hc : 0 ≤ c) {κ : ℝ≥0} (hκpos : 0 < κ) (hκ1 : κ ≤ 1) |
| (hκae : ∀ᵐ ω ∂environmentLaw ν, ∀ y u, κ ≤ (ω y).1 u) |
| (hg : 0 < g) (hg1 : g ≤ 1) (hε : 0 ≤ ε) |
| (hdata : BufferChanceData ν e f α c ε g R₀) (root : AdaptedBufferNode e) |
| (hr : R₀ ≤ root.radius) (hroot : BufferNodeValid e f z₀ κ root) |
| (hactive : root.active=Set.univ) (n : ℕ) : |
| let B := bufferTreeSuccess e f hef R₀ z₀ ε α g κ (positiveBufferPlan ν e f) |
| (positiveBufferPlan_pos ν e f) root |
| c*(∫ ω, failureWeight B c n ω ∂environmentLaw ν) ≤ |
| ∫ ω in B n, failureWeight B c n ω ∂environmentLaw ν := by |
| let B := bufferTreeSuccess e f hef R₀ z₀ ε α g κ (positiveBufferPlan ν e f) |
| (positiveBufferPlan_pos ν e f) root |
| have hBm := bufferTreeSuccess_measurable e f hef R₀ z₀ ε α g κ |
| (positiveBufferPlan ν e f) (positiveBufferPlan_pos ν e f) root |
| have hi := failureWeight_integrable (environmentLaw ν) B hBm c n |
| have hp (ω) : 0 ≤ failureWeight B c n ω := (Real.exp_pos _).le |
| have hh := bufferTree_weighted_success ν e f hef hR₀ z₀ ε α g c hκpos hκ1 hκae hg hg1 hε hdata root hr hroot hactive c n |
| change ENNReal.ofReal c*(∫⁻ ω, ENNReal.ofReal (failureWeight B c n ω) ∂environmentLaw ν) ≤ |
| ∫⁻ ω in B n, ENNReal.ofReal (failureWeight B c n ω) ∂environmentLaw ν at hh |
| rw [←ofReal_integral_eq_lintegral_ofReal hi (ae_of_all _ hp), |
| ←ofReal_integral_eq_lintegral_ofReal hi.restrict (ae_of_all _ hp),←ENNReal.ofReal_mul hc] at hh |
| exact (ENNReal.ofReal_le_ofReal_iff (integral_nonneg hp)).mp hh |
| end DirectionalTransience |
|
|
| end |
|
|
| end |
|
|
| end OAI |
|
|
