| import Mathlib |
| import OAI.Probability.Ballisticity.Coupling.SeedRadius |
| import OAI.Probability.Ballisticity.Coupling.StageSeedProfile |
|
|
| namespace OAI |
|
|
| section |
|
|
| open MeasureTheory ProbabilityTheory |
| open scoped ENNReal BigOperators Classical |
| namespace DirectionalTransience |
|
|
| lemma fresh_random_budget_mass_lt {d k : ℕ} (ν : Measure (Row d)) [IsProbabilityMeasure ν] |
| (e f : Direction d) (a G r s : ℝ) (H : ℕ) |
| (π : Environment d → BudgetProfile (k:=k) e f a G) |
| (hπ : @Measurable _ _ (rowSigma (BelowHeight (realPosition (step e)) a)) _ π) |
| (A : Set (Environment d)) (hA : MeasurableSet[rowSigma (BelowHeight (realPosition (step e)) a)] A) |
| (c : ℝ≥0∞) |
| (hbound : ∀ η ∈ A, environmentLaw ν {ω | relativeBudgetMass e f H r (π η).val ω<s}≤c) : |
| environmentLaw ν (A∩{ω | relativeBudgetMass e f H r (π ω).val ω<s})≤c*environmentLaw ν A := by |
| let S := BelowHeight (realPosition (step e)) a |
| let T : Set (Lattice d) := {y | a≤dot (realPosition y) (realPosition (step e))} |
| let B : Set (Environment d×Environment d) := {p | relativeBudgetMass e f H r (π p.1).val p.2<s} |
| have hQ := budgetProfileMass_joint_rows (k:=k) e f a G H r T (fun x hx j y hy => by |
| change a≤dot (realPosition y) (realPosition (step e)) |
| simpa only [hx j] using hy.1) |
| have hB : @MeasurableSet (Environment d×Environment d) (MeasurableSpace.prod (rowSigma S) (rowSigma T)) B := |
| measurableSet_lt (hQ.comp ((hπ.comp measurable_fst).prodMk measurable_snd)) measurable_const |
| have hV : @Measurable (Environment d) (Environment d) inferInstance (rowSigma T) id := |
| measurable_id.mono le_rfl (rowSigma_le T) |
| have hb (η : Environment d) (hη : η∈A) : |
| (@Measure.map (Environment d) (Environment d) inferInstance (rowSigma T) id (environmentLaw ν)) |
| {ω | (η,ω)∈B}≤c := by |
| rw [show {ω | (η,ω)∈B}=Prod.mk η ⁻¹' B from rfl, |
| Measure.map_apply hV (hB.preimage measurable_prodMk_left)] |
| exact hbound η hη |
| apply @fresh_rows_random_test_bound d (Environment d) (Environment d) (rowSigma S) (rowSigma T) |
| ν _ S T _ id id measurable_id measurable_id A hA B hB c hb |
| exact Set.disjoint_left.mpr fun x hx hy => |
| (not_lt.mpr (show a≤dot (realPosition x) (realPosition (step e)) from hy)) |
| (show dot (realPosition x) (realPosition (step e))<a from hx) |
|
|
| end DirectionalTransience |
|
|
| end |
|
|
| section |
|
|
| open MeasureTheory ProbabilityTheory |
| open scoped ENNReal Classical |
| namespace DirectionalTransience |
|
|
| lemma fresh_layer_diagonal_bound {d : ℕ} (ν : Measure (Row d)) [IsProbabilityMeasure ν] |
| (e : Direction d) (a : ℝ) (A : Set (Environment d)) |
| (hA : MeasurableSet[rowSigma (BelowHeight (realPosition (step e)) a)] A) |
| (B : Set (Environment d×Environment d)) |
| (hB : @MeasurableSet (Environment d×Environment d) |
| (MeasurableSpace.prod (rowSigma (BelowHeight (realPosition (step e)) a)) |
| (rowSigma {y | a≤dot (realPosition y) (realPosition (step e))})) B) |
| (c : ℝ≥0∞) (hbound : ∀ η∈A, environmentLaw ν {ω | (η,ω)∈B}≤c) : |
| environmentLaw ν (A∩{ω | (ω,ω)∈B})≤c*environmentLaw ν A := by |
| let S := BelowHeight (realPosition (step e)) a |
| let T : Set (Lattice d) := {y | a≤dot (realPosition y) (realPosition (step e))} |
| have hV : @Measurable (Environment d) (Environment d) inferInstance (rowSigma T) id := |
| measurable_id.mono le_rfl (rowSigma_le T) |
| have hb (η : Environment d) (hη : η∈A) : |
| (@Measure.map (Environment d) (Environment d) inferInstance (rowSigma T) id (environmentLaw ν)) |
| {ω | (η,ω)∈B}≤c := by |
| rw [show {ω | (η,ω)∈B}=Prod.mk η ⁻¹' B from rfl, |
| Measure.map_apply hV (hB.preimage measurable_prodMk_left)] |
| exact hbound η hη |
| apply @fresh_rows_random_test_bound d (Environment d) (Environment d) (rowSigma S) (rowSigma T) |
| ν _ S T _ id id measurable_id measurable_id A hA B hB c hb |
| exact Set.disjoint_left.mpr fun x hx hy => |
| (not_lt.mpr (show a≤dot (realPosition x) (realPosition (step e)) from hy)) |
| (show dot (realPosition x) (realPosition (step e))<a from hx) |
|
|
| lemma fresh_layer_profile_event_bound {d : ℕ} (ν : Measure (Row d)) [IsProbabilityMeasure ν] |
| (e : Direction d) (a : ℝ) {D : Type*} [MeasurableSpace D] |
| (b : D → Environment d → ℝ≥0∞) |
| (hb : @Measurable (D×Environment d) ℝ≥0∞ |
| (MeasurableSpace.prod inferInstance |
| (rowSigma {y | a≤dot (realPosition y) (realPosition (step e))})) _ |
| (fun p => b p.1 p.2)) |
| (π : Environment d → D) |
| (hπ : @Measurable _ _ (rowSigma (BelowHeight (realPosition (step e)) a)) _ π) |
| (A : Set (Environment d)) (hA : MeasurableSet[rowSigma (BelowHeight (realPosition (step e)) a)] A) |
| (g c : ℝ≥0∞) (hbound : ∀ η∈A, environmentLaw ν {ω | b (π η) ω<g}≤c) : |
| environmentLaw ν (A∩{ω | b (π ω) ω<g})≤c*environmentLaw ν A := by |
| apply fresh_layer_diagonal_bound ν e a A hA {p | b (π p.1) p.2<g} _ c hbound |
| exact measurableSet_lt (hb.comp ((hπ.comp measurable_fst).prodMk measurable_snd)) measurable_const |
|
|
| end DirectionalTransience |
|
|
| end |
|
|
| section |
|
|
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal BigOperators Classical |
| namespace DirectionalTransience |
|
|
| lemma seedSeparatedMass_joint_upper {d k : ℕ} (e f : Direction d) (a G : ℝ) (H : ℕ) : |
| @Measurable (LayerTupleProfile (k:=k) e a×Environment d) ℝ≥0∞ |
| (@Prod.instMeasurableSpace _ _ inferInstance |
| (rowSigma {y | a≤dot (realPosition y) (realPosition (step e))})) _ |
| (fun p => seedSeparatedRaw e f H G p.1.val p.2 Set.univ) := by |
| apply (Measure.measurable_coe MeasurableSet.univ).comp |
| apply seedSeparatedRaw_joint_rows |
| intro x hx j y hy |
| change a≤dot (realPosition y) (realPosition (step e)) |
| simpa only [hx j] using hy.1 |
|
|
| lemma fresh_random_seed_failure {d k : ℕ} (ν : Measure (Row d)) [IsProbabilityMeasure ν] |
| (e f : Direction d) (a G : ℝ) (H : ℕ) (g : ℝ≥0∞) |
| (π : Environment d → LayerTupleProfile (k:=k) e a) |
| (hπ : @Measurable _ _ (rowSigma (BelowHeight (realPosition (step e)) a)) _ π) |
| (A : Set (Environment d)) (hA : MeasurableSet[rowSigma (BelowHeight (realPosition (step e)) a)] A) |
| (c : ℝ≥0∞) |
| (hbound : ∀ η∈A, environmentLaw ν {ω | seedSeparatedRaw e f H G (π η).val ω Set.univ<g}≤c) : |
| environmentLaw ν (A∩{ω | seedSeparatedRaw e f H G (π ω).val ω Set.univ<g})≤c*environmentLaw ν A := by |
| exact fresh_layer_profile_event_bound ν e a (D:=LayerTupleProfile (k:=k) e a) |
| (fun (p : LayerTupleProfile (k:=k) e a) ω => seedSeparatedRaw e f H G p.val ω Set.univ) |
| (seedSeparatedMass_joint_upper (k:=k) e f a G H) π hπ A hA g c hbound |
|
|
| lemma seedSeparatedRaw_measurable {d k : ℕ} (e f : Direction d) (H : ℕ) (G a : ℝ) |
| (π : LayerTupleProfile (k:=k) e a) : Measurable (fun ω => seedSeparatedRaw e f H G π.val ω) := by |
| have hh := seedSeparatedRaw_joint_rows (k:=k) e f H G a Set.univ (fun _ _ _ => Set.subset_univ _) |
| exact (hh.comp (measurable_const.prodMk measurable_id)).mono (rowSigma_le _) le_rfl |
|
|
| theorem adapted_separated_seeds {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))) |
| (k : ℕ) (hk : 2≤k) (p : ℝ) (hp : 0<p) : |
| ∃ C c₀ g₀ : ℝ, 0<C ∧ 0<c₀ ∧ 0<g₀ ∧ ∃ H₀ : ℕ, ∀ H≥H₀, |
| ∀ (a : ℝ) (π : Environment d → LayerTupleProfile (k:=k) e a), |
| @Measurable _ _ (rowSigma (BelowHeight (realPosition (step e)) a)) _ π → |
| ∀ A : Set (Environment d), MeasurableSet[rowSigma (BelowHeight (realPosition (step e)) a)] A → |
| environmentLaw ν (A∩{ω | seedSeparatedRaw e f H |
| (c₀*fluctuationRadius (independentConditionedPairLaw ν (realPosition (step e))) |
| (commonIncrementProcess (realPosition (step e)) f 0) ((H:ℝ)/C)) |
| (π ω).val ω Set.univ<ENNReal.ofReal g₀})≤ENNReal.ofReal p*environmentLaw ν A := by |
| obtain ⟨C,c₀,g₀,hC,hc,hg,H₀,hh⟩ := separated_seeds ν hue e f hef htrans k hk p hp |
| refine ⟨C,c₀,g₀,hC,hc,hg,H₀,fun H hH a π hπ A hA => ?_⟩ |
| apply fresh_random_seed_failure ν e f a _ H _ π hπ A hA (ENNReal.ofReal p) |
| intro η _ |
| have hs := hh H hH ⟨(π η).val,(π η).property.1⟩ |
| let G := c₀*fluctuationRadius (independentConditionedPairLaw ν (realPosition (step e))) |
| (commonIncrementProcess (realPosition (step e)) f 0) ((H:ℝ)/C) |
| have hm : MeasurableSet {ω | seedSeparatedRaw e f H G (π η).val ω Set.univ<ENNReal.ofReal g₀} := |
| measurableSet_lt ((Measure.measurable_coe MeasurableSet.univ).comp |
| (seedSeparatedRaw_measurable e f H G a (π η))) measurable_const |
| have he : {ω | ENNReal.ofReal g₀≤rawTupleMixture (realPosition (step e)) H (π η).val ω {v | TupleSeparated f G v}} = |
| {ω | seedSeparatedRaw e f H G (π η).val ω Set.univ<ENNReal.ofReal g₀}ᶜ := by |
| ext ω; simp only [Set.mem_ofPred_eq,Set.mem_compl_iff,not_lt,seedSeparatedRaw_univ] |
| change 1-p < (environmentLaw ν).real {ω | ENNReal.ofReal g₀≤rawTupleMixture (realPosition (step e)) H (π η).val ω {v | TupleSeparated f G v}} at hs |
| rw [he,measureReal_compl hm,probReal_univ] at hs |
| rw [← ofReal_measureReal (measure_ne_top _ _)] |
| apply ENNReal.ofReal_le_ofReal |
| linarith |
|
|
| end DirectionalTransience |
|
|
| end |
|
|
| end OAI |
|
|
