| import Mathlib |
| import OAI.Probability.Ballisticity.Coupling.StageSeedProfile |
| import OAI.Probability.Ballisticity.Estimates.RetainedComposition |
|
|
| namespace OAI |
|
|
| section |
|
|
| open MeasureTheory ProbabilityTheory |
| open scoped ENNReal NNReal Classical |
| namespace DirectionalTransience |
|
|
| noncomputable def cellSeedMass {d k : ℕ} (e f : Direction d) (a G : ℝ) (Hs : ℕ) |
| (π : Environment d → LayerTupleProfile (k:=k) e a) (ω : Environment d) : ℝ≥0∞ := |
| seedSeparatedRaw e f Hs G (π ω).val ω Set.univ |
|
|
| noncomputable def cellEarlyMass {d k : ℕ} (e f : Direction d) (hef : e.1 ≠ f.1) |
| (a G : ℝ) (Hs He : ℕ) (π : Environment d → LayerTupleProfile (k:=k) e a) |
| (ω : Environment d) : ℝ := |
| relativeBudgetMass e f He (G/4) (seedSeparatedProfile e f hef a G Hs (π ω) ω).val ω |
|
|
| noncomputable def cellEarlyProfile {d k : ℕ} (e f : Direction d) (hef : e.1 ≠ f.1) |
| (a G : ℝ) (hG : 0 ≤ G) (Hs He : ℕ) (π : Environment d → LayerTupleProfile (k:=k) e a) |
| (ω : Environment d) : BudgetProfile (k:=k) e f (a+Hs+He) (3*G/4) := by |
| let q := nextBudgetProfile e f He (a+Hs) G (G/4) (by positivity) |
| (seedSeparatedProfile e f hef a G Hs (π ω) ω) ω |
| refine ⟨q.val,q.property.1,?_⟩ |
| have hg : G-G/4=3*G/4 := by ring |
| simpa only [hg] using q.property.2 |
|
|
| lemma cellSeedMass_measurable {d k : ℕ} (e f : Direction d) (a G : ℝ) (Hs : ℕ) |
| (π : Environment d → LayerTupleProfile (k:=k) e a) |
| (hπ : @Measurable _ _ (rowSigma (BelowHeight (realPosition (step e)) a)) _ π) : |
| @Measurable _ _ (rowSigma (BelowHeight (realPosition (step e)) (a+Hs))) _ |
| (cellSeedMass e f a G Hs π) := by |
| have hle : rowSigma (BelowHeight (realPosition (step e)) a) ≤ |
| rowSigma (BelowHeight (realPosition (step e)) (a+Hs)) := |
| rowSigma_mono (fun x hx => by |
| change dot (realPosition x) (realPosition (step e)) < _ at * |
| exact lt_of_lt_of_le hx (le_add_of_nonneg_right (Nat.cast_nonneg _))) |
| let : MeasurableSpace (Environment d) := |
| rowSigma (BelowHeight (realPosition (step e)) (a+Hs)) |
| unfold cellSeedMass |
| apply (Measure.measurable_coe MeasurableSet.univ).comp |
| apply (seedSeparatedRaw_joint_rows e f Hs G a _ ?_).comp |
| ((hπ.mono hle le_rfl).prodMk measurable_id) |
| intro x hx j y hy |
| change dot (realPosition y) (realPosition (step e))<a+Hs |
| simpa only [hx j] using hy.2 |
|
|
| lemma cellEarlyMass_measurable {d k : ℕ} (e f : Direction d) (hef : e.1 ≠ f.1) |
| (a G : ℝ) (Hs He : ℕ) (π : Environment d → LayerTupleProfile (k:=k) e a) |
| (hπ : @Measurable _ _ (rowSigma (BelowHeight (realPosition (step e)) a)) _ π) : |
| @Measurable _ _ (rowSigma (BelowHeight (realPosition (step e)) (a+Hs+He))) _ |
| (cellEarlyMass e f hef a G Hs He π) := by |
| have hle : rowSigma (BelowHeight (realPosition (step e)) (a+Hs)) ≤ |
| rowSigma (BelowHeight (realPosition (step e)) (a+Hs+He)) := |
| rowSigma_mono (fun x hx => by |
| change dot (realPosition x) (realPosition (step e)) < _ at * |
| exact lt_of_lt_of_le hx (le_add_of_nonneg_right (Nat.cast_nonneg _))) |
| let : MeasurableSpace (Environment d) := |
| rowSigma (BelowHeight (realPosition (step e)) (a+Hs+He)) |
| have hm := seedSeparatedProfile_measurable e f hef a G Hs π hπ |
| exact (budgetProfileMass_joint_rows e f (a+Hs) G He (G/4) _ (by |
| intro x hx j y hy |
| change dot (realPosition y) (realPosition (step e))<a+Hs+He |
| simpa only [hx j] using hy.2)).comp ((hm.mono hle le_rfl).prodMk measurable_id) |
|
|
| lemma cellEarlyProfile_measurable {d k : ℕ} (e f : Direction d) (hef : e.1 ≠ f.1) |
| (a G : ℝ) (hG : 0 ≤ G) (Hs He : ℕ) (π : Environment d → LayerTupleProfile (k:=k) e a) |
| (hπ : @Measurable _ _ (rowSigma (BelowHeight (realPosition (step e)) a)) _ π) : |
| @Measurable _ _ (rowSigma (BelowHeight (realPosition (step e)) (a+Hs+He))) _ |
| (cellEarlyProfile e f hef a G hG Hs He π) := by |
| let : MeasurableSpace (Environment d) := |
| rowSigma (BelowHeight (realPosition (step e)) (a+Hs+He)) |
| have hm := nextBudgetProfile_measurable_below e f (a+Hs) G (G/4) (by positivity) He |
| (fun ω => seedSeparatedProfile e f hef a G Hs (π ω) ω) |
| (seedSeparatedProfile_measurable e f hef a G Hs π hπ) |
| exact (measurable_subtype_coe.comp hm).subtype_mk |
|
|
| lemma cellEarlyProfile_retained {d k : ℕ} (e f : Direction d) (hef : e.1 ≠ f.1) |
| (a G : ℝ) (hG : 0 ≤ G) (Hs He : ℕ) (π : Environment d → LayerTupleProfile (k:=k) e a) |
| (ω : Environment d) : |
| (cellSeedMass e f a G Hs π ω * ENNReal.ofReal (cellEarlyMass e f hef a G Hs He π ω)) • |
| (cellEarlyProfile e f hef a G hG Hs He π ω).val ≤ |
| rawTupleMixture (realPosition (step e)) (Hs+He) (π ω).val ω := by |
| let p := seedSeparatedProfile e f hef a G Hs (π ω) ω |
| have hp : cellSeedMass e f a G Hs π ω • p.val=seedSeparatedRaw e f Hs G (π ω).val ω := |
| seedSeparatedProfile_mass e f hef a G Hs (π ω) ω |
| have hq : ENNReal.ofReal (cellEarlyMass e f hef a G Hs He π ω) • |
| (cellEarlyProfile e f hef a G hG Hs He π ω).val = |
| budgetEndpointMixture e f He (G/4) p.val ω := by |
| change ENNReal.ofReal (relativeBudgetMass e f He (G/4) p.val ω) • |
| (nextBudgetProfile e f He (a+Hs) G (G/4) (by positivity) p ω).val = _ |
| rw [←relativeBudgetMassENN_toReal, ENNReal.ofReal_toReal (lt_of_le_of_lt |
| (relativeBudgetMassENN_le_one e f He (G/4) p.val ω) ENNReal.one_lt_top).ne] |
| exact nextBudgetProfile_mass e f He (a+Hs) G (G/4) (by positivity) p ω |
| rw [mul_smul,hq] |
| calc |
| _ ≤ cellSeedMass e f a G Hs π ω • rawTupleMixture (realPosition (step e)) He p.val ω := by |
| exact smul_le_smul_left _ (budgetEndpointMixture_le_raw e f He (G/4) p.val ω) |
| _ = rawTupleMixture (realPosition (step e)) He |
| (cellSeedMass e f a G Hs π ω • p.val) ω := (rawTupleMixture_smul _ _ _ _ _).symm |
| _ ≤ rawTupleMixture (realPosition (step e)) He |
| (rawTupleMixture (realPosition (step e)) Hs (π ω).val ω) ω := by |
| rw [hp] |
| exact rawTupleMixture_mono _ _ _ Measure.restrict_le_self |
| _ ≤ _ := rawTupleMixture_comp_le e Hs He (π ω).val ω |
|
|
| end DirectionalTransience |
|
|
| end |
|
|
| section |
|
|
| open MeasureTheory ProbabilityTheory |
| open scoped ENNReal BigOperators Classical |
| namespace DirectionalTransience |
|
|
| def failedOpportunityBlocks {Ω : Type*} (F : ℕ → ℕ → Set Ω) (i : ℕ) : List ℕ → Set Ω |
| | [] => Set.univ |
| | n::ns => F i n ∩ failedOpportunityBlocks F (i+n) ns |
|
|
| lemma failedOpportunityBlocks_bound {Ω : Type*} [m : MeasurableSpace Ω] |
| (μ : Measure Ω) (ℱ : Filtration ℕ m) (F : ℕ → ℕ → Set Ω) |
| (hFm : ∀ i n, 0<n → MeasurableSet[ℱ (i+n)] (F i n)) (c : ℝ≥0∞) |
| (hfresh : ∀ i n, 0<n → ∀ A : Set Ω, MeasurableSet[ℱ i] A → |
| μ (A∩F i n)≤c^n*μ A) (ns : List ℕ) (hpos : ∀ n∈ns, 0<n) |
| (i : ℕ) (A : Set Ω) (hA : MeasurableSet[ℱ i] A) : |
| μ (A∩failedOpportunityBlocks F i ns)≤c^ns.sum*μ A := by |
| induction ns generalizing i A with |
| | nil => simp [failedOpportunityBlocks] |
| | cons n ns ih => |
| have hn : 0<n := hpos n (by simp) |
| have hAn : MeasurableSet[ℱ (i+n)] (A∩F i n) := |
| (ℱ.mono (Nat.le_add_right i n) A hA).inter (hFm i n hn) |
| have hi := ih (fun k hk => hpos k (by simp [hk])) (i+n) (A∩F i n) hAn |
| rw [failedOpportunityBlocks,← Set.inter_assoc] |
| apply hi.trans |
| calc |
| c^ns.sum*μ (A∩F i n) ≤ c^ns.sum*(c^n*μ A) := by gcongr; exact hfresh i n hn A hA |
| _ = c^(n::ns).sum*μ A := by simp [pow_add]; ring |
|
|
| theorem opportunity_exhaustion_probability {Ω : Type*} [m : MeasurableSpace Ω] |
| (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m) |
| (F : ℕ → ℕ → Set Ω) |
| (hFm : ∀ i n, 0<n → MeasurableSet[ℱ (i+n)] (F i n)) (c : ℝ≥0∞) |
| (hfresh : ∀ i n, 0<n → ∀ A : Set Ω, MeasurableSet[ℱ i] A → |
| μ (A∩F i n)≤c^n*μ A) (m : ℕ) : |
| μ (⋃ p : Composition m, failedOpportunityBlocks F 0 p.blocks)≤ |
| (2:ℝ≥0∞)^(m-1)*c^m := by |
| apply (measure_iUnion_le _).trans |
| calc |
| (∑' p : Composition m, μ (failedOpportunityBlocks F 0 p.blocks)) ≤ |
| ∑' _p : Composition m, c^m := by |
| apply ENNReal.tsum_le_tsum |
| intro p |
| have hh := failedOpportunityBlocks_bound μ ℱ F hFm c hfresh p.blocks |
| (fun n hn => p.blocks_pos hn) 0 Set.univ MeasurableSet.univ |
| simpa using hh |
| _ = (2:ℝ≥0∞)^(m-1)*c^m := by |
| rw [tsum_fintype] |
| simp [composition_card,nsmul_eq_mul] |
|
|
| end DirectionalTransience |
|
|
| end |
|
|
| end OAI |
|
|
