| import Mathlib |
| import OAI.Probability.Ballisticity.Estimates.BudgetEndpoint |
|
|
| namespace OAI |
|
|
| section |
|
|
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal BigOperators Topology Classical |
| namespace DirectionalTransience |
|
|
| noncomputable def normalizeOr {α : Type*} [MeasurableSpace α] (μ ρ : Measure α) : Measure α := |
| if μ Set.univ = 0 then ρ else normalizedMeasure μ |
|
|
| lemma normalizeOr_probability {α : Type*} [MeasurableSpace α] (μ ρ : Measure α) |
| [IsFiniteMeasure μ] [IsProbabilityMeasure ρ] : IsProbabilityMeasure (normalizeOr μ ρ) := by |
| by_cases h : μ Set.univ=0 |
| · simpa only [normalizeOr,ite_eq_left h] using (inferInstance : IsProbabilityMeasure ρ) |
| · simpa only [normalizeOr,ite_eq_right h] using normalizedMeasure_probability μ h |
|
|
| lemma normalizeOr_mass {α : Type*} [MeasurableSpace α] (μ ρ : Measure α) |
| [IsFiniteMeasure μ] : μ Set.univ • normalizeOr μ ρ = μ := by |
| by_cases h : μ Set.univ=0 |
| · rw [h,zero_smul]; exact (Measure.measure_univ_eq_zero.mp h).symm |
| · rw [normalizeOr,ite_eq_right h,mass_smul_normalizedMeasure] |
|
|
| lemma normalizeOr_ae {α : Type*} [MeasurableSpace α] (μ ρ : Measure α) (P : α → Prop) |
| (hμ : ∀ᵐ x ∂μ, P x) (hρ : ∀ᵐ x ∂ρ, P x) : ∀ᵐ x ∂normalizeOr μ ρ, P x := by |
| unfold normalizeOr |
| split |
| · exact hρ |
| · exact Measure.ae_smul_measure hμ _ |
|
|
| lemma measurable_normalizeOr {Ω α : Type*} [MeasurableSpace Ω] [MeasurableSpace α] |
| (μ ρ : Ω → Measure α) (hμ : Measurable μ) (hρ : Measurable ρ) : |
| Measurable (fun ω => normalizeOr (μ ω) (ρ ω)) := by |
| unfold normalizeOr normalizedMeasure |
| apply Measurable.ite (measurableSet_eq_fun ((Measure.measurable_coe MeasurableSet.univ).comp hμ) measurable_const) hρ |
| apply Measure.measurable_of_measurable_coe |
| intro U hU |
| simp only [Measure.smul_apply,smul_eq_mul] |
| exact (((Measure.measurable_coe MeasurableSet.univ).comp hμ).inv).mul |
| ((Measure.measurable_coe hU).comp hμ) |
|
|
| abbrev BudgetProfile {d k : ℕ} (e f : Direction d) (a G : ℝ) := |
| {π : Measure (Fin k → Lattice d) // IsProbabilityMeasure π ∧ |
| ∀ᵐ x ∂π, x ∈ TupleAtHeight (realPosition (step e)) a ∧ TupleSeparated f G x} |
|
|
| instance budgetProfile_probability {d k : ℕ} (e f : Direction d) (a G : ℝ) |
| (π : BudgetProfile (k:=k) e f a G) : IsProbabilityMeasure π.val := π.property.1 |
|
|
| def BudgetProfile.toSupported {d k : ℕ} {e f : Direction d} {a G : ℝ} |
| (π : BudgetProfile (k:=k) e f a G) : SupportedTupleMeasures (TupleAtHeight (k:=k) (realPosition (step e)) a) := |
| ⟨π.val,ae_iff.mp (π.property.2.mono fun _ h => h.1)⟩ |
|
|
| lemma BudgetProfile.measurable_toSupported {d k : ℕ} (e f : Direction d) (a G : ℝ) : |
| Measurable (BudgetProfile.toSupported (d:=d) (k:=k) (e:=e) (f:=f) (a:=a) (G:=G)) := |
| measurable_subtype_coe.subtype_mk |
|
|
| def tupleShiftUp {d k : ℕ} (e : Direction d) (H : ℕ) (x : Fin k → Lattice d) : Fin k → Lattice d := |
| fun j => x j + H • step e |
|
|
| lemma tupleShiftUp_support {d k : ℕ} (e f : Direction d) (H : ℕ) (a G : ℝ) |
| (x : Fin k → Lattice d) (hx : x ∈ TupleAtHeight (realPosition (step e)) a ∧ TupleSeparated f G x) : |
| tupleShiftUp e H x ∈ TupleAtHeight (realPosition (step e)) (a+H) ∧ |
| TupleSeparated f G (tupleShiftUp e H x) := by |
| have hs (g : Direction d) (y : Lattice d) : signedCoordinate g (y+H • step e) = |
| signedCoordinate g y + (H:ℝ)*signedCoordinate g (step e) := by |
| simp only [signedCoordinate,Pi.add_apply,nsmul_eq_mul,Pi.mul_apply,Pi.natCast_apply,Int.cast_add,Int.cast_mul,Int.cast_natCast] |
| split <;> ring |
| constructor |
| · intro j |
| change dot (realPosition (x j+H • step e)) (realPosition (step e))=a+H |
| rw [dot_signed_direction,hs,signedCoordinate_step_self,mul_one,← dot_signed_direction,hx.1 j] |
| · intro i j hij |
| change G ≤ |signedCoordinate f (x i+H • step e)-signedCoordinate f (x j+H • step e)| |
| rw [hs,hs,add_sub_add_right_eq_sub] |
| exact hx.2 i j hij |
|
|
| noncomputable def nextBudgetProfile {d k : ℕ} (e f : Direction d) (H : ℕ) (a G r : ℝ) |
| (hr : 0 ≤ r) (π : BudgetProfile (k:=k) e f a G) (ω : Environment d) : |
| BudgetProfile (k:=k) e f (a+H) (G-r) := by |
| let μ := budgetEndpointMixture e f H r π.val ω |
| let ρ := π.val.map (tupleShiftUp e H) |
| letI : IsProbabilityMeasure ρ := inferInstance |
| refine ⟨normalizeOr μ ρ,normalizeOr_probability μ ρ,?_⟩ |
| apply normalizeOr_ae |
| · exact budgetEndpointMixture_support e f H a G r π.val ω π.property.2 |
| · rw [ae_map_iff (measurable_of_countable _).aemeasurable (Set.to_countable _).measurableSet] |
| exact π.property.2.mono fun x hx => by |
| have h := tupleShiftUp_support e f H a G x hx |
| exact ⟨h.1,tupleSeparated_mono f (sub_le_self G hr) _ h.2⟩ |
|
|
| lemma nextBudgetProfile_mass {d k : ℕ} (e f : Direction d) (H : ℕ) (a G r : ℝ) |
| (hr : 0 ≤ r) (π : BudgetProfile (k:=k) e f a G) (ω : Environment d) : |
| relativeBudgetMassENN e f H r π.val ω • |
| (nextBudgetProfile e f H a G r hr π ω).val = budgetEndpointMixture e f H r π.val ω := by |
| change _ • normalizeOr _ _ = _ |
| rw [← budgetEndpointMixture_univ] |
| exact normalizeOr_mass _ _ |
|
|
| end DirectionalTransience |
|
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| end |
|
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| end OAI |
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|
