| import OAI.Analysis.CircleDomains.Probability.ResidualTailNull | |
| import OAI.Analysis.CircleDomains.Selection.SquareVisitTolls | |
| namespace OAI | |
| /-! The stored rational-interval constraints give null residual-incidence | |
| envelopes under the actual square bounds on visits. -/ | |
| open Set Filter Function MeasureTheory | |
| open scoped ENNReal Topology Classical | |
| noncomputable section | |
| namespace CircleDomainRigidity | |
| variable {X Y : Type*} [MetricSpace X] [MetricSpace Y] | |
| [MeasurableSpace X] [MeasurableSpace C(PathTime, X)] | |
| private def rationalStart : RationalPathTime := ⟨0, by norm_num⟩ | |
| private def rationalEnd : RationalPathTime := ⟨1, by norm_num⟩ | |
| private theorem rationalStart_eq : rationalStart.toTime = (⊥ : PathTime) := by | |
| apply Subtype.ext | |
| norm_num [rationalStart, RationalPathTime.toTime] | |
| private theorem rationalEnd_eq : rationalEnd.toTime = (⊤ : PathTime) := by | |
| apply Subtype.ext | |
| norm_num [rationalEnd, RationalPathTime.toTime] | |
| theorem baseCode_prior_bound (d : PathCodeData X Y) (Γ : BasePathCode d) (i n : ℕ) : | |
| ResidualIncidenceBound d.dust univ pathHitSet (d.priorDeletions i n) | |
| (d.priorLaws i) (priorHitMass (d.priorLaws i)) Γ.val := by | |
| have h := Γ.property.1.2.2.1 i n rationalStart rationalEnd | |
| simpa only [rationalStart_eq, rationalEnd_eq, Icc_bot_top] using h | |
| theorem baseCode_level_bound (d : PathCodeData X Y) (Γ : BasePathCode d) (i n : ℕ) : | |
| ResidualIncidenceBound d.dust univ (d.levels i) (d.flagDeletions n) | |
| volume (d.levelWeights i) Γ.val := by | |
| have h := Γ.property.1.2.2.2.1 i n rationalStart rationalEnd | |
| simpa only [rationalStart_eq, rationalEnd_eq, Icc_bot_top] using h | |
| variable [BorelSpace X] [SecondCountableTopology X] [BorelSpace C(PathTime, X)] | |
| theorem ae_code_boundaryToll_finite (d : PathCodeData X Y) | |
| (P : ProbabilityMeasure (BasePathCode d)) (m : X → ℝ≥0∞) | |
| (hm : (support m).Countable) | |
| (hweight : countingSquareNorm (fun x : support m => m x.val) < ∞) | |
| (hvisits : countingSquareNorm | |
| (fun x : support m => meanVisitCount (codePathLaw d P) x.val) < ∞) : | |
| ∀ᵐ Γ ∂(P : Measure (BasePathCode d)), boundaryToll Γ.val.1 m < ∞ := by | |
| have h := ae_boundaryToll_finite (codePathLaw d P) m hm hweight hvisits | |
| exact (ae_map_iff (measurable_codePath d).aemeasurable | |
| (measurableSet_lt (measurable_boundaryToll hm) measurable_const)).mp h | |
| theorem ae_code_prior_envelope_null (d : PathCodeData X Y) | |
| (P : ProbabilityMeasure (BasePathCode d)) (i : ℕ) | |
| (hm : (support (priorHitMass (d.priorLaws i))).Countable) | |
| (hweight : countingSquareNorm (fun x : support (priorHitMass (d.priorLaws i)) => | |
| priorHitMass (d.priorLaws i) x.val) < ∞) | |
| (hvisits : countingSquareNorm (fun x : support (priorHitMass (d.priorLaws i)) => | |
| meanVisitCount (codePathLaw d P) x.val) < ∞) | |
| (hcover : ∀ x ∈ support (priorHitMass (d.priorLaws i)), | |
| ∀ᶠ n in atTop, x ∈ d.priorDeletions i n) : | |
| ∀ᵐ Γ ∂(P : Measure (BasePathCode d)), (d.priorLaws i) | |
| (⋂ n, {η | (Γ.val, η) ∈ residualIncidence d.dust univ pathHitSet | |
| (d.priorDeletions i n)}) = 0 := by | |
| filter_upwards [ae_code_boundaryToll_finite d P _ hm hweight hvisits] with Γ hΓ | |
| exact residual_incidence_envelope_null d.dust univ pathHitSet _ _ hm _ hcover | |
| Γ.val hΓ (baseCode_prior_bound d Γ i) | |
| theorem ae_code_level_envelope_null (d : PathCodeData X Y) | |
| (P : ProbabilityMeasure (BasePathCode d)) (i : ℕ) | |
| (hm : (support (d.levelWeights i)).Countable) | |
| (hweight : countingSquareNorm (fun x : support (d.levelWeights i) => | |
| d.levelWeights i x.val) < ∞) | |
| (hvisits : countingSquareNorm (fun x : support (d.levelWeights i) => | |
| meanVisitCount (codePathLaw d P) x.val) < ∞) | |
| (hcover : ∀ x ∈ support (d.levelWeights i), | |
| ∀ᶠ n in atTop, x ∈ d.flagDeletions n) : | |
| ∀ᵐ Γ ∂(P : Measure (BasePathCode d)), volume | |
| (⋂ n, {t | (Γ.val, t) ∈ residualIncidence d.dust univ (d.levels i) | |
| (d.flagDeletions n)}) = 0 := by | |
| filter_upwards [ae_code_boundaryToll_finite d P _ hm hweight hvisits] with Γ hΓ | |
| exact residual_incidence_envelope_null d.dust univ (d.levels i) _ _ hm _ hcover | |
| Γ.val hΓ (baseCode_level_bound d Γ i) | |
| end CircleDomainRigidity | |
| end | |
| end OAI | |