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5711324 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 | 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
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