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9655094 | 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 | import OAI.Analysis.CircleDomains.Rigidity.CodeFreeTrace
namespace OAI
/-! Combining the actual residual constraints with null incidence sets
on the open tags needs no globally chosen circle trace. -/
open Set Function Filter MeasureTheory
open scoped Topology ENNReal
namespace CircleDomainRigidity
variable {X Y : Type*} [MetricSpace X] [MetricSpace Y]
[MeasurableSpace X] [BorelSpace X] [SecondCountableTopology X]
[MeasurableSpace C(PathTime, X)] [BorelSpace C(PathTime, X)]
theorem ae_code_free_level_incidence_null_of_tags
(d : PathCodeData X Y) (P : ProbabilityMeasure (BasePathCode d)) (i : ℕ)
(Free : Set X) (hFree : Free ⊆ d.dust)
(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)
(hdelete : ∀ n, Disjoint Free (d.flagDeletions n : Set X))
(htags : ∀ᵐ Γ ∂(P : Measure (BasePathCode d)), ∀ k,
volume {θ | ∃ t ∈ Ioo (tagLeft (Γ.val.2 k)) (tagRight (Γ.val.2 k)),
Γ.val.1 t ∈ Free ∧ (θ, Γ.val.1 t) ∈ d.levels i} = 0) :
∀ᵐ Γ ∂(P : Measure (BasePathCode d)),
volume {θ | ∃ x ∈ range Γ.val.1 ∩ Free, (θ, x) ∈ d.levels i} = 0 := by
filter_upwards [ae_code_level_envelope_null d P i hm hweight hvisits hcover, htags]
with Γ hres htag
apply measure_mono_null (t :=
(⋂ n, {θ | (Γ.val, θ) ∈ residualIncidence d.dust univ (d.levels i) (d.flagDeletions n)}) ∪
⋃ k, {θ | ∃ t ∈ Ioo (tagLeft (Γ.val.2 k)) (tagRight (Γ.val.2 k)),
Γ.val.1 t ∈ Free ∧ (θ, Γ.val.1 t) ∈ d.levels i}) _
(measure_union_null hres (measure_iUnion_null htag))
rintro θ ⟨x, ⟨⟨t, rfl⟩, hx⟩, hθ⟩
by_cases ht : ∃ k, t ∈ Ioo (tagLeft (Γ.val.2 k)) (tagRight (Γ.val.2 k))
· obtain ⟨k, hk⟩ := ht
exact Or.inr (mem_iUnion.mpr ⟨k, t, hk, hx, hθ⟩)
· apply Or.inl
apply mem_iInter.mpr
intro n
refine ⟨t, ?_, mem_univ _, hθ, fun hp => disjoint_left.mp (hdelete n) hx hp⟩
rw [residualTimes_eq]
exact ⟨hFree hx, fun h => ht (mem_iUnion.mp h)⟩
end CircleDomainRigidity
end OAI
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