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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 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 | import OAI.Analysis.CircleDomains.Rigidity.CodeRealFreeTrace
import OAI.Analysis.CircleDomains.Selection.RealDomainIntersections
import OAI.Analysis.CircleDomains.Selection.FiniteFlagIntersections
import OAI.Analysis.CircleDomains.Probability.BandMeasure
namespace OAI
/-! Generic prior-level intersections for real coordinates under a base-code
law, with physical, flagged, and free contributions. -/
open Set Function Filter MeasureTheory
open scoped Topology ENNReal NNReal
noncomputable section
namespace CircleDomainRigidity
attribute [local instance] sphereMetric sphereBorel
local instance : BorelSpace Sphere := ⟨rfl⟩
variable [MeasurableSpace C(PathTime, Sphere)] [BorelSpace C(PathTime, Sphere)]
theorem ae_code_real_level_intersections_finite
(hcoarea : PlanarLipschitzFiberCoarea)
(hsard : SmoothPlanarSard) (hsch : PlanarSchoenflies)
(d : PathCodeData Sphere Sphere)
(hG : IsOpen d.physicalDomain) (hU : IsOpen d.quotientDomain)
(hdense : Dense d.quotientDomain) (hdust : d.dust = d.quotientDomainᶜ)
(π : Sphere → Sphere) (hπ : Continuous π) (hclosed : IsClosedMap π) (hsurj : Surjective π)
(hr : ∀ x y, π x = π y ↔ x = y ∨
(x ∈ d.physicalDomainᶜ ∧ y ∈ connectedComponentIn d.physicalDomainᶜ x))
(he : ∀ y : d.physicalDomain, (d.domainEquiv y).val = π y.val)
(hTD : IsTotallyDisconnected d.quotientDomainᶜ)
(hbad : ∀ p ∈ d.physicalDomainᶜ, π p ∈ d.quotientDomainᶜ)
(Flags : Set Sphere) (hFlags : Flags.Countable) (hFlagsB : Flags ⊆ d.quotientDomainᶜ)
(f : Sphere → ℝ) {M : ℝ} (hM : ∀ z ∈ d.quotientDomain, |f z| ≤ M)
(hf : ContinuousOn f d.quotientDomain)
(hv : ContDiffOn ℝ 1 (fun z : ℂ => f (π (z : Sphere)))
(((↑) : ℂ → Sphere) ⁻¹' d.physicalDomain))
(henergy : (∫⁻ z in ((↑) : ℂ → Sphere) ⁻¹' d.physicalDomain,
ENNReal.ofReal ‖fderiv ℝ (fun z : ℂ => f (π (z : Sphere))) z‖ ^ 2) < ∞)
{K : Set ℂ} (hK : IsCompact K)
(hzero : ∀ z : ℂ, (z : Sphere) ∈ d.physicalDomain → z ∉ K → f (π (z : Sphere)) = 0)
(hinfty : OnePoint.infty ∈ d.physicalDomain → f (π OnePoint.infty) = 0)
(P : ProbabilityMeasure (BasePathCode d))
(ρ : Sphere → ℝ≥0∞) (hρ : Measurable ρ)
(hocc : meanCodeOccupation d P = sphericalArea.withDensity ρ)
(hρ2 : (∫⁻ y, ρ y ^ 2 ∂sphericalArea) < ∞)
(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) < ∞)
(hband : ∀ z ∈ d.quotientDomainᶜ, ENNReal.ofReal
(limsup f (𝓝[d.quotientDomain] z) - liminf f (𝓝[d.quotientDomain] z)) ≤
d.levelWeights i z)
(hmFree : ∀ z ∈ d.quotientDomainᶜ \ Flags, d.levelWeights i z = 0)
(hcover : ∀ x ∈ support (d.levelWeights i), ∀ᶠ n in atTop, x ∈ d.flagDeletions n)
(hdeletions : ∀ n, (d.flagDeletions n : Set Sphere) ⊆ Flags)
(hH : ∀ j, IsClosed (d.levels j))
(hP : OrderedLevelNullCompatibility d.levels (d.quotientDomainᶜ \ Flags))
(hunique : ∀ x ∈ d.quotientDomainᶜ \ Flags, ∀ j a b,
(a, x) ∈ d.levels j → (b, x) ∈ d.levels j → a = b)
(ω : TupleType → ℝ≥0∞) [SFinite (tupleReference ω)]
(hτ : meanCodeTuple d P ≪ tupleReference ω)
(hlevel : ∀ θ x, (θ, x) ∈ d.levels i ↔ (x, θ) ∈ circleBandGraph d.quotientDomain f) :
∀ᵐ Γ ∂(P : Measure (BasePathCode d)), ∀ᵐ θ : UnitAddCircle ∂volume,
(range Γ.val.1 ∩ {x | (θ, x) ∈ d.levels i}).Finite := by
have hfree := ae_code_real_free_level_incidence_null hsard hsch d hG hU hdense hdust
π hπ hclosed hsurj hr he hTD hbad Flags hFlags hFlagsB f hM hf hv henergy P ρ hρ hocc hρ2
i hm hweight hvisits hband hmFree hcover hdeletions hH hP hunique ω hτ hlevel
have hfinite : ∀ᵐ γ ∂(codePathLaw d P : Measure C(PathTime, Sphere)),
physicalDomainLength d.domainEquiv d.basePoint γ < ∞ := by
change ∀ᵐ γ ∂(P : Measure (BasePathCode d)).map (codePath d), _
apply (ae_map_iff (measurable_codePath d).aemeasurable
(measurableSet_finitePhysicalDomainLength d.domainEquiv hU d.basePoint)).mpr
exact Eventually.of_forall fun Γ => Γ.property.1.1
have hdom := ae_real_domain_circle_count_finite hcoarea hG d.domainEquiv hU d.basePoint π hπ he
f hf hv henergy hK hzero hinfty (codePathLaw d P) hfinite ρ hρ
((meanPhysicalOccupation_codePathLaw d hU P).trans hocc) hρ2
have hvol : ∀ x ∈ Flags, volume {θ : UnitAddCircle | (θ, x) ∈ d.levels i} ≤ d.levelWeights i x := by
intro x hx
have heq : {θ : UnitAddCircle | (θ, x) ∈ d.levels i} =
{θ : UnitAddCircle | (x, θ) ∈ circleBandGraph d.quotientDomain f} :=
Set.ext fun θ => hlevel θ x
rw [heq]
exact (volume_circleBandGraph_incidence_le_budget hM
(by rw [hdense.closure_eq]; exact mem_univ x) le_rfl).trans (hband x (hFlagsB hx))
have hflag := ae_flag_intersection_count_finite_of_support (hH i) Flags hFlags
(d.levelWeights i) hvol (codePathLaw d P) hm hweight hvisits
have hdomP := ae_of_ae_map (measurable_codePath d).aemeasurable hdom
have hflagP := ae_of_ae_map (measurable_codePath d).aemeasurable hflag
filter_upwards [hfree, hdomP, hflagP] with Γ hΓfree hΓdom hΓflag
have hfreet : ∀ᵐ θ : UnitAddCircle ∂volume,
¬ ∃ x ∈ range Γ.val.1 ∩ (d.quotientDomainᶜ \ Flags), (θ, x) ∈ d.levels i :=
ae_iff.mpr (by simpa only [not_not] using hΓfree)
filter_upwards [hfreet, hΓdom, hΓflag] with θ hθfree hθdom hθflag
apply finite_level_of_no_free_hits d.quotientDomain Flags {x | (θ, x) ∈ d.levels i}
Γ.val.1 hθfree
· simpa only [hlevel, codePath] using hθdom
· exact hθflag
end CircleDomainRigidity
end
end OAI
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