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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 | import OAI.Analysis.CircleDomains.Rigidity.CodeRealFreeTrace
import OAI.Analysis.CircleDomains.Selection.RealDomainIntersections
import OAI.Analysis.CircleDomains.Selection.FiniteFlagIntersections
import OAI.Analysis.CircleDomains.Probability.BandMeasure
import OAI.Analysis.CircleDomains.Selection.MeanLevelAssembly
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 mean_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) :
(∫⁻ θ : UnitAddCircle, ∫⁻ Γ : BasePathCode d,
pathImageCount {x | (θ, x) ∈ d.levels i} Γ.val.1 ∂(P : Measure _)) < ∞ := 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 := mean_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 := mean_flag_intersection_count_finite_of_support (hH i) Flags hFlags
(d.levelWeights i) hvol (codePathLaw d P) hm hweight hvisits
have hσU : IsSigmaCompact d.quotientDomain := by
let : LocallyCompactSpace d.quotientDomain := hU.locallyCompactSpace
exact isSigmaCompact_iff_sigmaCompactSpace.mpr inferInstance
have hfree' : ∀ᵐ Γ ∂(P : Measure (BasePathCode d)), ∀ᵐ θ : UnitAddCircle ∂volume,
¬ ∃ x ∈ range (codePath d Γ) ∩ (d.quotientDomainᶜ \ Flags), (θ, x) ∈ d.levels i := by
filter_upwards [hfree] with Γ hΓ
exact ae_iff.mpr (by simpa only [not_not, codePath] using hΓ)
have hb := mean_parameter_level_count_le_of_no_free_hits (hH i)
d.quotientDomain Flags hσU hFlags P (codePath d) (measurable_codePath d) hfree'
have hDmeas := measurable_sigmaCompact_restricted_relation_path_count hσU
((hH i).preimage continuous_swap)
have hFmeas := measurable_flag_intersection_count (hH i) Flags hFlags
have hdint (θ : UnitAddCircle) :
(∫⁻ Γ : BasePathCode d, pathImageCount (d.quotientDomain ∩ {x | (θ, x) ∈ d.levels i})
(codePath d Γ) ∂(P : Measure _)) =
∫⁻ γ, pathImageCount (d.quotientDomain ∩ {x | (θ, x) ∈ d.levels i}) γ
∂(codePathLaw d P : Measure _) := by
have hmeas : Measurable (fun γ : C(PathTime, Sphere) =>
pathImageCount (d.quotientDomain ∩ {x | (θ, x) ∈ d.levels i}) γ) :=
hDmeas.comp (measurable_id.prodMk measurable_const)
rw [codePathLaw, ProbabilityMeasure.toMeasure_map]
exact (lintegral_map hmeas (measurable_codePath d)).symm
have hfint (θ : UnitAddCircle) :
(∫⁻ Γ : BasePathCode d, pathImageCount (Flags ∩ {x | (θ, x) ∈ d.levels i})
(codePath d Γ) ∂(P : Measure _)) =
∫⁻ γ, pathImageCount (Flags ∩ {x | (θ, x) ∈ d.levels i}) γ
∂(codePathLaw d P : Measure _) := by
have hmeas : Measurable (fun γ : C(PathTime, Sphere) =>
pathImageCount (Flags ∩ {x | (θ, x) ∈ d.levels i}) γ) :=
hFmeas.comp (measurable_id.prodMk measurable_const)
rw [codePathLaw, ProbabilityMeasure.toMeasure_map]
exact (lintegral_map hmeas (measurable_codePath d)).symm
apply hb.trans_lt
apply ENNReal.add_lt_top.mpr
constructor
· simp_rw [hdint]
simpa only [hlevel] using hdom
· simpa only [hfint] using hflag
end CircleDomainRigidity
end
end OAI
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