| import OAI.Analysis.CircleDomains.Probability.CodeLawCandidates |
| import OAI.Analysis.CircleDomains.Probability.DensityMassBound |
| import OAI.Analysis.CircleDomains.Probability.CountableVisitNorm |
|
|
| namespace OAI |
|
|
| /-! The quantitative outputs of countable modulus extraction are members of |
| one fixed candidate family, independent of the prescribed countable set. -/ |
|
|
| open Set MeasureTheory |
| open scoped ENNReal |
|
|
| noncomputable section |
| namespace CircleDomainRigidity |
|
|
| variable {X Y S : Type*} [MetricSpace X] [MetricSpace Y] |
| [MeasurableSpace X] [BorelSpace X] [SecondCountableTopology X] |
| [MeasurableSpace Y] [BorelSpace Y] [SecondCountableTopology Y] |
| [MeasurableSpace C(PathTime, X)] [BorelSpace C(PathTime, X)] |
| [MeasurableSpace C(PathTime, Y)] [BorelSpace C(PathTime, Y)] |
|
|
| omit [MeasurableSpace Y] [BorelSpace Y] [SecondCountableTopology Y] |
| [MeasurableSpace C(PathTime, Y)] [BorelSpace C(PathTime, Y)] in |
| theorem meanVisitCount_codePathLaw_eq_integral (d : PathCodeData X Y) |
| (P : ProbabilityMeasure (BasePathCode d)) (a : S → X) (s : S) : |
| meanVisitCount (codePathLaw d P) (a s) = |
| ∫⁻ Γ, codeVisitWeight d a s Γ ∂(P : Measure (BasePathCode d)) := by |
| have hv : Measurable (fun γ : C(PathTime, X) => visitCount γ (a s)) := |
| measurable_visitCount.comp (measurable_id.prodMk measurable_const) |
| unfold meanVisitCount codePathLaw |
| rw [ProbabilityMeasure.toMeasure_map, lintegral_map hv (measurable_codePath d)] |
| rfl |
|
|
| omit [MeasurableSpace Y] [BorelSpace Y] [SecondCountableTopology Y] |
| [MeasurableSpace C(PathTime, Y)] [BorelSpace C(PathTime, Y)] in |
| theorem countableSquareLawBounds_of_visit_norm (d : PathCodeData X Y) |
| (P : ProbabilityMeasure (BasePathCode d)) (a : S → X) {C : ℝ≥0∞} |
| (h : countingSquareNorm (fun s => meanVisitCount (codePathLaw d P) (a s)) ≤ C) : |
| CountableSquareLawBounds (codeVisitWeight d a) (C ^ 2) P := by |
| intro F |
| simp_rw [← meanVisitCount_codePathLaw_eq_integral] |
| have he : countingSquareNorm (fun s => meanVisitCount (codePathLaw d P) (a s)) ^ 2 = |
| ∑' s, meanVisitCount (codePathLaw d P) (a s) ^ 2 := by |
| rw [countingSquareNorm, ← ENNReal.rpow_two, ← ENNReal.rpow_mul] |
| norm_num |
| exact (ENNReal.sum_le_tsum F).trans (he ▸ pow_le_pow_left' h 2) |
|
|
| omit [SecondCountableTopology Y] [MeasurableSpace C(PathTime, Y)] |
| [BorelSpace C(PathTime, Y)] in |
| theorem codeLawCandidate_of_extracted_bounds (d : PathCodeData X Y) |
| (a : S → X) (μ : Measure Y) (τ : Measure LevelTuple) |
| (P : ProbabilityMeasure (BasePathCode d)) |
| {g : Y → ℝ≥0∞} {q : LevelTuple → ℝ≥0∞} (hg : Measurable g) (hq : Measurable q) |
| (hocc : meanCodeOccupation d P = μ.withDensity g) |
| (htuple : meanCodeTuple d P = τ.withDensity q) |
| {E C : ℝ≥0∞} (hgE : (∫⁻ y, g y ^ 2 ∂μ) ≤ E) |
| (hqE : (∫⁻ t, q t ^ 2 ∂τ) ≤ E) |
| (hv : countingSquareNorm (fun s => meanVisitCount (codePathLaw d P) (a s)) ≤ C) : |
| CodeLawCandidate d a μ τ (densityMassBound μ E) (densityMassBound τ E) E (C ^ 2) P := |
| ⟨boundedDensityBudget_of_square_bound μ _ hg hocc hgE, |
| countableSquareLawBounds_of_visit_norm d P a hv, |
| boundedDensityBudget_of_square_bound τ _ hq htuple hqE⟩ |
|
|
| end CircleDomainRigidity |
|
|
| end |
|
|
| end OAI |
|
|
