openAI-math-dataset / lean /OAI /Analysis /CircleDomains /Probability /CodeLawCandidateConsequences.lean
| import OAI.Analysis.CircleDomains.Probability.CodeLawCandidateBounds | |
| import OAI.Analysis.CircleDomains.Probability.SlotLawMeasures | |
| namespace OAI | |
| /-! Unpack the fused candidate's quantitative estimates and slot laws. -/ | |
| open Set MeasureTheory Filter | |
| 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] | |
| [MeasurableSpace C(PathTime, X)] [BorelSpace C(PathTime, X)] | |
| omit [MeasurableSpace Y] [BorelSpace Y] in | |
| theorem countableSquareLawBounds_visit_norm (d : PathCodeData X Y) | |
| (P : ProbabilityMeasure (BasePathCode d)) (a : S → X) {C : ℝ≥0∞} | |
| (h : CountableSquareLawBounds (codeVisitWeight d a) (C ^ 2) P) : | |
| countingSquareNorm (fun s => meanVisitCount (codePathLaw d P) (a s)) ≤ C := by | |
| have hs : (∑' s, meanVisitCount (codePathLaw d P) (a s) ^ 2) ≤ C ^ 2 := by | |
| rw [ENNReal.tsum_eq_iSup_sum] | |
| apply iSup_le | |
| intro F | |
| simpa only [meanVisitCount_codePathLaw_eq_integral] using h F | |
| calc | |
| _ ≤ (C ^ 2) ^ (1 / 2 : ℝ) := ENNReal.rpow_le_rpow hs (by norm_num) | |
| _ = C := by rw [← ENNReal.rpow_two, ← ENNReal.rpow_mul]; norm_num | |
| omit [BorelSpace X] [SecondCountableTopology X] [BorelSpace C(PathTime, X)] in | |
| theorem CodeLawCandidate.slot_absolutelyContinuous | |
| (d : PathCodeData X Y) (a : S → X) (μ : Measure Y) (τ : Measure LevelTuple) | |
| (M N E V : ℝ≥0∞) (P : ProbabilityMeasure (BasePathCode d)) | |
| (h : CodeLawCandidate d a μ τ M N E V P) (k : ℕ) : slotLaw d P k ≪ τ := by | |
| obtain ⟨q, _, hq, _⟩ := h.2.2.2 | |
| exact slotLaw_absolutelyContinuous d P τ (hq ▸ withDensity_absolutelyContinuous τ q) k | |
| omit [MeasurableSpace Y] [BorelSpace Y] in | |
| theorem ae_finite_prescribed_visits [Countable S] | |
| (d : PathCodeData X Y) (P : ProbabilityMeasure (BasePathCode d)) (a : S → X) | |
| {V : ℝ≥0∞} (hV : V < ∞) | |
| (h : CountableSquareLawBounds (codeVisitWeight d a) V P) : | |
| ∀ᵐ Γ ∂(P : Measure (BasePathCode d)), ∀ s, visitCount (codePath d Γ) (a s) < ∞ := by | |
| classical | |
| apply ae_all_iff.mpr | |
| intro s | |
| have hb : (∫⁻ Γ, codeVisitWeight d a s Γ ∂(P : Measure (BasePathCode d))) ^ 2 < ∞ := by | |
| simpa using (h {s}).trans_lt hV | |
| apply ae_lt_top (measurable_codeVisitWeight d a s) | |
| intro hh | |
| rw [hh] at hb | |
| simp at hb | |
| end CircleDomainRigidity | |
| end | |
| end OAI | |