openAI-math-dataset / lean /OAI /Analysis /CircleDomains /Probability /CodeLawPhysicalOccupation.lean
| import OAI.Analysis.CircleDomains.Topology.SphereLevels | |
| import OAI.Analysis.CircleDomains.Probability.PhysicalOccupationCode | |
| import OAI.Analysis.CircleDomains.Probability.CodePathLaws | |
| import OAI.Analysis.CircleDomains.Probability.CodeLawMeasures | |
| namespace OAI | |
| /-! Forgetting tags preserves the actual mean physical occupation. -/ | |
| open Set Function Filter MeasureTheory | |
| open scoped ENNReal Topology | |
| noncomputable section | |
| namespace CircleDomainRigidity | |
| attribute [local instance] sphereMetric sphereBorel | |
| local instance : BorelSpace Sphere := ⟨rfl⟩ | |
| variable [MeasurableSpace C(PathTime, Sphere)] [BorelSpace C(PathTime, Sphere)] | |
| theorem meanPhysicalOccupation_codePathLaw (d : PathCodeData Sphere Sphere) | |
| (hU : IsOpen d.quotientDomain) (P : ProbabilityMeasure (BasePathCode d)) : | |
| meanPhysicalOccupation d.domainEquiv d.basePoint (codePathLaw d P) = | |
| meanCodeOccupation d P := by | |
| apply Measure.ext | |
| intro S hS | |
| have hm : Measurable (fun γ => physicalOccupationCode d.domainEquiv d.basePoint γ S) := | |
| (Measure.measurable_coe hS).comp (measurable_physicalOccupationCode d.domainEquiv hU d.basePoint) | |
| rw [meanPhysicalOccupation_apply _ hU _ _ hS, meanCodeOccupation_apply d hU P hS, | |
| codePathLaw, ProbabilityMeasure.toMeasure_map, lintegral_map hm (measurable_codePath d)] | |
| apply lintegral_congr | |
| intro Γ | |
| exact congrArg (fun ν : Measure Sphere => ν S) | |
| (physicalOccupationCode_eq d.domainEquiv d.basePoint (baseCodePhysicalPath d Γ)) | |
| theorem codePathLaw_collar_endpoints (d : PathCodeData Sphere Sphere) | |
| (hU : IsOpen d.quotientDomain) (P : ProbabilityMeasure (BasePathCode d)) | |
| (V : Set Sphere) (hV : IsOpen V) (hanchor : Disjoint d.anchor V) (hb : d.target ∈ V) : | |
| ∀ᵐ γ : C(PathTime, Sphere) ∂(codePathLaw d P : Measure C(PathTime, Sphere)), | |
| γ ⊥ ∉ V ∧ γ ⊤ ∈ V ∧ γ ⊤ = d.target ∧ 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 ?_).mpr | |
| · exact Eventually.of_forall fun Γ => | |
| ⟨fun hv => disjoint_left.mp hanchor Γ.property.2.1.1 hv, | |
| by change Γ.val.1 ⊤ ∈ V; rw [Γ.property.2.1.2.1]; exact hb, | |
| Γ.property.2.1.2.1, Γ.property.1.1⟩ | |
| · exact (((continuous_eval_const ⊥).measurable hV.measurableSet).compl).inter | |
| (((continuous_eval_const ⊤).measurable hV.measurableSet).inter | |
| ((measurableSet_eq_fun (continuous_eval_const ⊤).measurable measurable_const).inter | |
| (measurableSet_finitePhysicalDomainLength d.domainEquiv hU d.basePoint))) | |
| end CircleDomainRigidity | |
| end | |
| end OAI | |