| import OAI.Analysis.CircleDomains.Probability.CodeLawMixtures | |
| import OAI.Analysis.CircleDomains.Probability.DensityBudgetMixture | |
| import OAI.Analysis.CircleDomains.Probability.CountableSquareLawBounds | |
| import OAI.Analysis.CircleDomains.Selection.VisitCounts | |
| namespace OAI | |
| /-! The quantitative family of code laws used in dust fusion is Borel and | |
| closed under arbitrary Borel probability mixtures. -/ | |
| open Set Filter MeasureTheory | |
| open scoped Topology ENNReal | |
| noncomputable section | |
| namespace CircleDomainRigidity | |
| variable {X Y S Z : 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)] | |
| def codeVisitWeight (d : PathCodeData X Y) (a : S → X) (s : S) (Γ : BasePathCode d) : ℝ≥0∞ := | |
| visitCount (codePath d Γ) (a s) | |
| omit [MeasurableSpace Y] [BorelSpace Y] [SecondCountableTopology Y] | |
| [MeasurableSpace C(PathTime, Y)] [BorelSpace C(PathTime, Y)] in | |
| theorem measurable_codeVisitWeight (d : PathCodeData X Y) (a : S → X) (s : S) : | |
| Measurable (codeVisitWeight d a s) := | |
| measurable_visitCount.comp ((measurable_codePath d).prodMk measurable_const) | |
| def CodeLawCandidate (d : PathCodeData X Y) (a : S → X) (μ : Measure Y) | |
| (τ : Measure LevelTuple) (massOcc massTuple energy visitBudget : ℝ≥0∞) | |
| (P : ProbabilityMeasure (BasePathCode d)) : Prop := | |
| BoundedDensityBudget μ (meanCodeOccupation d P) massOcc energy ∧ | |
| CountableSquareLawBounds (codeVisitWeight d a) visitBudget P ∧ | |
| BoundedDensityBudget τ (meanCodeTuple d P) massTuple energy | |
| theorem measurableSet_codeLawCandidate [Countable S] | |
| (d : PathCodeData X Y) (hU : IsOpen d.quotientDomain) (a : S → X) | |
| (μ : Measure Y) [IsFiniteMeasure μ] (τ : Measure LevelTuple) [IsFiniteMeasure τ] | |
| (massOcc massTuple energy visitBudget : ℝ≥0∞) | |
| (hOcc : massOcc < ∞) (hTuple : massTuple < ∞) : | |
| MeasurableSet {P | CodeLawCandidate d a μ τ massOcc massTuple energy visitBudget P} := by | |
| exact (measurableSet_boundedDensityBudget (meanCodeOccupation d) | |
| (measurable_meanCodeOccupation d hU) μ massOcc energy hOcc).inter | |
| ((measurableSet_countableSquareLawBounds (codeVisitWeight d a) | |
| (measurable_codeVisitWeight d a) visitBudget).inter | |
| (measurableSet_boundedDensityBudget (meanCodeTuple d) (measurable_meanCodeTuple d) | |
| τ massTuple energy hTuple)) | |
| theorem CodeLawCandidate.mixture [MeasurableSpace Z] | |
| (d : PathCodeData X Y) (hU : IsOpen d.quotientDomain) (a : S → X) | |
| (μ : Measure Y) [IsFiniteMeasure μ] (τ : Measure LevelTuple) [IsFiniteMeasure τ] | |
| (massOcc massTuple energy visitBudget : ℝ≥0∞) | |
| (hOcc : massOcc < ∞) (hTuple : massTuple < ∞) | |
| (P : ProbabilityMeasure Z) (law : Z → ProbabilityMeasure (BasePathCode d)) | |
| (hlaw : Measurable law) | |
| (h : ∀ z, CodeLawCandidate d a μ τ massOcc massTuple energy visitBudget (law z)) : | |
| CodeLawCandidate d a μ τ massOcc massTuple energy visitBudget | |
| (probabilityMixture P law hlaw) := by | |
| refine ⟨?_, ?_, ?_⟩ | |
| · rw [meanCodeOccupation_probabilityMixture d hU P law hlaw] | |
| exact boundedDensityBudget_bind (fun z => meanCodeOccupation d (law z)) | |
| ((measurable_meanCodeOccupation d hU).comp hlaw) μ massOcc energy hOcc | |
| (fun z => (h z).1) (P : Measure Z) | |
| · exact CountableSquareLawBounds.mixture (codeVisitWeight d a) | |
| (measurable_codeVisitWeight d a) visitBudget P law hlaw | |
| (Eventually.of_forall fun z => (h z).2.1) | |
| · rw [meanCodeTuple_probabilityMixture d P law hlaw] | |
| exact boundedDensityBudget_bind (fun z => meanCodeTuple d (law z)) | |
| ((measurable_meanCodeTuple d).comp hlaw) τ massTuple energy hTuple | |
| (fun z => (h z).2.2) (P : Measure Z) | |
| end CircleDomainRigidity | |
| end | |
| end OAI | |