openAI-math-dataset / lean /OAI /NumberTheory /DukePrimeDegree /Entropy /CountableFutureFiberEntropy.lean
| import OAI.NumberTheory.DukePrimeDegree.Entropy.CountableFutureSigma | |
| import OAI.NumberTheory.DukePrimeDegree.Geometry.CountableRootPrefixAtoms | |
| import OAI.NumberTheory.DukePrimeDegree.Geometry.RootPrefixProductKernel | |
| namespace OAI | |
| /-! # Actual leaf entropy conditioned on a countable future | |
| Finite time-and-label approximations identify the conditional entropy of | |
| any finite present label with the entropy of the actual leaf kernel | |
| restricted to its positive-mass countable future atom. | |
| -/ | |
| noncomputable section | |
| open MeasureTheory Set | |
| namespace DukePrimeDegree | |
| open PrimeDegreePackets | |
| theorem conditionalEntropy_countableFutureFiber | |
| {S V Y : Type*} [MeasurableSpace S] [MeasurableSpace V] | |
| [StandardBorelSpace V] [Nonempty V] [MeasurableSpace Y] | |
| (ρ : Measure (S × V)) [IsFiniteMeasure ρ] | |
| {M : ℕ} {P : Y → Fin M} (hP : Measurable P) | |
| {Q : Y → ℕ} (hQ : Measurable Q) {T : Y → Y} (hT : Measurable T) | |
| {F : S × V → Y} (hF : Measurable F) | |
| (hpos : ∀ᵐ z ∂ρ, ρ.condKernel z.1 (countableFutureFiber T Q F z) ≠ 0) : | |
| conditionalSigmaEntropy ρ (P ∘ F) | |
| (MeasurableSpace.comap Prod.fst (inferInstance : MeasurableSpace S) ⊔ | |
| MeasurableSpace.comap F (⨆ k : ℕ, observationSigma (fun y => Q (T^[k+1] y)))) = | |
| ∫ z, observationEntropy | |
| ((anchorNormalize (ρ.condKernel z.1) (countableFutureFiber T Q F z)).restrict | |
| (countableFutureFiber T Q F z)) (fun t => P (F (z.1,t))) ∂ρ := by | |
| rw [←transverseCountableFutureSigma_eq_sup T Q F] | |
| simpa only [countableFutureFiber_eq_iInter,Function.comp_def] using | |
| conditionalEntropy_infinite_label_restriction ρ (hP.comp hF) | |
| (fun m z => countableFutureCode T Q m (F z)) | |
| (fun m => (countableFutureCode_measurable hT hQ m).comp hF) | |
| (transverse_countableFutureCode_monotone T Q F) | |
| (countableFutureCode_fibers_antitone T Q F) | |
| (by simpa only [countableFutureFiber_eq_iInter] using hpos) | |
| namespace RootPrefixBox | |
| open scoped Topology | |
| variable {d : ℕ} {P₀ : Finset (Fin (d+1) × Fin (d+1))} {D : Finset (Fin (d+1))} | |
| {hP₀ : InitialRootPattern P₀} {c : EliminationChain P₀ D} | |
| {hf : c.finalEntries = allPositiveRoots (d+1)} | |
| {V : Type*} [TopologicalSpace V] [SecondCountableTopology V] | |
| [MeasurableSpace V] [BorelSpace V] | |
| {C : RootPrefixCoordinates (d+1) P₀ V} {g : G (d+1)} | |
| theorem conditionalEntropy_countableFuture_intrinsicProduct | |
| (B : RootPrefixBox C c hP₀ hf g) (R : ℝ) (hBR : B.time ⊆ c.remainingWindow R 0) | |
| (μ : Measure (X (d+1))) [IsProbabilityMeasure μ] | |
| [ErgodicSMul (Multiplicative (LogCoordinates d)) (X (d+1)) μ] | |
| {M : ℕ} {P : X (d+1) → Fin M} (hP : Measurable P) | |
| {Q : X (d+1) → ℕ} (hQ : Measurable Q) | |
| {T : X (d+1) → X (d+1)} (hT : Measurable T) | |
| (hgood : ∀ᵐ x ∂μ, ∀ᶠ v in 𝓝 (completeRootChain (d+1)).zeroTime, ∀ k : ℕ, | |
| Q (T^[k] (rightAction x ((completeRootChain (d+1)).word v)))=Q (T^[k] x)) : | |
| let F := C.timeParam c hP₀ hf g | |
| conditionalSigmaEntropy (B.coordinateMeasure μ) (P ∘ F) | |
| (MeasurableSpace.comap Prod.fst (inferInstance : MeasurableSpace (UpperRootTransverse (d+1) × V)) ⊔ | |
| MeasurableSpace.comap F (⨆ k : ℕ, observationSigma (fun y => Q (T^[k+1] y)))) = | |
| ∫ z, observationEntropy | |
| ((anchorNormalize (B.intrinsicProductConditional μ z) (countableFutureFiber T Q F z)).restrict | |
| (countableFutureFiber T Q F z)) (fun t => P (F (z.1,t))) ∂B.coordinateMeasure μ := by | |
| let F := C.timeParam c hP₀ hf g | |
| have hpos : ∀ᵐ z ∂B.coordinateMeasure μ, | |
| (B.coordinateMeasure μ).condKernel z.1 (countableFutureFiber T Q F z) ≠ 0 := by | |
| filter_upwards [B.ae_wholeFutureAtom_pos μ T Q hgood] with z hz | |
| apply ne_of_gt | |
| apply hz.trans_le | |
| apply measure_mono | |
| intro t ht k | |
| exact ht (k+1) | |
| have he := conditionalEntropy_countableFutureFiber (B.coordinateMeasure μ) hP hQ hT | |
| (C.timeParam_localHomeomorph c hP₀ hf g).continuous.measurable hpos | |
| refine he.trans (integral_congr_ae ?_) | |
| filter_upwards [B.ae_conditionalKernel_eq_intrinsicProduct R hBR μ] with z hz | |
| change (B.coordinateMeasure μ).condKernel z.1 = B.intrinsicProductConditional μ z at hz | |
| rw [hz] | |
| end RootPrefixBox | |
| end DukePrimeDegree | |
| end | |
| end OAI | |