openAI-math-dataset / lean /OAI /NumberTheory /DukePrimeDegree /Entropy /ConditionalInformationLowerBound.lean
| import OAI.NumberTheory.DukePrimeDegree.Entropy.ConditionalInformationClipping | |
| import OAI.NumberTheory.PrimePackets.FutureConditionalEntropy | |
| namespace OAI | |
| /-! # Whole-future conditional information is bounded by metric entropy | |
| Finite present clips have their own, smaller future sigma algebra. | |
| Conditioning on the full countable future and further transverse data | |
| only decreases their entropy. Passing to the proved clipping limit gives | |
| the bound without a finite Shannon entropy assumption on the countable | |
| observation. | |
| -/ | |
| noncomputable section | |
| open MeasureTheory Set Filter | |
| open scoped Topology ENNReal | |
| namespace DukePrimeDegree | |
| open PrimeDegreePackets | |
| variable {Y : Type*} | |
| lemma clipped_future_sigma_le (T : Y → Y) (Q : Y → ℕ) | |
| (m : MeasurableSpace Y) | |
| (hfuture : (⨆ k : ℕ, observationSigma (fun y => Q (T^[k+1] y))) ≤ m) | |
| (N : ℕ) : (⨆ k, observationSigma (futureName T (heightClip Q N) k)) ≤ m := by | |
| let : MeasurableSpace Y := m | |
| have hcoord (k : ℕ) : Measurable (fun y => Q (T^[k+1] y)) := | |
| (comap_measurable _).mono ((le_iSup _ k).trans hfuture) le_rfl | |
| apply iSup_le | |
| intro k | |
| apply observationSigma_le | |
| intro w | |
| have hName : Measurable (futureName T (heightClip Q N) k) := by | |
| apply measurable_pi_iff.mpr | |
| intro i | |
| have heq : (fun y => futureName T (heightClip Q N) k y i) = | |
| heightClip (fun y => Q (T^[i.val+1] y)) N := by | |
| funext y | |
| apply Fin.ext | |
| simp only [futureName, orbitName, heightClip, Function.iterate_succ_apply] | |
| rw [heq] | |
| exact heightClip_measurable (hcoord i.val) N | |
| exact hName (measurableSet_singleton w) | |
| theorem conditionalInformation_integral_le_metricEntropy | |
| (m : MeasurableSpace Y) [mY : MeasurableSpace Y] | |
| (μ : Measure Y) [IsProbabilityMeasure μ] | |
| {T : Y → Y} (hT : MeasurePreserving T μ μ) | |
| (Q : Y → ℕ) (hQ : Measurable Q) (hm : m ≤ mY) | |
| (hfuture : (⨆ k : ℕ, observationSigma (fun y => Q (T^[k+1] y))) ≤ m) | |
| (hpos : ∀ᵐ y ∂μ, | |
| 0 < (μ[({z | Q z = Q y}.indicator (fun _ => (1 : ℝ))) | m]) y) | |
| (hint : Integrable (conditionalInformation μ Q m) μ) : | |
| ENNReal.ofReal (∫ y, conditionalInformation μ Q m y ∂μ) ≤ measureEntropy μ T := by | |
| have hclip := conditionalSigmaEntropy_heightClip_tendsto m μ Q hQ hm hpos hint | |
| apply le_of_tendsto (ENNReal.continuous_ofReal.continuousAt.tendsto.comp hclip) | |
| apply Filter.Eventually.of_forall | |
| intro N | |
| have hP := heightClip_measurable hQ N | |
| have hle : conditionalSigmaEntropy μ (heightClip Q N) m ≤ | |
| observationEntropyRate μ T (heightClip Q N) := by | |
| rw [observationEntropyRate_eq_futureConditional μ hT _ | |
| (fun a => hP (measurableSet_singleton a))] | |
| exact conditionalSigmaEntropy_antitone μ _ | |
| (fun a => hP (measurableSet_singleton a)) | |
| (clipped_future_sigma_le T Q m hfuture N) hm | |
| apply (ENNReal.ofReal_le_ofReal hle).trans | |
| unfold measureEntropy | |
| exact le_iSup_of_le (N+1) (le_iSup_of_le (heightClip Q N) | |
| (le_iSup_of_le (fun a => hP (measurableSet_singleton a)) le_rfl)) | |
| end DukePrimeDegree | |
| end | |
| end OAI | |