openAI-math-dataset / lean /OAI /NumberTheory /DukePrimeDegree /Entropy /CountableFutureConditioning.lean
| import OAI.NumberTheory.DukePrimeDegree.Entropy.FutureSigmaAlgebra | |
| import OAI.NumberTheory.DukePrimeDegree.Entropy.ConditionalSigmaLimit | |
| namespace OAI | |
| /-! # Finite approximation of a countable future observation | |
| The countable geometric observations can be used for conditioning without | |
| assuming finite Shannon entropy. Their conditioning sigma algebra is the | |
| increasing limit of finite observations, and finite-label conditional | |
| entropy converges along this limit. | |
| -/ | |
| noncomputable section | |
| open Set Filter MeasureTheory | |
| open scoped Topology | |
| namespace DukePrimeDegree | |
| open PrimeDegreePackets | |
| variable {Y : Type*} [mY : MeasurableSpace Y] | |
| def finiteCountableObservation (Q : Y → ℕ) (m : ℕ) : | |
| Y → Fin (Fintype.card (Fin m → Bool)) := | |
| finiteGeneratingObservation ℕ m ∘ Q | |
| lemma finiteCountableObservation_measurable {Q : Y → ℕ} (hQ : Measurable Q) (m : ℕ) : | |
| Measurable (finiteCountableObservation Q m) := | |
| (finiteGeneratingObservation_measurable ℕ m).comp hQ | |
| omit mY in | |
| lemma finiteCountableObservation_monotone (Q : Y → ℕ) : | |
| Monotone (fun m => observationSigma (finiteCountableObservation Q m)) := by | |
| intro m n hmn | |
| simp only [finiteCountableObservation,observationSigma_comp_eq_comap] | |
| exact MeasurableSpace.comap_mono (finiteGeneratingObservation_monotone ℕ hmn) | |
| omit mY in | |
| lemma finiteCountableObservation_generate (Q : Y → ℕ) : | |
| (⨆ m, observationSigma (finiteCountableObservation Q m)) = observationSigma Q := by | |
| simp only [finiteCountableObservation,observationSigma_comp_eq_comap] | |
| rw [←MeasurableSpace.comap_iSup,finiteGeneratingObservation_generate] | |
| rfl | |
| omit mY in | |
| lemma finiteCountableFuture_generate (T : Y → Y) (Q : Y → ℕ) : | |
| (⨆ m, ⨆ k : ℕ, observationSigma | |
| (fun x => finiteCountableObservation Q m (T^[k+1] x))) = | |
| ⨆ k : ℕ, observationSigma (fun x => Q (T^[k+1] x)) := by | |
| rw [iSup_comm] | |
| congr 1 | |
| funext k | |
| have H := finiteCountableObservation_generate (fun x => Q (T^[k+1] x)) | |
| exact H | |
| /-- Only the observed finite label needs finite entropy. The extra future | |
| observation may take countably many values. -/ | |
| theorem conditionalSigmaEntropy_countableFuture_limit {α : Type*} [Fintype α] | |
| (μ : Measure Y) [IsFiniteMeasure μ] (P : Y → α) | |
| {T : Y → Y} (hT : Measurable T) {Q : Y → ℕ} (hQ : Measurable Q) | |
| (m₀ : MeasurableSpace Y) (hm₀ : m₀ ≤ mY) : | |
| Tendsto (fun m => conditionalSigmaEntropy (mY := mY) μ P | |
| (m₀ ⊔ ⨆ k : ℕ, observationSigma | |
| (fun x => finiteCountableObservation Q m (T^[k+1] x)))) atTop | |
| (𝓝 (conditionalSigmaEntropy (mY := mY) μ P | |
| (m₀ ⊔ ⨆ k : ℕ, observationSigma (fun x => Q (T^[k+1] x))))) := by | |
| have hmono : Monotone (fun m => m₀ ⊔ ⨆ k : ℕ, observationSigma | |
| (fun x => finiteCountableObservation Q m (T^[k+1] x))) := by | |
| intro m n hmn | |
| apply sup_le_sup le_rfl | |
| apply iSup_le | |
| intro k | |
| apply le_iSup_of_le k | |
| exact finiteCountableObservation_monotone (fun x => Q (T^[k+1] x)) hmn | |
| have hle (m : ℕ) : (m₀ ⊔ ⨆ k : ℕ, observationSigma | |
| (fun x => finiteCountableObservation Q m (T^[k+1] x))) ≤ mY := by | |
| apply sup_le hm₀ | |
| apply iSup_le | |
| intro k | |
| exact observationSigma_le (mY := mY) _ (fun a => | |
| ((finiteCountableObservation_measurable (mY := mY) hQ m).comp (hT.iterate (k+1))) | |
| (measurableSet_singleton a)) | |
| have H := conditionalSigmaEntropy_tendsto_iSup (mY := mY) μ P _ hmono hle | |
| rw [iSup_sup_eq,iSup_const,finiteCountableFuture_generate] at H | |
| exact H | |
| end DukePrimeDegree | |
| end | |
| end OAI | |