openAI-math-dataset / lean /OAI /NumberTheory /DukePrimeDegree /Entropy /CountableConditioningChain.lean
| import OAI.NumberTheory.DukePrimeDegree.Entropy.CountableBoundaryEntropy | |
| namespace OAI | |
| /-! # Chain rules with one countable conditioning label | |
| Finite clipping of the conditioning label and the martingale limit extend | |
| the finite chain rule without requiring finite entropy of that label. | |
| -/ | |
| noncomputable section | |
| open MeasureTheory Set Filter | |
| open scoped Topology | |
| namespace DukePrimeDegree | |
| open PrimeDegreePackets | |
| lemma observationSigma_pair_eq {Y α β : Type*} [Countable α] [Countable β] | |
| (P : Y → α) (Q : Y → β) : | |
| observationSigma (fun y => (P y,Q y))=observationSigma P ⊔ observationSigma Q := by | |
| let : MeasurableSpace α := ⊤ | |
| let : MeasurableSpace β := ⊤ | |
| have hprod : (inferInstance : MeasurableSpace (α × β))=⊤ := by | |
| apply top_unique | |
| intro s _ | |
| exact (Set.to_countable s).measurableSet | |
| unfold observationSigma | |
| rw [←hprod] | |
| exact MeasurableSpace.comap_prodMk P Q | |
| theorem conditionalSigmaEntropy_chain_nat {Y α β : Type*} | |
| [MeasurableSpace Y] [Fintype α] [Fintype β] | |
| (μ : Measure Y) [IsProbabilityMeasure μ] | |
| (P : Y → α) (hP : ∀ a, MeasurableSet {y | P y=a}) | |
| (R : Y → β) (hR : ∀ b, MeasurableSet {y | R y=b}) | |
| (F : Y → ℕ) (hF : Measurable F) : | |
| conditionalSigmaEntropy μ (fun y => (P y,R y)) | |
| (MeasurableSpace.comap F inferInstance) = | |
| conditionalSigmaEntropy μ P (MeasurableSpace.comap F inferInstance) + | |
| conditionalSigmaEntropy μ R | |
| (MeasurableSpace.comap F inferInstance ⊔ observationSigma P) := by | |
| let C := fun L : ℕ => fun y : Y => (heightClip F L y,P y) | |
| have hC (L : ℕ) (b : Fin (L+1) × α) : MeasurableSet {y | C L y=b} := | |
| observationPair_measurable_fiber _ _ | |
| (fun a => heightClip_measurable hF L (measurableSet_singleton a)) hP b | |
| have hCm : Monotone (fun L => observationSigma (C L)) := by | |
| intro L J hLJ | |
| simp only [C,observationSigma_pair_eq] | |
| exact sup_le_sup_right (heightClip_sigma_monotone F hLJ) _ | |
| have hCg : (⨆ L, observationSigma (C L))= | |
| MeasurableSpace.comap F inferInstance ⊔ observationSigma P := by | |
| simp only [C,observationSigma_pair_eq] | |
| rw [iSup_sup_eq,heightClip_sigma_generate,iSup_const] | |
| have h₁ := conditional_heightClip_tendsto μ (fun y => (P y,R y)) | |
| (observationPair_measurable_fiber P R hP hR) F hF | |
| have h₂ := conditional_heightClip_tendsto μ P hP F hF | |
| have h₃ := conditionalObservationEntropy_tendsto_sigma μ R hR C hC hCm | |
| rw [hCg] at h₃ | |
| apply tendsto_nhds_unique h₁ | |
| apply (h₂.add h₃).congr | |
| intro L | |
| exact (conditionalObservationEntropy_chain μ P R (heightClip F L)).symm | |
| theorem conditionalSigmaEntropy_comp_preserving_nat {Y α : Type*} | |
| [MeasurableSpace Y] [Fintype α] | |
| (μ : Measure Y) [IsProbabilityMeasure μ] | |
| {T : Y → Y} (hT : MeasurePreserving T μ μ) | |
| (P : Y → α) (hP : ∀ a, MeasurableSet {y | P y=a}) | |
| (F : Y → ℕ) (hF : Measurable F) : | |
| conditionalSigmaEntropy μ (P ∘ T) (MeasurableSpace.comap (F ∘ T) inferInstance) = | |
| conditionalSigmaEntropy μ P (MeasurableSpace.comap F inferInstance) := by | |
| have h₁ := conditional_heightClip_tendsto μ (P ∘ T) | |
| (fun a => (hP a).preimage hT.measurable) (F ∘ T) (hF.comp hT.measurable) | |
| have h₂ := conditional_heightClip_tendsto μ P hP F hF | |
| apply tendsto_nhds_unique h₁ | |
| apply h₂.congr | |
| intro L | |
| symm | |
| exact conditionalObservationEntropy_comp_preserving μ P (heightClip F L) hP | |
| (fun a => heightClip_measurable hF L (measurableSet_singleton a)) hT | |
| end DukePrimeDegree | |
| end | |
| end OAI | |