openAI-math-dataset / lean /OAI /NumberTheory /DukePrimeDegree /Entropy /ConditionalFiberRecoding.lean
| import OAI.NumberTheory.DukePrimeDegree.Entropy.ConditionalMapEntropy | |
| import OAI.NumberTheory.DukePrimeDegree.Entropy.BilateralConditionalSupport | |
| import OAI.NumberTheory.DukePrimeDegree.Entropy.PredecessorSymbols | |
| namespace OAI | |
| /-! # Finite recoding within the fibers of a countable observation -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Set | |
| namespace DukePrimeDegree | |
| open PrimeDegreePackets | |
| lemma observationEntropy_eq_of_ae_eq {Y α : Type*} [MeasurableSpace Y] | |
| [Fintype α] (ν : Measure Y) (P R : Y → α) (h : P =ᵐ[ν] R) : | |
| observationEntropy ν P=observationEntropy ν R := by | |
| unfold observationEntropy | |
| congr 1 | |
| funext a | |
| apply congrArg ENNReal.toReal | |
| apply measure_congr | |
| filter_upwards [h] with y hy | |
| simp only [hy] | |
| theorem conditionalSigmaEntropy_le_fiber_recoding | |
| {Y S α β : Type*} [MeasurableSpace Y] [StandardBorelSpace Y] [Nonempty Y] | |
| [MeasurableSpace S] [MeasurableEq S] | |
| [MeasurableSpace α] [MeasurableSingletonClass α] [Fintype α] | |
| [MeasurableSpace β] [MeasurableSingletonClass β] [Fintype β] | |
| (μ : Measure Y) [IsProbabilityMeasure μ] | |
| (F : Y → S) (hF : Measurable F) | |
| (P : Y → α) (hP : Measurable P) (B : Y → β) (hB : Measurable B) | |
| (f : S → β → α) (hf : ∀ x, P x=f (F x) (B x)) : | |
| conditionalSigmaEntropy μ P (MeasurableSpace.comap F inferInstance) ≤ | |
| conditionalSigmaEntropy μ B (MeasurableSpace.comap F inferInstance) := by | |
| rw [conditionalSigmaEntropy_comap_eq_integral μ F hF P | |
| (fun a => hP (measurableSet_singleton a)), | |
| conditionalSigmaEntropy_comap_eq_integral μ F hF B | |
| (fun a => hB (measurableSet_singleton a))] | |
| apply integral_mono_ae | |
| (integrable_kernel_entropy (μ.map F) (condDistrib id F μ) P | |
| (fun a => hP (measurableSet_singleton a))) | |
| (integrable_kernel_entropy (μ.map F) (condDistrib id F μ) B | |
| (fun a => hB (measurableSet_singleton a))) | |
| filter_upwards [ae_condDistrib_self_fiber μ F hF] with s hs | |
| have he : P =ᵐ[condDistrib id F μ s] (f s ∘ B) := by | |
| filter_upwards [hs] with y hy | |
| simpa only [Function.comp_apply,hy] using hf y | |
| rw [observationEntropy_eq_of_ae_eq _ P (f s ∘ B) he] | |
| apply observationEntropy_le_of_refines _ _ B | |
| (fun a => ((measurable_of_finite (f s)).comp hB) (measurableSet_singleton a)) | |
| (fun a => hB (measurableSet_singleton a)) | |
| intro x y hxy | |
| exact congrArg (f s) hxy | |
| theorem predecessor_clip_conditionalEntropy_le | |
| {Y : Type*} [MeasurableSpace Y] [StandardBorelSpace Y] [Nonempty Y] | |
| (μ : Measure Y) [IsProbabilityMeasure μ] | |
| (T : Y → Y) (hT : Measurable T) (Q : Y → ℕ) (hQ : Measurable Q) | |
| {M : ℕ} (B : Y → Fin M) (hB : Measurable B) | |
| (decode : ℕ → Fin M → ℕ) (hdecode : ∀ x, decode (Q (T x)) (B x)=Q x) | |
| (L : ℕ) : | |
| conditionalSigmaEntropy μ (heightClip Q L) | |
| (MeasurableSpace.comap (Q ∘ T) inferInstance) ≤ | |
| conditionalSigmaEntropy μ B (MeasurableSpace.comap (Q ∘ T) inferInstance) := by | |
| apply conditionalSigmaEntropy_le_fiber_recoding μ (Q ∘ T) (hQ.comp hT) | |
| (heightClip Q L) (heightClip_measurable hQ L) B hB | |
| (fun q b => heightClip id L (decode q b)) | |
| intro x | |
| change heightClip id L (Q x)=heightClip id L (decode (Q (T x)) (B x)) | |
| rw [hdecode] | |
| end DukePrimeDegree | |
| end | |
| end OAI | |