| import OAI.NumberTheory.DukePrimeDegree.Geometry.RemainingAtomicProduct |
| import OAI.NumberTheory.DukePrimeDegree.Entropy.ObservationInformation |
|
|
| namespace OAI |
|
|
| /-! # Vanishing transverse conditional entropy for atomic root products -/ |
|
|
| noncomputable section |
| open MeasureTheory Set |
| namespace DukePrimeDegree |
| open PrimeDegreePackets |
|
|
| lemma observationEntropy_dirac_eq_zero {Y α : Type*} [MeasurableSpace Y] |
| [MeasurableSingletonClass Y] [Fintype α] (y : Y) (Q : Y → α) : |
| observationEntropy (Measure.dirac y) Q = 0 := by |
| classical |
| unfold observationEntropy finiteShannon |
| apply Finset.sum_eq_zero |
| intro a _ |
| by_cases ha : Q y=a <;> simp [Measure.dirac_real_apply,ha] |
|
|
| theorem conditionalSigmaEntropy_nonneg {Y α : Type*} [mY : MeasurableSpace Y] [Fintype α] |
| (μ : Measure Y) (P : Y → α) (m : MeasurableSpace Y) : |
| 0 ≤ conditionalSigmaEntropy (mY := mY) μ P m := by |
| apply integral_nonneg_of_ae |
| filter_upwards [ae_all_iff.mpr (fun a => |
| conditionalProbability_unit_bounds (mY := mY) μ {z | P z=a} m)] with z hz |
| exact Finset.sum_nonneg fun a _ => Real.negMulLog_nonneg (hz a).1 (hz a).2 |
|
|
| namespace RootPrefixBox |
| 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)} |
| {T : Type*} [TopologicalSpace T] [SecondCountableTopology T] |
| [MeasurableSpace T] [BorelSpace T] |
| {C : RootPrefixCoordinates (d+1) P₀ T} {g : G (d+1)} |
|
|
| theorem conditionalEntropy_fst_eq_zero_of_atomic_roots |
| (A : RootPrefixBox C c hP₀ hf g) (R : ℝ) (hAR : A.time ⊆ c.remainingWindow R 0) |
| (μ : Measure (X (d+1))) [IsProbabilityMeasure μ] |
| [ErgodicSMul (Multiplicative (LogCoordinates d)) (X (d+1)) μ] |
| (h : ∀ p : ↥(allPositiveRoots (d+1) \ P₀), ∀ᵐ x ∂μ, |
| intrinsicRootLeaf (ne_of_lt (Finset.mem_filter.mp (Finset.mem_sdiff.mp p.prop).1).2) μ x = |
| Measure.dirac 0) |
| {α : Type*} [Fintype α] |
| (Q : (UpperRootTransverse (d+1) × T) × c.Time → α) |
| (hQ : ∀ a, MeasurableSet {z | Q z=a}) : |
| conditionalSigmaEntropy (A.coordinateMeasure μ) Q |
| (MeasurableSpace.comap Prod.fst |
| (inferInstance : MeasurableSpace (UpperRootTransverse (d+1) × T))) = 0 := by |
| rw [conditionalSigmaEntropy_fst (A.coordinateMeasure μ) Q hQ] |
| have he := integral_map (μ := A.coordinateMeasure μ) measurable_fst.aemeasurable |
| (measurable_fiber_observationEntropy (A.conditionalKernel μ) Q hQ).aestronglyMeasurable |
| calc |
| _ = ∫ z, observationEntropy (A.conditionalKernel μ z.1) (fun t => Q (z.1,t)) |
| ∂A.coordinateMeasure μ := he |
| _ = 0 := by |
| apply integral_eq_zero_of_ae |
| filter_upwards [A.ae_conditionalKernel_eq_dirac_of_atomic_roots R hAR μ h] with z hz |
| simp only [hz,observationEntropy_dirac_eq_zero,Pi.zero_apply] |
|
|
| theorem conditionalEntropy_eq_zero_of_atomic_roots |
| (A : RootPrefixBox C c hP₀ hf g) (R : ℝ) (hAR : A.time ⊆ c.remainingWindow R 0) |
| (μ : Measure (X (d+1))) [IsProbabilityMeasure μ] |
| [ErgodicSMul (Multiplicative (LogCoordinates d)) (X (d+1)) μ] |
| (h : ∀ p : ↥(allPositiveRoots (d+1) \ P₀), ∀ᵐ x ∂μ, |
| intrinsicRootLeaf (ne_of_lt (Finset.mem_filter.mp (Finset.mem_sdiff.mp p.prop).1).2) μ x = |
| Measure.dirac 0) |
| {α : Type*} [Fintype α] |
| (Q : (UpperRootTransverse (d+1) × T) × c.Time → α) |
| (hQ : ∀ a, MeasurableSet {z | Q z=a}) |
| (m : MeasurableSpace ((UpperRootTransverse (d+1) × T) × c.Time)) |
| (hm : m ≤ Prod.instMeasurableSpace) |
| (hfst : MeasurableSpace.comap Prod.fst |
| (inferInstance : MeasurableSpace (UpperRootTransverse (d+1) × T)) ≤ m) : |
| conditionalSigmaEntropy (mY := Prod.instMeasurableSpace) (A.coordinateMeasure μ) Q m = 0 := by |
| apply le_antisymm |
| · exact (conditionalSigmaEntropy_antitone (mY := Prod.instMeasurableSpace) |
| (A.coordinateMeasure μ) Q hQ hfst hm).trans_eq |
| (A.conditionalEntropy_fst_eq_zero_of_atomic_roots R hAR μ h Q hQ) |
| · exact conditionalSigmaEntropy_nonneg (mY := Prod.instMeasurableSpace) |
| (A.coordinateMeasure μ) Q m |
|
|
| end RootPrefixBox |
| end DukePrimeDegree |
|
|
| end |
|
|
| end OAI |
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|
