| import OAI.NumberTheory.DukePrimeDegree.Entropy.FinitePatchEntropy |
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| namespace OAI |
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| /-! # A countable cover by neutral future patches has zero conditional entropy rate -/ |
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| noncomputable section |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped Topology BigOperators |
| namespace DukePrimeDegree |
| open PrimeDegreePackets |
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| theorem exists_finite_patch_complement_lt |
| {Y : Type*} [MeasurableSpace Y] (μ : Measure Y) [IsFiniteMeasure μ] |
| (E : ℕ → Set Y) (hE : ∀ i, MeasurableSet (E i)) |
| (hfull : ∀ᵐ y ∂μ, y ∈ ⋃ i, E i) {ε : ℝ} (hε : 0 < ε) : |
| ∃ N : ℕ, μ.real (⋃ i : Fin (N+1), E i.val)ᶜ < ε := by |
| have hz : μ (⋂ i, (E i)ᶜ)=0 := by |
| rw [←compl_iUnion] |
| exact ae_iff.mp hfull |
| have ht := tendsto_measure_iInter_le (μ := μ) (s := fun i => (E i)ᶜ) |
| (fun i => (hE i).compl.nullMeasurableSet) ⟨0,measure_ne_top _ _⟩ |
| rw [hz] at ht |
| have ht' : Tendsto (fun N : ℕ => μ.real (⋂ i ≤ N, (E i)ᶜ)) atTop (𝓝 0) := by |
| simpa only [Measure.real,ENNReal.toReal_zero,Function.comp_def] using |
| (ENNReal.tendsto_toReal ENNReal.zero_ne_top).comp ht |
| obtain ⟨N,hN⟩ := (ht'.eventually (gt_mem_nhds hε)).exists |
| refine ⟨N,?_⟩ |
| have he : (⋃ i : Fin (N+1), E i.val)ᶜ = ⋂ i ≤ N, (E i)ᶜ := by |
| ext y |
| simp only [mem_compl_iff,mem_iUnion,not_exists,mem_iInter] |
| constructor |
| · intro h i hi |
| exact h ⟨i,by omega⟩ |
| · intro h i |
| exact h i.val (by omega) |
| rw [he] |
| exact hN |
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| theorem countable_patch_name_entropy_sublinear {n k : ℕ} |
| (ell : Fin n → ℝ) (hell : ∑ i, ell i=0) |
| (Q : X n → ℕ) (hQ : Measurable Q) |
| (E : ℕ → Set (X n)) (hE : ∀ i, MeasurableSet (E i)) |
| (hpair : ∀ i, ∀ x ∈ E i, ∀ y ∈ E i, |
| fullFutureObservation |
| (fun z => rightAction z (diagonalOneParameter ell hell (-1))) Q 0 x = |
| fullFutureObservation |
| (fun z => rightAction z (diagonalOneParameter ell hell (-1))) Q 0 y → |
| ∃ g ∈ matrixBall n 1, rightAction x g=y ∧ |
| ∀ t : ℝ, Commute g (diagonalOneParameter ell hell t)) |
| (μ : Measure (X n)) [IsProbabilityMeasure μ] |
| (hfull : ∀ᵐ y ∂μ, y ∈ ⋃ i, E i) |
| (hT : MeasurePreserving |
| (fun z => rightAction z (diagonalOneParameter ell hell (-1))) μ μ) |
| (P : X n → Fin k) (hP : Measurable P) |
| {ε : ℝ} (hε : 0 < ε) : |
| ∃ C : ℝ, ∀ N : ℕ, |
| conditionalSigmaEntropy μ |
| (orbitName (fun z => rightAction z (diagonalOneParameter ell hell (-1))) P N) |
| (fullFutureSigma |
| (fun z => rightAction z (diagonalOneParameter ell hell (-1))) Q N) ≤ C+N*ε := by |
| obtain ⟨δ,η,hδ,hδ1,hη,hcost⟩ := rareLabelCost_small k hε |
| obtain ⟨B,r,R,hB,hsmall,hr,hrobust⟩ := exists_robust_finite_observation μ P hP |
| (ENNReal.ofReal (η/2)) (by simp only [ne_eq,ENNReal.ofReal_eq_zero]; linarith) |
| have hsmall' : μ.real B < η/2 := by |
| have h := (ENNReal.toReal_lt_toReal (measure_ne_top μ B) ENNReal.ofReal_ne_top).mpr hsmall |
| simpa only [Measure.real,ENNReal.toReal_ofReal (show 0 ≤ η/2 by positivity)] using h |
| obtain ⟨p,hp⟩ := exists_finite_patch_complement_lt μ E hE hfull |
| (show 0 < η/2 by positivity) |
| obtain ⟨M,hM⟩ := finite_patch_name_entropy_bound ell hell Q hQ |
| (fun i : Fin (p+1) => E i.val) (fun i => hE i.val) |
| (fun i => hpair i.val) μ hT P hP B hB R hr hrobust hδ hδ1 |
| refine ⟨Real.log M,fun N => (hM N).trans ?_⟩ |
| apply add_le_add le_rfl |
| apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg N) |
| exact (hcost _ (add_nonneg measureReal_nonneg measureReal_nonneg) |
| (by linarith)).le |
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| end DukePrimeDegree |
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| end |
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| end OAI |
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