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a44267b | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 | import OAI.NumberTheory.DukePrimeDegree.Entropy.FinitePatchEntropy
namespace OAI
/-! # A countable cover by neutral future patches has zero conditional entropy rate -/
noncomputable section
open MeasureTheory ProbabilityTheory Set Filter
open scoped Topology BigOperators
namespace DukePrimeDegree
open PrimeDegreePackets
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
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
end DukePrimeDegree
end
end OAI
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