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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 85 86 87 88 89 90 91 92 93 94 95 96 97 98 | import OAI.NumberTheory.DukePrimeDegree.Entropy.SparseFineObservation
import OAI.NumberTheory.DukePrimeDegree.Entropy.FinitePrefixChartCodes
import Mathlib.Data.Nat.Pairing
namespace OAI
/-! # One low-entropy observation for a countable coordinate atlas -/
noncomputable section
open Set MeasureTheory
open scoped ENNReal
namespace DukePrimeDegree
variable {Y : Type*} [TopologicalSpace Y] [T2Space Y]
[SecondCountableTopology Y] [MeasurableSpace Y] [BorelSpace Y]
theorem exists_countable_sparse_fine_observation
(μ : Measure Y) [IsProbabilityMeasure μ]
(e : Y ≃ₜ Y) (he : MeasurePreserving e μ μ)
(A : ℕ → Set Y) (hA : ∀ i, MeasurableSet (A i))
(hfree : ∀ i x, x ∈ A i → ∀ n : ℕ, 0 < n → (e^n) x ≠ x)
{d : ℕ} (f : ℕ → Y → Fin d → ℝ) (hf : ∀ i, Measurable (f i))
(R : ℕ → ℝ) (hR : ∀ i, 0 ≤ R i)
(hbound : ∀ i x, x ∈ A i → ∀ j, |f i x j| ≤ R i)
{ε : ℝ} (hε : 0 < ε) :
∃ Q : Y → ℕ, Measurable Q ∧
Summable (fun n => Real.negMulLog (μ.real {x | Q x=n})) ∧
(∑' n, Real.negMulLog (μ.real {x | Q x=n})) < ε ∧
∀ᵐ x ∂μ, ∀ i, x ∈ A i → ∀ y,
(∀ k : ℕ, Q ((e^(k+1)) y)=Q ((e^(k+1)) x)) →
∀ η : ℝ, 0 < η → ∃ k : ℕ, (e^(k+1)) x ∈ A i ∧ (e^(k+1)) y ∈ A i ∧
∀ j, |f i ((e^(k+1)) y) j-f i ((e^(k+1)) x) j| < η := by
obtain ⟨K,B,hB,hclose⟩ := exists_finite_prefix_chart_codes A hA f hf R hR hbound
obtain ⟨δ,hδ,hδs,_,Hδ⟩ := exists_lastActiveObservation_entropy_budget
(Y := Y) K hε
choose S hS hSA hSm hcover using fun n : ℕ =>
exists_small_complete_orbit_section μ e he (hA n.unpair.1)
(hfree n.unpair.1) (hδ n)
have hsμ : (∑' n, μ (S n)) ≠ ∞ := by
apply ne_top_of_le_ne_top hδs.tsum_ofReal_ne_top
apply ENNReal.tsum_le_tsum
intro n
exact (ENNReal.le_ofReal_iff_toReal_le (measure_ne_top μ _) (hδ n).le).mpr (hSm n).le
have hfinite := ae_finite_setOfPred_mem hsμ
have hfiniteAll : ∀ᵐ x ∂μ, ∀ k : ℕ, {n | (e^k) x ∈ S n}.Finite :=
ae_all_iff.mpr fun k => (measurePreserving_homeomorph_pow μ e he k).quasiMeasurePreserving.ae hfinite
have hvisitAll : ∀ᵐ x ∂μ, ∀ n, x ∈ A n.unpair.1 → ∀ M : ℕ,
∃ k : ℕ, M ≤ k ∧ (e^k) x ∈ S n :=
ae_all_iff.mpr fun n => ae_future_visits_of_orbit_section μ e he (hS n) (hcover n)
let I := Option (Σ n : ℕ, Fin (K n))
let : MeasurableSpace I := ⊤
let Q : Y → ℕ := fun x => Encodable.encode (lastActiveObservation S B x)
have hQ : Measurable Q := (measurable_of_countable Encodable.encode).comp
(measurable_lastActiveObservation hS B hB)
have hcost := Hδ μ S hS (fun n => (hSm n).le) B hB
have hencode := encoded_observation_information μ (lastActiveObservation S B)
refine ⟨Q,hQ,hencode.1.mpr hcost.1,hencode.2.trans_lt hcost.2,?_⟩
filter_upwards [hfiniteAll,hvisitAll] with x hxfin hxvisit
intro i hxA y hxy η hη
obtain ⟨p,hp⟩ := exists_nat_gt (2/η)
let n := Nat.pair i p
have hindex : n.unpair.1=i := by simp only [n,Nat.unpair_pair]
have hxA' : x ∈ A n.unpair.1 := by rwa [hindex]
obtain ⟨k,hk,hkx⟩ := hxvisit n hxA' 1
obtain ⟨m,hnm,hm⟩ := lastActiveObservation_scale_ge S B (hxfin k) hkx
have him : i ≤ m := (Nat.left_le_pair i p).trans hnm
have hpm : p ≤ m := (Nat.right_le_pair i p).trans hnm
have hkcode : lastActiveObservation S B ((e^k) y)=lastActiveObservation S B ((e^k) x) := by
apply Encodable.encode_injective
have hh := hxy (k-1)
simpa only [Nat.sub_add_cancel hk] using hh
have hycode : lastActiveObservation S B ((e^k) y)=some ⟨m,B m ((e^k) x)⟩ := hkcode.trans hm
have hmy : (e^k) y ∈ lastActiveObservationAtom S B (some ⟨m,B m ((e^k) x)⟩) := by
rw [←lastActiveObservation_fiber]
exact hycode
have hxmem : (e^k) x ∈ A i := by
have H := hSA n hkx
rwa [hindex] at H
obtain ⟨hymem,hfine⟩ := hclose m i him _ hxmem _ hmy.2
refine ⟨k-1,by simpa only [Nat.sub_add_cancel hk] using hxmem,
by simpa only [Nat.sub_add_cancel hk] using hymem,?_⟩
intro j
have hpmR : (p:ℝ) ≤ m := by exact_mod_cast hpm
have hlarge : 2 < (p:ℝ)*η := (div_lt_iff₀ hη).mp hp
have hsmall : 2*(1/(m+1:ℝ)) < η := by
have hd : (0:ℝ) < m+1 := by positivity
rw [mul_one_div]
apply (div_lt_iff₀ hd).mpr
nlinarith
have H := (hfine j).trans hsmall
simpa only [Nat.sub_add_cancel hk] using H
end DukePrimeDegree
end
end OAI
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