openAI-math-dataset / lean /OAI /NumberTheory /DukePrimeDegree /Entropy /CountableSparseFineObservation.lean
| 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 | |