openAI-math-dataset / lean /OAI /NumberTheory /DukePrimeDegree /Entropy /CountableBoundaryEntropy.lean
| import OAI.NumberTheory.DukePrimeDegree.Entropy.CountableObservationInformation | |
| import OAI.NumberTheory.DukePrimeDegree.Entropy.ConditionalSigmaLimit | |
| import OAI.NumberTheory.PrimePackets.FiniteGeneratingObservations | |
| namespace OAI | |
| /-! # A countable boundary label has sublinear information cost | |
| Finite clipping controls the information in a single countable label, | |
| even when that label has infinite Shannon entropy. The clipping error | |
| depends only on its tail probability and the alphabet of the observed | |
| word, rather than on the entropy of the countable label. | |
| -/ | |
| noncomputable section | |
| open MeasureTheory Set Filter | |
| open scoped Topology BigOperators ENNReal | |
| namespace DukePrimeDegree | |
| open PrimeDegreePackets | |
| theorem heightClip_sigma_monotone {Y : Type*} (Q : Y → ℕ) : | |
| Monotone (fun L => observationSigma (heightClip Q L)) := by | |
| intro L J hLJ | |
| let f : Fin (J+1) → Fin (L+1) := fun a => | |
| ⟨min a.val L,Nat.lt_succ_of_le (min_le_right _ _)⟩ | |
| have he : heightClip Q L = f ∘ heightClip Q J := by | |
| funext y | |
| apply Fin.ext | |
| change min (Q y) L=min (min (Q y) J) L | |
| omega | |
| change observationSigma (heightClip Q L) ≤ observationSigma (heightClip Q J) | |
| rw [he] | |
| exact observationSigma_comp_le (heightClip Q J) f | |
| theorem heightClip_sigma_generate {Y : Type*} (Q : Y → ℕ) : | |
| (⨆ L, observationSigma (heightClip Q L)) = | |
| MeasurableSpace.comap Q inferInstance := by | |
| apply le_antisymm | |
| · let : MeasurableSpace Y := MeasurableSpace.comap Q inferInstance | |
| have hQ : Measurable Q := comap_measurable Q | |
| exact iSup_le fun L => observationSigma_le _ | |
| (fun a => heightClip_measurable hQ L (measurableSet_singleton a)) | |
| · let : MeasurableSpace Y := ⨆ L, observationSigma (heightClip Q L) | |
| have hQ : Measurable Q := by | |
| apply measurable_to_countable' | |
| intro a | |
| have hc : Measurable (heightClip Q (a+1)) := by | |
| apply measurable_to_countable' | |
| intro b | |
| exact (le_iSup (fun L => observationSigma (heightClip Q L)) (a+1)) | |
| _ ⟨{b},trivial,rfl⟩ | |
| have he : {y | Q y=a} = | |
| {y | (heightClip Q (a+1) y).val=a} := by | |
| ext y | |
| change Q y=a ↔ min (Q y) (a+1)=a | |
| omega | |
| change MeasurableSet {y | Q y=a} | |
| rw [he] | |
| exact measurableSet_eq_fun | |
| ((measurable_of_finite (fun b : Fin (a+2) => b.val)).comp hc) measurable_const | |
| exact hQ.comap_le | |
| theorem conditional_heightClip_tendsto {Y α : Type*} | |
| [MeasurableSpace Y] [Fintype α] | |
| (μ : Measure Y) [IsProbabilityMeasure μ] | |
| (P : Y → α) (hP : ∀ a, MeasurableSet {y | P y=a}) | |
| (Q : Y → ℕ) (hQ : Measurable Q) : | |
| Tendsto (fun L => conditionalObservationEntropy μ P (heightClip Q L)) atTop | |
| (𝓝 (conditionalSigmaEntropy μ P (MeasurableSpace.comap Q inferInstance))) := by | |
| have H := conditionalObservationEntropy_tendsto_sigma μ P hP (heightClip Q) | |
| (fun L a => heightClip_measurable hQ L (measurableSet_singleton a)) | |
| (heightClip_sigma_monotone Q) | |
| rwa [heightClip_sigma_generate] at H | |
| private theorem conditional_heightClip_le_larger {Y α : Type*} | |
| [MeasurableSpace Y] [Fintype α] | |
| (μ : Measure Y) [IsProbabilityMeasure μ] | |
| (P : Y → α) (hP : ∀ a, MeasurableSet {y | P y=a}) | |
| (Q : Y → ℕ) (hQ : Measurable Q) {L J : ℕ} (hLJ : L ≤ J) : | |
| conditionalObservationEntropy μ P (heightClip Q L) ≤ | |
| conditionalObservationEntropy μ P (heightClip Q J) + | |
| μ.real {y | L ≤ Q y} * Real.log (Fintype.card α) := by | |
| let F (M : ℕ) (y : Y) := | |
| finiteShannon (fun a => observationConditionalProbability μ P (heightClip Q M) a y) | |
| have hFint (M : ℕ) : Integrable (F M) μ := | |
| integrable_finite_values μ (heightClip Q M) | |
| (fun a => heightClip_measurable hQ M (measurableSet_singleton a)) | |
| (fun b => finiteShannon (fun a => | |
| μ.real {y | (heightClip Q M y,P y)=(b,a)} / | |
| μ.real {y | heightClip Q M y=b})) | |
| have hB : MeasurableSet {y | L ≤ Q y} := hQ measurableSet_Ici | |
| have hbound : F L ≤ᵐ[μ] fun y => F J y + | |
| {y | L ≤ Q y}.indicator (fun _ => Real.log (Fintype.card α)) y := by | |
| filter_upwards [observationConditionalProbability_ae_sum μ P (heightClip Q L) hP | |
| (fun a => heightClip_measurable hQ L (measurableSet_singleton a)), | |
| observationConditionalProbability_ae_sum μ P (heightClip Q J) hP | |
| (fun a => heightClip_measurable hQ J (measurableSet_singleton a))] with y hyL hyJ | |
| by_cases hy : L ≤ Q y | |
| · rw [Set.indicator_of_mem (show y ∈ {y | L ≤ Q y} from hy)] | |
| have hlow : 0 ≤ F J y := finiteShannon_nonneg | |
| (fun a => observationConditionalProbability_nonneg μ P (heightClip Q J) a y) hyJ | |
| have hu : F L y ≤ Real.log (Fintype.card α) := finiteShannon_le_log_card | |
| (fun a => observationConditionalProbability_nonneg μ P (heightClip Q L) a y) hyL | |
| linarith | |
| · have hyL' : Q y < L := Nat.lt_of_not_ge hy | |
| have hyJ' : Q y < J := lt_of_lt_of_le hyL' hLJ | |
| have he : F L y=F J y := by | |
| apply congrArg finiteShannon | |
| funext a | |
| unfold observationConditionalProbability | |
| have hnum (M : ℕ) (hM : Q y < M) : | |
| {z | (heightClip Q M z,P z)=(heightClip Q M y,a)} = | |
| {z | Q z=Q y ∧ P z=a} := by | |
| ext z | |
| simp only [Prod.mk.injEq,Set.mem_ofPred_eq] | |
| have hh := Set.ext_iff.mp (heightClip_atom_eq_of_lt Q y hM) z | |
| exact and_congr_left fun _ => hh | |
| rw [hnum L hyL',hnum J hyJ',heightClip_atom_eq_of_lt Q y hyL', | |
| heightClip_atom_eq_of_lt Q y hyJ'] | |
| rw [Set.indicator_of_notMem (show y ∉ {y | L ≤ Q y} from hy),he,add_zero] | |
| rw [conditionalObservationEntropy_eq_integral μ P (heightClip Q L) hP | |
| (fun a => heightClip_measurable hQ L (measurableSet_singleton a)), | |
| conditionalObservationEntropy_eq_integral μ P (heightClip Q J) hP | |
| (fun a => heightClip_measurable hQ J (measurableSet_singleton a))] | |
| have H := integral_mono_ae (hFint L) | |
| ((hFint J).add ((integrable_const _).indicator hB)) hbound | |
| simp only [Pi.add_apply] at H | |
| rw [integral_add (hFint J) ((integrable_const _).indicator hB), | |
| integral_indicator_const _ hB,smul_eq_mul] at H | |
| exact H | |
| /-- Removing a finite clipping of the conditioning label changes entropy | |
| only on the clipped tail. -/ | |
| theorem conditional_heightClip_le_countable {Y α : Type*} | |
| [MeasurableSpace Y] [Fintype α] | |
| (μ : Measure Y) [IsProbabilityMeasure μ] | |
| (P : Y → α) (hP : ∀ a, MeasurableSet {y | P y=a}) | |
| (Q : Y → ℕ) (hQ : Measurable Q) (L : ℕ) : | |
| conditionalObservationEntropy μ P (heightClip Q L) ≤ | |
| conditionalSigmaEntropy μ P (MeasurableSpace.comap Q inferInstance) + | |
| μ.real {y | L ≤ Q y} * Real.log (Fintype.card α) := by | |
| apply ge_of_tendsto ((conditional_heightClip_tendsto μ P hP Q hQ).add_const _) | |
| filter_upwards [eventually_ge_atTop L] with J hLJ | |
| exact conditional_heightClip_le_larger μ P hP Q hQ hLJ | |
| theorem observationEntropy_le_countable_conditioning {Y α : Type*} | |
| [MeasurableSpace Y] [Fintype α] | |
| (μ : Measure Y) [IsProbabilityMeasure μ] | |
| (P : Y → α) (hP : ∀ a, MeasurableSet {y | P y=a}) | |
| (Q : Y → ℕ) (hQ : Measurable Q) (L : ℕ) : | |
| observationEntropy μ P ≤ Real.log (L+1) + | |
| conditionalSigmaEntropy μ P (MeasurableSpace.comap Q inferInstance) + | |
| μ.real {y | L ≤ Q y} * Real.log (Fintype.card α) := by | |
| have hC a := heightClip_measurable hQ L (measurableSet_singleton a) | |
| have hjoint := observationEntropy_le_add_conditional μ P (heightClip Q L) hP hC | |
| have hcard := finiteShannon_le_log_card (fun a => | |
| measureReal_nonneg (μ := μ) (s := {y | heightClip Q L y=a})) | |
| (observation_distribution_sum μ (heightClip Q L) hC) | |
| have htail := conditional_heightClip_le_countable μ P hP Q hQ L | |
| simp only [Fintype.card_fin,Nat.cast_add,Nat.cast_one] at hcard | |
| change observationEntropy μ (heightClip Q L) ≤ _ at hcard | |
| linarith | |
| theorem conditional_countable_le_observationEntropy {Y α : Type*} | |
| [MeasurableSpace Y] [Fintype α] | |
| (μ : Measure Y) [IsProbabilityMeasure μ] | |
| (P : Y → α) (hP : ∀ a, MeasurableSet {y | P y=a}) | |
| (Q : Y → ℕ) (hQ : Measurable Q) : | |
| conditionalSigmaEntropy μ P (MeasurableSpace.comap Q inferInstance) ≤ | |
| observationEntropy μ P := by | |
| apply le_of_tendsto (conditional_heightClip_tendsto μ P hP Q hQ) | |
| exact Eventually.of_forall fun L => conditionalObservationEntropy_le μ P | |
| (heightClip Q L) hP | |
| (fun a => heightClip_measurable hQ L (measurableSet_singleton a)) | |
| theorem countable_tail_measureReal_tendsto_zero {Y : Type*} [MeasurableSpace Y] | |
| (μ : Measure Y) [IsFiniteMeasure μ] (Q : Y → ℕ) (hQ : Measurable Q) : | |
| Tendsto (fun L => μ.real {y | L ≤ Q y}) atTop (𝓝 0) := by | |
| have he : (⋂ L : ℕ, {y | L ≤ Q y}) = ∅ := by | |
| ext y | |
| simp only [mem_iInter,mem_ofPred_eq,mem_empty_iff_false,iff_false] | |
| intro h | |
| exact Nat.not_succ_le_self (Q y) (h (Q y+1)) | |
| have H := tendsto_measure_iInter_atTop | |
| (μ := μ) (fun L => (hQ measurableSet_Ici).nullMeasurableSet) | |
| (show Antitone (fun L => {y | L ≤ Q y}) from fun L J h y hy => h.trans hy) | |
| ⟨0,measure_ne_top _ _⟩ | |
| change Tendsto (fun L => μ {y | L ≤ Q y}) atTop | |
| (𝓝 (μ (⋂ L : ℕ, {y | L ≤ Q y}))) at H | |
| rw [he,measure_empty] at H | |
| simpa only [measureReal_def,ENNReal.toReal_zero,Function.comp_def] using | |
| (ENNReal.continuousAt_toReal (by simp : (0 : ℝ≥0∞) ≠ ∞)).tendsto.comp H | |
| /-- A single countable observation at the end of an orbit word carries | |
| sublinear information about that word. Its Shannon entropy may be infinite. -/ | |
| theorem countable_boundary_information_sublinear {Y α : Type*} | |
| [MeasurableSpace Y] [Fintype α] | |
| (μ : Measure Y) [IsProbabilityMeasure μ] | |
| {T : Y → Y} (hT : MeasurePreserving T μ μ) | |
| (P : Y → α) (hP : ∀ a, MeasurableSet {y | P y=a}) | |
| (Q : Y → ℕ) (hQ : Measurable Q) : | |
| Tendsto (fun N : ℕ => | |
| (observationEntropy μ (orbitName T P N) - | |
| conditionalSigmaEntropy μ (orbitName T P N) | |
| (MeasurableSpace.comap (Q ∘ T^[N]) inferInstance)) / N) | |
| atTop (𝓝 0) := by | |
| let E (N : ℕ) := observationEntropy μ (orbitName T P N) - | |
| conditionalSigmaEntropy μ (orbitName T P N) | |
| (MeasurableSpace.comap (Q ∘ T^[N]) inferInstance) | |
| have hE (N : ℕ) : 0 ≤ E N := sub_nonneg.mpr | |
| (conditional_countable_le_observationEntropy μ (orbitName T P N) | |
| (orbitName_measurable_fiber hT.measurable hP N) (Q ∘ T^[N]) | |
| (hQ.comp (hT.measurable.iterate N))) | |
| have hbound (L N : ℕ) : | |
| E N ≤ Real.log (L+1) + μ.real {y | L ≤ Q y} * | |
| ((N : ℝ) * Real.log (Fintype.card α)) := by | |
| have H := observationEntropy_le_countable_conditioning μ (orbitName T P N) | |
| (orbitName_measurable_fiber hT.measurable hP N) (Q ∘ T^[N]) | |
| (hQ.comp (hT.measurable.iterate N)) L | |
| have htail : μ.real {y | L ≤ (Q ∘ T^[N]) y} = μ.real {y | L ≤ Q y} := | |
| (hT.iterate N).measureReal_preimage (hQ measurableSet_Ici).nullMeasurableSet | |
| rw [htail,Fintype.card_fun,Fintype.card_fin,Nat.cast_pow,Real.log_pow] at H | |
| dsimp only [E] | |
| linarith | |
| apply Metric.tendsto_nhds.mpr | |
| intro ε hε | |
| have ht : Tendsto (fun L => μ.real {y | L ≤ Q y} * | |
| Real.log (Fintype.card α)) atTop (𝓝 0) := by | |
| simpa using (countable_tail_measureReal_tendsto_zero μ Q hQ).mul_const | |
| (Real.log (Fintype.card α)) | |
| obtain ⟨L,hL⟩ := (ht.eventually (gt_mem_nhds (show 0 < ε/2 by positivity))).exists | |
| have hl : Tendsto (fun N : ℕ => Real.log (L+1)/(N : ℝ)) atTop (𝓝 0) := | |
| tendsto_const_nhds.div_atTop tendsto_natCast_atTop_atTop | |
| filter_upwards [hl.eventually (gt_mem_nhds (show 0 < ε/2 by positivity)), | |
| eventually_ge_atTop (1 : ℕ)] with N hlog hN | |
| have hn : 0 < (N : ℝ) := by exact_mod_cast (show 0 < N by omega) | |
| have he : E N/(N : ℝ) ≤ Real.log (L+1)/(N : ℝ) + | |
| μ.real {y | L ≤ Q y} * Real.log (Fintype.card α) := by | |
| apply (div_le_iff₀ hn).mpr | |
| have hh := hbound L N | |
| have heq : (Real.log (L+1)/(N : ℝ) + | |
| μ.real {y | L ≤ Q y} * Real.log (Fintype.card α)) * N = | |
| Real.log (L+1) + μ.real {y | L ≤ Q y} * | |
| ((N : ℝ) * Real.log (Fintype.card α)) := by | |
| field_simp | |
| rwa [heq] | |
| change dist (E N/(N : ℝ)) 0 < ε | |
| rw [Real.dist_eq,sub_zero,abs_of_nonneg (div_nonneg (hE N) hn.le)] | |
| linarith | |
| theorem conditional_boundary_entropy_rate {Y α : Type*} | |
| [MeasurableSpace Y] [Fintype α] | |
| (μ : Measure Y) [IsProbabilityMeasure μ] | |
| {T : Y → Y} (hT : MeasurePreserving T μ μ) | |
| (P : Y → α) (hP : ∀ a, MeasurableSet {y | P y=a}) | |
| (Q : Y → ℕ) (hQ : Measurable Q) : | |
| Tendsto (fun N : ℕ => conditionalSigmaEntropy μ (orbitName T P N) | |
| (MeasurableSpace.comap (Q ∘ T^[N]) inferInstance) / N) atTop | |
| (𝓝 (observationEntropyRate μ T P)) := by | |
| have H := (tendsto_observationEntropyRate μ hT P hP).sub | |
| (countable_boundary_information_sublinear μ hT P hP Q hQ) | |
| simp only [sub_zero] at H | |
| convert H using 1 | |
| funext N | |
| rw [sub_div] | |
| ring | |
| end DukePrimeDegree | |
| end | |
| end OAI | |