| import OAI.Probability.Localization.Termination |
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| namespace OAI |
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| open MeasureTheory ProbabilityTheory Set |
| open scoped BigOperators ENNReal |
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| namespace RepeatedLocalization |
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| namespace FiniteEntropy |
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| noncomputable def terminalKL {n : ℕ} (P : Measure (Coordinate n)) (Q : Cell n) (z : Coordinate n) : ℝ≥0∞ := |
| InformationTheory.klDiv (terminalLaw P Q (terminalLog P Q z)) |
| (uniformCell (terminalCell P Q (terminalLog P Q z))) |
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| lemma measurable_terminalKL {n : ℕ} (hn : 0 < n) (P : Measure (Coordinate n)) (Q : Cell n) : |
| Measurable (terminalKL P Q) := |
| (measurable_of_countable (fun h : Log n => InformationTheory.klDiv (terminalLaw P Q h) |
| (uniformCell (terminalCell P Q h)))).comp (measurable_terminalLog hn P Q) |
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| lemma measurable_depth_real {n : ℕ} (hn : 0 < n) (P : Measure (Coordinate n)) (Q : Cell n) : |
| Measurable (fun z => (depth P Q z:ℝ)) := |
| (measurable_of_countable (fun h : Log n => (((terminalCell P Q h).1-Q.1:ℕ):ℝ))).comp |
| (measurable_terminalLog hn P Q) |
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| lemma terminal_historyKL {n : ℕ} (hn : 0 < n) (P : Measure (Coordinate n)) (Q : Cell n) |
| (hroot : ∀ᵐ z ∂P,z ∈ cellSet Q) {z : Coordinate n} (hz : z ∈ cellSet Q) (ht : Terminates P Q z) : |
| terminalKL P Q z = historyKL P Q (terminalLog P Q z) := by |
| unfold terminalKL historyKL terminalLaw terminalCell |
| rw [cond_congr_ae P (terminal_fiber_ae_event hn P Q hroot hz ht)] |
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| lemma terminal_cost_budget {n : ℕ} (hn : 32 ≤ n) (P : Measure (Coordinate n)) [IsProbabilityMeasure P] |
| (Q : Cell n) (hac : P ≪ uniformCell Q) (hK : InformationTheory.klDiv P (uniformCell Q) ≠ ∞) : |
| Integrable (fun z => (terminalKL P Q z).toReal + entropySlope n/2*(depth P Q z:ℝ)) P ∧ |
| (∫ z,(terminalKL P Q z).toReal + entropySlope n/2*(depth P Q z:ℝ) ∂P) ≤ |
| (InformationTheory.klDiv P (uniformCell Q)).toReal + 2/Real.exp 1 := by |
| have hn0 : 0 < n := by omega |
| have hs : 0 ≤ entropySlope n/2 := le_of_lt (div_pos (entropySlope_pos hn0) (by norm_num)) |
| apply frozen_nonnegative_bound P |
| (fun N z => (historyKL P Q (run P Q N z)).toReal + entropySlope n/2*(truncatedDepth P Q N z:ℝ)) |
| _ _ (budget_nonneg P Q) |
| · intro N |
| exact (truncated_KL_integrable hn0 P Q hac hK N).add |
| ((truncated_depth_integrable hn0 P Q hac hK N).const_mul _) |
| · intro N z |
| exact add_nonneg ENNReal.toReal_nonneg (mul_nonneg hs (Nat.cast_nonneg _)) |
| · exact ((measurable_terminalKL hn0 P Q).ennreal_toReal.add |
| ((measurable_depth_real hn0 P Q).const_mul _)).aestronglyMeasurable |
| · intro z |
| exact add_nonneg ENNReal.toReal_nonneg (mul_nonneg hs (Nat.cast_nonneg _)) |
| · have hroot := ae_mem_cell_of_ac hac |
| filter_upwards [hroot,almost_sure_termination hn P Q hac hK] with z hz ht |
| refine ⟨attempts P Q z,fun M hM => ?_⟩ |
| have he := run_eventually_terminal P Q ht M hM |
| simp only [truncatedDepth,he,← terminal_historyKL hn0 P Q hroot hz ht,depth,terminalCell] |
| · exact truncated_final_budget hn P Q hac hK |
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| lemma terminal_depth_integrable {n : ℕ} (hn : 32 ≤ n) (P : Measure (Coordinate n)) [IsProbabilityMeasure P] |
| (Q : Cell n) (hac : P ≪ uniformCell Q) (hK : InformationTheory.klDiv P (uniformCell Q) ≠ ∞) : |
| Integrable (fun z => (depth P Q z:ℝ)) P := by |
| have hn0 : 0 < n := by omega |
| have hc := (terminal_cost_budget hn P Q hac hK).1 |
| apply Integrable.mono' (hc.div_const (entropySlope n/2)) (measurable_depth_real hn0 P Q).aestronglyMeasurable |
| filter_upwards [] with z |
| rw [Real.norm_eq_abs,abs_of_nonneg (Nat.cast_nonneg _),le_div_iff₀ (div_pos (entropySlope_pos hn0) (by norm_num))] |
| nlinarith [ENNReal.toReal_nonneg (a := terminalKL P Q z)] |
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| lemma terminal_KL_integrable {n : ℕ} (hn : 32 ≤ n) (P : Measure (Coordinate n)) [IsProbabilityMeasure P] |
| (Q : Cell n) (hac : P ≪ uniformCell Q) (hK : InformationTheory.klDiv P (uniformCell Q) ≠ ∞) : |
| Integrable (fun z => (terminalKL P Q z).toReal) P := by |
| have hc := (terminal_cost_budget hn P Q hac hK).1 |
| have hd := terminal_depth_integrable hn P Q hac hK |
| simpa only [add_sub_cancel_right] using |
| hc.sub' (hd.const_mul (entropySlope n/2)) |
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| lemma terminal_attempts_budget {n : ℕ} (hn : 32 ≤ n) (P : Measure (Coordinate n)) [IsProbabilityMeasure P] |
| (Q : Cell n) (hac : P ≪ uniformCell Q) (hK : InformationTheory.klDiv P (uniformCell Q) ≠ ∞) : |
| Integrable (fun z => (attempts P Q z:ℝ)) P ∧ |
| (∫ z,(attempts P Q z:ℝ) ∂P) ≤ 2*(1+∫ z,(depth P Q z:ℝ) ∂P) := by |
| have hn0 : 0 < n := by omega |
| have hroot := ae_mem_cell_of_ac hac |
| have hterm := almost_sure_termination hn P Q hac hK |
| have hd := terminal_depth_integrable hn P Q hac hK |
| apply frozen_nonnegative_bound P (fun N z => (truncatedAttempts P Q N z:ℝ)) _ _ |
| · positivity |
| · exact integrable_truncatedAttempts hn0 P Q |
| · intro N z |
| exact Nat.cast_nonneg _ |
| · exact ((measurable_of_countable (fun a : ℕ => (a:ℝ))).comp (measurable_attempts hn0 P Q)).aestronglyMeasurable |
| · intro z |
| exact Nat.cast_nonneg _ |
| · filter_upwards [hterm] with z ht |
| refine ⟨attempts P Q z,fun M hM => ?_⟩ |
| simp only [truncatedAttempts,run_eventually_terminal P Q ht M hM,terminalLog_length P Q ht] |
| · intro N |
| have he : ∀ᵐ z ∂P,(truncatedDepth P Q N z:ℝ) ≤ (depth P Q z:ℝ) := by |
| filter_upwards [hroot,hterm] with z hz ht |
| have hm := truncatedDepth_mono hn0 P Q hz (Nat.le_max_left N (attempts P Q z)) |
| have hf := run_eventually_terminal P Q ht (max N (attempts P Q z)) (Nat.le_max_right _ _) |
| simp only [truncatedDepth,hf] at hm |
| exact_mod_cast hm |
| have hi := integral_mono_ae (truncated_depth_integrable hn0 P Q hac hK N) hd he |
| exact (truncated_attempts_integral hn0 P Q hac hK N).trans (by linarith) |
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| lemma terminal_ennreal_budget {n : ℕ} (hn : 32 ≤ n) (P : Measure (Coordinate n)) [IsProbabilityMeasure P] |
| (Q : Cell n) (hac : P ≪ uniformCell Q) (hK : InformationTheory.klDiv P (uniformCell Q) ≠ ∞) : |
| (∫⁻ z,terminalKL P Q z ∂P) + ENNReal.ofReal ((Real.log 2/4)*n*(∫ z,(depth P Q z:ℝ) ∂P)) ≤ |
| InformationTheory.klDiv P (uniformCell Q) + ENNReal.ofReal (2/Real.exp 1) := by |
| have hd := terminal_depth_integrable hn P Q hac hK |
| have hk := terminal_KL_integrable hn P Q hac hK |
| have hc := (terminal_cost_budget hn P Q hac hK).2 |
| rw [integral_add hk (hd.const_mul _),integral_const_mul] at hc |
| have hfin : ∀ᵐ z ∂P,terminalKL P Q z ≠ ∞ := by |
| filter_upwards [ae_fiber_positive P (terminalLog P Q)] with z hz |
| exact (terminal_regular hn P Q hac hK (terminalLog P Q z) hz).2.2 |
| have he : (∫⁻ z,terminalKL P Q z ∂P) = ENNReal.ofReal (∫ z,(terminalKL P Q z).toReal ∂P) := by |
| rw [ofReal_integral_eq_lintegral_ofReal hk (ae_of_all _ (fun _ => ENNReal.toReal_nonneg))] |
| exact lintegral_congr_ae (hfin.mono (fun z hz => (ENNReal.ofReal_toReal hz).symm)) |
| have hs : (Real.log 2/4)*(n:ℝ) = entropySlope n/2 := by unfold entropySlope; ring |
| rw [he,hs,← ENNReal.ofReal_add (integral_nonneg (fun _ => ENNReal.toReal_nonneg))] |
| calc |
| _ ≤ ENNReal.ofReal ((InformationTheory.klDiv P (uniformCell Q)).toReal+2/Real.exp 1) := ENNReal.ofReal_le_ofReal hc |
| _ = _ := by |
| rw [ENNReal.ofReal_add ENNReal.toReal_nonneg,ENNReal.ofReal_toReal hK] |
| positivity |
| · exact mul_nonneg (le_of_lt (div_pos (entropySlope_pos (by omega)) (by norm_num))) |
| (integral_nonneg (fun _ => Nat.cast_nonneg _)) |
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| end FiniteEntropy |
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| end RepeatedLocalization |
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| end OAI |
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