| import OAI.Probability.Localization.Rewards |
| import OAI.Probability.Localization.CountableIntegrals |
| import OAI.Probability.Localization.EntropyChain |
|
|
| namespace OAI |
|
|
| open MeasureTheory ProbabilityTheory Set |
| open scoped BigOperators ENNReal |
|
|
| namespace RepeatedLocalization |
|
|
| namespace FiniteEntropy |
|
|
| noncomputable def rewardStep {n : ℕ} (P : Measure (Coordinate n)) (Q : Cell n) |
| (r : State n → Letter n → ℝ) (N : ℕ) (z : Coordinate n) : ℝ := |
| let s := state P Q (run P Q N z) |
| r s (attemptLetter (ProbabilityTheory.cond P s.event) s.cell z) |
|
|
| lemma reward_only_none {n : ℕ} (r : State n → Letter n → ℝ) |
| (hr : ∀ s R,r s (some R) = 0) (s : State n) (a : Letter n) : |
| r s a = if a = none then r s none else 0 := by |
| cases a <;> simp [hr] |
|
|
| lemma integrable_fixed_reward {n : ℕ} (hn : 0 < n) (P : Measure (Coordinate n)) |
| [IsFiniteMeasure P] (r : State n → Letter n → ℝ) (hr : ∀ s R,r s (some R) = 0) |
| (s : State n) : |
| Integrable (fun z => r s (attemptLetter (ProbabilityTheory.cond P s.event) s.cell z)) P := by |
| classical |
| have he : (fun z => r s (attemptLetter (ProbabilityTheory.cond P s.event) s.cell z)) = |
| {z | attemptLetter (ProbabilityTheory.cond P s.event) s.cell z = none}.indicator (fun _ => r s none) := by |
| funext z |
| rw [reward_only_none r hr,Set.indicator_apply] |
| rfl |
| rw [he] |
| exact (integrable_const _).indicator ((measurable_attemptLetter hn _ _) (measurableSet_singleton none)) |
|
|
| lemma integral_fixed_reward_event {n : ℕ} (hn : 0 < n) (P : Measure (Coordinate n)) |
| [IsFiniteMeasure P] (r : State n → Letter n → ℝ) (hr : ∀ s R,r s (some R) = 0) |
| (s : State n) (hmeas : MeasurableSet s.event) (hsub : s.event ⊆ cellSet s.cell) |
| (hp : P s.event ≠ 0) : |
| (∫ z in s.event,r s (attemptLetter (ProbabilityTheory.cond P s.event) s.cell z) ∂P) = |
| (P s.event).toReal * (complementMass P s * r s none) := by |
| classical |
| let B := residual (ProbabilityTheory.cond P s.event) s.cell |
| have hB : MeasurableSet B := measurableSet_residual _ _ |
| have he : ∀ z ∈ s.event,r s (attemptLetter (ProbabilityTheory.cond P s.event) s.cell z) = |
| B.indicator (fun _ => r s none) z := by |
| intro z hz |
| rw [reward_only_none r hr,Set.indicator_apply] |
| have hh : attemptLetter (ProbabilityTheory.cond P s.event) s.cell z = none ↔ z ∈ B := |
| attemptLetter_eq_iff hn _ _ (hsub hz) none |
| exact if_congr hh rfl rfl |
| rw [setIntegral_congr_fun hmeas he,integral_indicator hB,Measure.restrict_restrict hB, |
| Set.inter_comm B s.event,integral_const,Measure.real,Measure.restrict_apply_univ,smul_eq_mul, |
| cond_event_mul P hmeas hp] |
| exact mul_assoc _ _ _ |
|
|
| lemma exists_run_fiber_root {n : ℕ} (P : Measure (Coordinate n)) (Q : Cell n) |
| (hroot : ∀ᵐ z ∂P,z ∈ cellSet Q) (N : ℕ) (h : Log n) |
| (hp : P {z | run P Q N z = h} ≠ 0) : |
| ∃ z,z ∈ cellSet Q ∧ run P Q N z = h := by |
| have he : ({z | run P Q N z = h} ∩ cellSet Q : Set (Coordinate n)) =ᵐ[P] {z | run P Q N z = h} := by |
| filter_upwards [hroot] with z hz |
| change (run P Q N z = h ∧ z ∈ cellSet Q) = (run P Q N z = h) |
| simp only [hz,and_true] |
| have hp' : P ({z | run P Q N z = h} ∩ cellSet Q) ≠ 0 := by |
| rw [measure_congr he]; exact hp |
| obtain ⟨z,hz⟩ := nonempty_of_measure_ne_zero hp' |
| exact ⟨z,hz.2,hz.1⟩ |
|
|
| lemma rewardStep_fiber_integrable {n : ℕ} (hn : 0 < n) (P : Measure (Coordinate n)) |
| [IsFiniteMeasure P] (Q : Cell n) (r : State n → Letter n → ℝ) |
| (hr : ∀ s R,r s (some R) = 0) (N : ℕ) (h : Log n) : |
| IntegrableOn (rewardStep P Q r N) {z | run P Q N z = h} P := by |
| apply (integrable_fixed_reward hn P r hr (state P Q h)).integrableOn.congr_fun |
| (s := {z | run P Q N z = h}) |
| · intro z hz |
| change run P Q N z = h at hz |
| simp only [rewardStep,hz] |
| · exact (measurable_run hn P Q N) (measurableSet_singleton h) |
|
|
| lemma rewardStep_fiber_integral {n : ℕ} (hn : 0 < n) (P : Measure (Coordinate n)) |
| [IsFiniteMeasure P] (Q : Cell n) (hroot : ∀ᵐ z ∂P,z ∈ cellSet Q) |
| (r : State n → Letter n → ℝ) (hr : ∀ s R,r s (some R) = 0) (N : ℕ) (h : Log n) : |
| (∫ z in {z | run P Q N z = h},rewardStep P Q r N z ∂P) = |
| (P {z | run P Q N z = h}).toReal * (complementMass P (state P Q h) * r (state P Q h) none) := by |
| by_cases hp : P {z | run P Q N z = h} = 0 |
| · rw [Measure.restrict_eq_zero.mpr hp,hp,integral_zero_measure,ENNReal.toReal_zero,zero_mul] |
| obtain ⟨z,hz,hzh⟩ := exists_run_fiber_root P Q hroot N h hp |
| have he : {w | run P Q N w = h} =ᵐ[P] (state P Q h).event := by |
| simpa only [hzh] using run_fiber_ae_event hn P Q hroot N hz |
| have hpe : P (state P Q h).event ≠ 0 := by rwa [← measure_congr he] |
| have hm : MeasurableSet {z | run P Q N z = h} := (measurable_run hn P Q N) (measurableSet_singleton h) |
| rw [setIntegral_congr_fun hm |
| (g := fun z => r (state P Q h) (attemptLetter (ProbabilityTheory.cond P (state P Q h).event) (state P Q h).cell z)) |
| (fun w hw => by change run P Q N w = h at hw; simp only [rewardStep,hw])] |
| rw [Measure.restrict_congr_set he,measure_congr he] |
| exact integral_fixed_reward_event hn P r hr _ (state_event_measurable P Q h) |
| (state_event_subset_cell P Q h) hpe |
|
|
| lemma integrable_activeIndicator {n : ℕ} (P : Measure (Coordinate n)) [IsFiniteMeasure P] |
| (Q : Cell n) (hn : 0 < n) (N : ℕ) : Integrable (activeIndicator P Q N) P := by |
| have he : activeIndicator P Q N = {z | (state P Q (run P Q N z)).stopped = false}.indicator (fun _ => (1:ℝ)) := by |
| funext z |
| simp only [activeIndicator,Set.indicator_apply,Set.mem_ofPred_eq] |
| by_cases hs : (state P Q (run P Q N z)).stopped = true |
| · simp [hs] |
| · simp [Bool.eq_false_iff.mpr hs] |
| rw [he] |
| exact (integrable_const _).indicator |
| (((measurable_of_countable (fun h : Log n => (state P Q h).stopped)).comp |
| (measurable_run hn P Q N)) (measurableSet_singleton false)) |
|
|
| lemma rewardStep_integral_bound {n : ℕ} (hn : 0 < n) (P : Measure (Coordinate n)) |
| [IsFiniteMeasure P] (Q : Cell n) (hroot : ∀ᵐ z ∂P,z ∈ cellSet Q) |
| (r : State n → Letter n → ℝ) (hr0 : ∀ s a,0 ≤ r s a) |
| (hr : ∀ s R,r s (some R) = 0) (b : ℝ) |
| (hb : ∀ s, complementMass P s * r s none ≤ b*(if s.stopped then 0 else 1)) (N : ℕ) : |
| Integrable (rewardStep P Q r N) P ∧ |
| (∫ z,rewardStep P Q r N z ∂P) ≤ b*(∫ z,activeIndicator P Q N z ∂P) := by |
| have hb' (h : Log n) : (∫ z in {z | run P Q N z = h},rewardStep P Q r N z ∂P) ≤ |
| ∫ z in {z | run P Q N z = h},b*activeIndicator P Q N z ∂P := by |
| rw [rewardStep_fiber_integral hn P Q hroot r hr N h] |
| change _ ≤ ∫ z in {z | run P Q N z = h}, |
| (fun h : Log n => b*(if (state P Q h).stopped then 0 else 1)) (run P Q N z) ∂P |
| rw [integral_comp_fiber P (run P Q N) |
| (fun h : Log n => b*(if (state P Q h).stopped then 0 else 1)) h |
| ((measurable_run hn P Q N) (measurableSet_singleton h))] |
| exact mul_le_mul_of_nonneg_left (hb _) ENNReal.toReal_nonneg |
| have hi := partition_integral_bound P (run P Q N) (measurable_run hn P Q N) |
| (fun z => hr0 _ _) (rewardStep_fiber_integrable hn P Q r hr N) |
| ((integrable_activeIndicator P Q hn N).const_mul b) hb' |
| rw [integral_const_mul] at hi |
| exact hi |
|
|
| lemma complementReward_some {n : ℕ} (P : Measure (Coordinate n)) (s : State n) (R : Cell n) : |
| complementReward P s (some R) = 0 := by simp [complementReward] |
|
|
| lemma continuingReward_some {n : ℕ} (P : Measure (Coordinate n)) (s : State n) (R : Cell n) : |
| continuingReward P s (some R) = 0 := by simp [continuingReward] |
|
|
| lemma complementStep_bound {n : ℕ} (hn : 0 < n) (P : Measure (Coordinate n)) |
| [IsFiniteMeasure P] (Q : Cell n) (hroot : ∀ᵐ z ∂P,z ∈ cellSet Q) (N : ℕ) : |
| Integrable (rewardStep P Q (complementReward P) N) P ∧ |
| (∫ z,rewardStep P Q (complementReward P) N z ∂P) ≤ |
| (1/Real.exp 1)*(∫ z,activeIndicator P Q N z ∂P) := by |
| apply rewardStep_integral_bound hn P Q hroot _ (complementReward_nonneg P) (complementReward_some P) |
| intro s |
| by_cases hs : s.stopped = true |
| · simp [complementReward,hs] |
| · simpa [complementReward,hs] using neg_log_mul_le (complementMass_nonneg P s) |
|
|
| lemma continuingStep_bound {n : ℕ} (hn : 0 < n) (P : Measure (Coordinate n)) |
| [IsFiniteMeasure P] (Q : Cell n) (hroot : ∀ᵐ z ∂P,z ∈ cellSet Q) (N : ℕ) : |
| Integrable (rewardStep P Q (continuingReward P) N) P ∧ |
| (∫ z,rewardStep P Q (continuingReward P) N z ∂P) ≤ |
| (1/2:ℝ)*(∫ z,activeIndicator P Q N z ∂P) := by |
| apply rewardStep_integral_bound hn P Q hroot _ (continuingReward_nonneg P) (continuingReward_some P) |
| intro s |
| by_cases hs : s.stopped = true |
| · simp [continuingReward,hs] |
| · by_cases hq : complementMass P s < 1/2 |
| · simpa only [continuingReward,hs,Bool.false_eq_true,ite_false,true_and,ite_eq_left hq,mul_one] using hq.le |
| · simp only [continuingReward,hs,Bool.false_eq_true,ite_false,true_and,ite_eq_right hq,mul_zero,mul_one] |
| norm_num |
|
|
| end FiniteEntropy |
|
|
| end RepeatedLocalization |
|
|
| end OAI |
|
|
