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dd14c26 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 | import OAI.Probability.Localization.RewardBounds
namespace OAI
open MeasureTheory ProbabilityTheory Set
open scoped BigOperators ENNReal
namespace RepeatedLocalization
namespace FiniteEntropy
lemma rewardLog_run_succ {n : ℕ} (P : Measure (Coordinate n)) (Q : Cell n)
(r : State n → Letter n → ℝ) (hr : ∀ s a,s.stopped = true → r s a = 0)
(N : ℕ) (z : Coordinate n) :
rewardLog P Q r (run P Q (N+1) z) = rewardLog P Q r (run P Q N z) + rewardStep P Q r N z := by
by_cases hs : (state P Q (run P Q N z)).stopped = true
· simp only [run,hs,ite_true,rewardStep,hr _ _ hs,add_zero]
· simp only [run,hs,Bool.false_eq_true,ite_false,rewardLog_append,rewardStep]
lemma complementTotal_succ {n : ℕ} (P : Measure (Coordinate n)) (Q : Cell n) (N : ℕ) (z : Coordinate n) :
complementTotal P Q (N+1) z = complementTotal P Q N z + rewardStep P Q (complementReward P) N z :=
rewardLog_run_succ P Q _ (fun s a hs => by simp [complementReward,hs]) N z
lemma continuingTotal_succ {n : ℕ} (P : Measure (Coordinate n)) (Q : Cell n) (N : ℕ) (z : Coordinate n) :
continuingTotal P Q (N+1) z = continuingTotal P Q N z + rewardStep P Q (continuingReward P) N z :=
rewardLog_run_succ P Q _ (fun s a hs => by simp [continuingReward,hs]) N z
lemma truncatedAttempts_succ {n : ℕ} (P : Measure (Coordinate n)) (Q : Cell n) (N : ℕ) (z : Coordinate n) :
(truncatedAttempts P Q (N+1) z : ℝ) = truncatedAttempts P Q N z + activeIndicator P Q N z := by
by_cases hs : (state P Q (run P Q N z)).stopped = true
· simp [truncatedAttempts,activeIndicator,run,hs]
· simp [truncatedAttempts,activeIndicator,run,hs,List.length_append]
lemma measurable_truncatedAttempts {n : ℕ} (hn : 0 < n) (P : Measure (Coordinate n)) (Q : Cell n) (N : ℕ) :
Measurable (fun z => (truncatedAttempts P Q N z : ℝ)) :=
(measurable_of_countable (fun h : Log n => (h.length:ℝ))).comp (measurable_run hn P Q N)
lemma integrable_truncatedAttempts {n : ℕ} (hn : 0 < n) (P : Measure (Coordinate n))
[IsFiniteMeasure P] (Q : Cell n) (N : ℕ) : Integrable (fun z => (truncatedAttempts P Q N z : ℝ)) P := by
apply Integrable.mono' (integrable_const (N:ℝ)) (measurable_truncatedAttempts hn P Q N).aestronglyMeasurable
exact ae_of_all _ (fun z => by
rw [Real.norm_eq_abs,abs_of_nonneg (Nat.cast_nonneg _)]
exact_mod_cast run_length_le P Q N z)
lemma truncated_reward_bounds {n : ℕ} (hn : 0 < n) (P : Measure (Coordinate n))
[IsFiniteMeasure P] (Q : Cell n) (hroot : ∀ᵐ z ∂P,z ∈ cellSet Q) (N : ℕ) :
Integrable (complementTotal P Q N) P ∧ Integrable (continuingTotal P Q N) P ∧
(∫ z,complementTotal P Q N z ∂P) ≤ (1/Real.exp 1)*(∫ z,(truncatedAttempts P Q N z : ℝ) ∂P) ∧
(∫ z,continuingTotal P Q N z ∂P) ≤ (1/2:ℝ)*(∫ z,(truncatedAttempts P Q N z : ℝ) ∂P) := by
induction N with
| zero =>
have hJ : complementTotal P Q 0 = fun _ => 0 := rfl
have hC : continuingTotal P Q 0 = fun _ => 0 := rfl
simp [hJ,hC,truncatedAttempts,run]
| succ N ih =>
have hJ := complementStep_bound hn P Q hroot N
have hC := continuingStep_bound hn P Q hroot N
have eqJ : complementTotal P Q (N+1) = complementTotal P Q N + rewardStep P Q (complementReward P) N :=
funext (complementTotal_succ P Q N)
have eqC : continuingTotal P Q (N+1) = continuingTotal P Q N + rewardStep P Q (continuingReward P) N :=
funext (continuingTotal_succ P Q N)
rw [eqJ,eqC]
refine ⟨ih.1.add hJ.1,ih.2.1.add hC.1,?_,?_⟩
· simp only [Pi.add_apply]
simp_rw [truncatedAttempts_succ]
rw [integral_add ih.1 hJ.1,integral_add (integrable_truncatedAttempts hn P Q N) (integrable_activeIndicator P Q hn N)]
nlinarith [ih.2.2.1,hJ.2]
· simp only [Pi.add_apply]
simp_rw [truncatedAttempts_succ]
rw [integral_add ih.2.1 hC.1,integral_add (integrable_truncatedAttempts hn P Q N) (integrable_activeIndicator P Q hn N)]
nlinarith [ih.2.2.2,hC.2]
lemma truncated_attempts_path {n : ℕ} (hn : 0 < n) (P : Measure (Coordinate n)) (Q : Cell n)
(N : ℕ) {z : Coordinate n} (hz : z ∈ cellSet Q) :
(truncatedAttempts P Q N z : ℝ) ≤ truncatedDepth P Q N z + continuingTotal P Q N z +
(if (state P Q (run P Q N z)).stopped then (1:ℝ) else 0) := by
induction N with
| zero => simp [truncatedAttempts,truncatedDepth,continuingTotal,rewardLog,run,state_nil,accumulate]
| succ N ih =>
have hzE := run_mem_event P Q N hz
have hdesc := state_descendant_of_mem P Q (run P Q N z) hzE
by_cases hs : (state P Q (run P Q N z)).stopped = true
· simpa only [truncatedAttempts,truncatedDepth,continuingTotal,run,hs,ite_true] using ih
let s := state P Q (run P Q N z)
let a := attemptLetter (ProbabilityTheory.cond P s.event) s.cell z
have hs' : s.stopped = false := Bool.eq_false_iff.mpr hs
have erun : run P Q (N+1) z = run P Q N z ++ [a] := by simp only [run,hs,Bool.false_eq_true,ite_false]; rfl
have eqA := truncatedAttempts_succ P Q N z
have eqC := continuingTotal_succ P Q N z
have hih : (truncatedAttempts P Q N z : ℝ) ≤ truncatedDepth P Q N z + continuingTotal P Q N z := by
simpa only [hs,Bool.false_eq_true,ite_false,add_zero] using ih
rw [eqA,eqC]
simp only [activeIndicator,hs,Bool.false_eq_true,ite_false,rewardStep]
change _ ≤ truncatedDepth P Q (N+1) z + (continuingTotal P Q N z + continuingReward P s a) +
(if (state P Q (run P Q (N+1) z)).stopped then 1 else 0)
rw [erun,state_append]
change _ ≤ truncatedDepth P Q (N+1) z + (continuingTotal P Q N z + continuingReward P s a) +
(if (update P s a).stopped then 1 else 0)
cases ha : a with
| none =>
have eqD : truncatedDepth P Q (N+1) z = truncatedDepth P Q N z := by
simp only [truncatedDepth,erun,state_append,ha,update,hs,Bool.false_eq_true,ite_false,Option.getD_none]
rw [eqD]
have hstop : (update P s none).stopped = decide ((1/2:ℝ) ≤ complementMass P s) := by
simp [update,hs',branch,complementMass]
rw [hstop]
by_cases hq : complementMass P s < 1/2
· have hq' : ¬(1/2:ℝ) ≤ complementMass P s := not_le.mpr hq
simp only [continuingReward,hs',Bool.false_eq_true,ite_false,true_and,ite_eq_left hq,
decide_eq_true_eq,ite_eq_right hq',add_zero]
linarith
· have hq' : (1/2:ℝ) ≤ complementMass P s := le_of_not_gt hq
simp only [continuingReward,hs',Bool.false_eq_true,ite_false,true_and,ite_eq_right hq,
decide_eq_true_eq,ite_eq_left hq',add_zero]
linarith
| some R =>
have hR : FirstHeavy (ProbabilityTheory.cond P s.event) s.cell R :=
((attemptLetter_some_iff hn _ _ R z).mp ha).1
have eqD : truncatedDepth P Q (N+1) z = truncatedDepth P Q N z + (R.1-s.cell.1) := by
simp only [truncatedDepth,erun,state_append,ha,update,hs,Bool.false_eq_true,ite_false,Option.getD_some]
have hbase : Q.1 ≤ s.cell.1 := hdesc.1
have hnext : s.cell.1 ≤ R.1 := hR.1.1.1
change R.1-Q.1 = s.cell.1-Q.1+(R.1-s.cell.1)
omega
rw [eqD,Nat.cast_add]
simp only [update,hs',Bool.false_eq_true,ite_false,reduceCtorEq,continuingReward,false_and,add_zero]
have hd : (1:ℝ) ≤ (R.1-s.cell.1 : ℕ) := by exact_mod_cast Nat.sub_pos_of_lt hR.1.2.1
linarith
end FiniteEntropy
end RepeatedLocalization
end OAI
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