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f205175 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 | import Mathlib
import OAI.Probability.CriticalPercolation.Exploration.QueryDelta
import OAI.Probability.CriticalPercolation.ProductMeasure.ExpectClosedFlip
namespace OAI
section
section
open Set MeasureTheory ProbabilityTheory
open Set
open scoped BigOperators
open scoped Classical
namespace CriticalPercolation.SearchTree
open scoped Classical
open FiniteProduct LazyDFS
variable {E R : Type*} [Fintype E] [DecidableEq E]
noncomputable def closedPivotWeight (T : SearchTree E R) (p : ℝ) (F : Set E → Prop)
(w : E → Prop → ℝ) (ω : Set E) (e : E) : ℝ :=
if ¬ω e ∧ openPivot F e (Function.update ω e True) then p*T.queryDelta F w ω e else 0
theorem contact_flip_identity (T : SearchTree E R) (used : Finset E) (hT : T.Valid used)
{p : ℝ} (hp : p ∈ Set.Ioo 0 1) (F : Set E → Prop) (w : E → Prop → ℝ)
(hw : ∀ e, ∑ b, w e b = 1) (h0 : ∀ e, w e False = 1-p) (h1 : ∀ e, w e True = p)
(ψ : ℝ → ℝ) :
expect w (fun ω => ∑ e, T.closedPivotWeight p F w ω e *
ψ (T.score p F w (Function.update ω e True))) =
expect w (fun ω => T.score p F w ω * ψ (T.score p F w ω)) := by
have hsingle (e : E) :
expect w (fun ω => T.closedPivotWeight p F w ω e *
ψ (T.score p F w (Function.update ω e True))) =
expect w (fun ω => (if openPivot F e ω then (1-p)*T.queryDelta F w ω e else 0) *
ψ (T.score p F w ω)) := by
have h := expect_closed_flip_decidable w hw e hp (h0 e) (h1 e)
(fun ω => (if openPivot F e ω then T.queryDelta F w ω e else 0) * ψ (T.score p F w ω))
convert h using 1
· congr 1
funext ω
let σ : Set E := ω
change T.closedPivotWeight p F w σ e * ψ (T.score p F w (Function.update σ e True)) =
if ¬σ e then p*((if openPivot F e (Function.update σ e True) then
T.queryDelta F w (Function.update σ e True) e else 0) *
ψ (T.score p F w (Function.update σ e True))) else 0
rw [queryDelta_update T F w σ e True]
simp only [closedPivotWeight]
by_cases hc : σ e <;>
by_cases hpiv : openPivot F e (Function.update σ e True) <;> simp [hc,hpiv,mul_assoc]
· congr 1
funext ω
by_cases hpiv : openPivot F e ω
· simp only [ite_eq_left hpiv,ite_eq_left hpiv.1]; ring
· simp only [ite_eq_right hpiv,mul_zero,zero_mul,ite_self]
rw [expect_sum]
simp_rw [hsingle]
rw [← expect_sum]
congr 1
funext ω
rw [← Finset.sum_mul]
exact congrArg (fun x => x * ψ (T.score p F w ω))
(score_eq_queryDelta_sum T used hT p F w ω).symm
end CriticalPercolation.SearchTree
namespace CriticalPercolation.SearchTree
open scoped Classical
open FiniteProduct LazyDFS
variable {E R : Type*} [Fintype E] [DecidableEq E]
noncomputable def prefixScore (T : SearchTree E R) (p : ℝ) (F : Set E → Prop)
(w : E → Prop → ℝ) (ω : Set E) (e : E) : ℝ :=
match T with
| .done _ => 0
| .ask f l r => (if openPivot F f ω then (1-p)*delta w F f else 0) +
if e = f then 0 else
if ω f then r.prefixScore p F (pin w f True) ω e
else l.prefixScore p F (pin w f False) ω e
lemma prefixScore_le (T : SearchTree E R) (used : Finset E) (hT : T.Valid used)
{p : ℝ} (hp : p ∈ Set.Ioo 0 1) (F : Set E → Prop)
(hF : ∀ s t, s ⊆ t → F s → F t) (w : E → Prop → ℝ)
(hw : ∀ e, ∑ b, w e b = 1) (hn : ∀ e b, 0 ≤ w e b)
(hf : Fresh w p used) (ω : Set E) (e : E) :
T.prefixScore p F w ω e ≤ T.score p F w ω := by
induction T generalizing used w with
| done _ => exact le_rfl
| ask f l r hl hr =>
simp only [prefixScore,score]
apply add_le_add_right
by_cases he : e = f
· rw [ite_eq_left he]
by_cases ho : ω f
· rw [ite_eq_left ho]
exact score_nonneg r _ hT.2.2 hp F hF _ (pin_normalized w hw f True)
(pin_nonneg w hn f True) (fresh_pin w p used hf f True) ω
· rw [ite_eq_right ho]
exact score_nonneg l _ hT.2.1 hp F hF _ (pin_normalized w hw f False)
(pin_nonneg w hn f False) (fresh_pin w p used hf f False) ω
· rw [ite_eq_right he]
by_cases ho : ω f
· rw [ite_eq_left ho,ite_eq_left ho]
exact hr _ hT.2.2 _ (pin_normalized w hw f True) (pin_nonneg w hn f True)
(fresh_pin w p used hf f True)
· rw [ite_eq_right ho,ite_eq_right ho]
exact hl _ hT.2.1 _ (pin_normalized w hw f False) (pin_nonneg w hn f False)
(fresh_pin w p used hf f False)
lemma closedPivotWeight_nonneg (T : SearchTree E R) (used : Finset E) (hT : T.Valid used)
{p : ℝ} (hp : p ∈ Set.Ioo 0 1) (F : Set E → Prop)
(hF : ∀ s t, s ⊆ t → F s → F t) (w : E → Prop → ℝ)
(hw : ∀ e, ∑ b, w e b = 1) (hn : ∀ e b, 0 ≤ w e b)
(hf : Fresh w p used) (ω : Set E) (e : E) : 0 ≤ T.closedPivotWeight p F w ω e := by
unfold closedPivotWeight
split_ifs
· exact mul_nonneg hp.1.le (queryDelta_nonneg T used hT hp F hF w hw hn hf ω e)
· exact le_rfl
theorem marked_prefix_bound (T : SearchTree E R) (used : Finset E) (hT : T.Valid used)
{p : ℝ} (hp : p ∈ Set.Ioo 0 1) (F : Set E → Prop)
(hF : ∀ s t, s ⊆ t → F s → F t) (w : E → Prop → ℝ)
(hw : ∀ e, ∑ b, w e b = 1) (hn : ∀ e b, 0 ≤ w e b)
(hf : Fresh w p used) (h0 : ∀ e, w e False = 1-p) (h1 : ∀ e, w e True = p) :
expect w (fun ω => ∑ e, T.closedPivotWeight p F w ω e *
(1 + T.prefixScore p F w (Function.update ω e True) e)) ≤
expect w (T.score p F w) + expect w (fun ω => (T.score p F w ω)^2) := by
calc
_ ≤ expect w (fun ω => ∑ e, T.closedPivotWeight p F w ω e *
(1 + T.score p F w (Function.update ω e True))) := by
apply expect_mono hn
intro ω
apply Finset.sum_le_sum
intro e _
exact mul_le_mul_of_nonneg_left
(add_le_add_right (prefixScore_le T used hT hp F hF w hw hn hf _ e) 1)
(closedPivotWeight_nonneg T used hT hp F hF w hw hn hf ω e)
_ = expect w (fun ω => T.score p F w ω * (1 + T.score p F w ω)) :=
contact_flip_identity T used hT hp F w hw h0 h1 (fun x => 1+x)
_ = expect w (fun ω => T.score p F w ω + (T.score p F w ω)^2) := by
congr 1
funext ω
ring
_ = _ := expect_add w _ _
end CriticalPercolation.SearchTree
end
end
end OAI
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