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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Exploration.ClosedPivotWeight
import OAI.Probability.CriticalPercolation.Modular.Mass
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 reverseGuess (T : SearchTree E R) (p : ℝ) (F : Set E → Prop)
(w : E → Prop → ℝ) (ω : Set E) (s : ℝ) (e : E) : ℝ :=
(if openPivot F e ω then (1-p)*T.queryDelta F w ω e else 0) /
(1+s+T.prefixScore p F w ω e)^2
noncomputable def reverseMass (T : SearchTree E R) (p : ℝ) (F : Set E → Prop)
(w : E → Prop → ℝ) (ω : Set E) (s : ℝ) : ℝ :=
match T with
| .done _ => 0
| .ask f l r =>
let b := if openPivot F f ω then (1-p)*delta w F f else 0
b/(1+s+b)^2 + if ω f then r.reverseMass p F (pin w f True) ω (s+b)
else l.reverseMass p F (pin w f False) ω (s+b)
lemma reverseMass_eq_sum (T : SearchTree E R) (used : Finset E) (hT : T.Valid used)
(p : ℝ) (F : Set E → Prop) (w : E → Prop → ℝ) (ω : Set E) (s : ℝ) :
T.reverseMass p F w ω s = ∑ e, T.reverseGuess p F w ω s e := by
induction T generalizing used w s with
| done _ => simp [reverseMass,reverseGuess,queryDelta,prefixScore]
| ask f l r hl hr =>
let b := if openPivot F f ω then (1-p)*delta w F f else 0
have hsum (T' : SearchTree E R) (w' : E → Prop → ℝ) (hT' : T'.Valid (insert f used)) :
(∑ e, (if openPivot F e ω then
(1-p)*(if e = f then delta w F f else T'.queryDelta F w' ω e) else 0) /
(1+s+(b+if e=f then 0 else T'.prefixScore p F w' ω e))^2) =
b/(1+s+b)^2 + ∑ e, T'.reverseGuess p F w' ω (s+b) e := by
have hz := queryDelta_zero_used T' _ hT' F w' ω f (Finset.mem_insert_self _ _)
have heq (e : E) : (if openPivot F e ω then
(1-p)*(if e = f then delta w F f else T'.queryDelta F w' ω e) else 0) /
(1+s+(b+if e=f then 0 else T'.prefixScore p F w' ω e))^2 =
(if e=f then b/(1+s+b)^2 else 0) + T'.reverseGuess p F w' ω (s+b) e := by
by_cases he : e=f
· subst e
simp [reverseGuess,hz,b]
· simp only [ite_eq_right he,reverseGuess,zero_add]
congr 2; ring
simp only [heq,Finset.sum_add_distrib]
simp
rw [reverseMass]
by_cases hf : ω f
· simp only [ite_eq_left hf]
rw [hr _ hT.2.2 (pin w f True),← hsum r (pin w f True) hT.2.2]
simp only [reverseGuess,queryDelta,prefixScore,ite_eq_left hf]
rfl
· simp only [ite_eq_right hf]
rw [hl _ hT.2.1 (pin w f False),← hsum l (pin w f False) hT.2.1]
simp only [reverseGuess,queryDelta,prefixScore,ite_eq_right hf]
rfl
lemma reverseMass_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 : ∀ a b, a ⊆ b → F a → F b) (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) {s : ℝ} (hs : 0 ≤ s) :
0 ≤ T.reverseMass p F w ω s ∧ T.reverseMass p F w ω s ≤ 1/(1+s) := by
induction T generalizing used w s with
| done _ => exact ⟨le_rfl, div_nonneg zero_le_one (by linarith)⟩
| ask f l r hl hr =>
let b := if openPivot F f ω then (1-p)*delta w F f else 0
have hf0 : w f False ≠ 0 := by rw [(hf f hT.1).1]; linarith [hp.2]
have hf1 : w f True ≠ 0 := by rw [(hf f hT.1).2]; exact hp.1.ne'
have hb : 0 ≤ b := by
dsimp [b]
split_ifs
· exact mul_nonneg (sub_nonneg.mpr hp.2.le) (delta_unit w hw hn f hf0 hf1 F hF).1
· exact le_rfl
have hb' : 0 ≤ b/(1+s+b)^2 := div_nonneg hb (sq_nonneg _)
have hs' : 0 ≤ s+b := add_nonneg hs hb
have step {x : ℝ} (hx : 0 ≤ x ∧ x ≤ 1/(1+(s+b))) :
0 ≤ b/(1+s+b)^2+x ∧ b/(1+s+b)^2+x ≤ 1/(1+s) := by
refine ⟨add_nonneg hb' hx.1, ?_⟩
calc
_ ≤ b/(1+s+b)^2 + 1/(1+(s+b)) := add_le_add_right hx.2 _
_ ≤ _ := by simpa only [add_assoc] using ReverseGuess.step_bound hs hb
simp only [reverseMass]
by_cases hbit : ω f
· simp only [ite_eq_left hbit]
exact step (hr _ hT.2.2 _ (pin_normalized w hw f True)
(pin_nonneg w hn f True) (fresh_pin w p used hf f True) hs')
· simp only [ite_eq_right hbit]
exact step (hl _ hT.2.1 _ (pin_normalized w hw f False)
(pin_nonneg w hn f False) (fresh_pin w p used hf f False) hs')
theorem reverseGuess_subprobability (T : SearchTree E R) (used : Finset E) (hT : T.Valid used)
{p : ℝ} (hp : p ∈ Set.Ioo 0 1) (F : Set E → Prop)
(hF : ∀ a b, a ⊆ b → F a → F b) (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, 0 ≤ T.reverseGuess p F w ω 0 e) ∧ (∑ e, T.reverseGuess p F w ω 0 e) ≤ 1 := by
constructor
· intro e
apply div_nonneg _ (sq_nonneg _)
split_ifs
· exact mul_nonneg (sub_nonneg.mpr hp.2.le)
(queryDelta_nonneg T used hT hp F hF w hw hn hf ω e)
· exact le_rfl
· rw [← reverseMass_eq_sum T used hT]
simpa using (reverseMass_bound T used hT hp F hF w hw hn hf ω (s := 0) le_rfl).2
end CriticalPercolation.SearchTree
end
end
end OAI
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