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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Exploration.ClosedFresh
namespace OAI
section
section
noncomputable section
namespace CriticalPercolation.SearchTree
open scoped Classical
open FiniteProduct
variable {E R : Type*}
def queryCount : SearchTree E R → (E → Prop) → ℕ
| .done _, _ => 0
| .ask e l r, ω => 1 + if ω e then queryCount r ω else queryCount l ω
def centeredSum (p : ℝ) : SearchTree E R → (E → Prop) → ℝ
| .done _, _ => 0
| .ask e l r, ω => if ω e then (1-p)+centeredSum p r ω else -p+centeredSum p l ω
def truncate : SearchTree E R → ℕ → SearchTree E Unit
| _, 0 => .done ()
| .done _, _+1 => .done ()
| .ask e l r, n+1 => .ask e (truncate l n) (truncate r n)
lemma truncate_valid [DecidableEq E] (t : SearchTree E R) (n : ℕ) (used : Finset E)
(h : t.Valid used) : (t.truncate n).Valid used := by
induction n generalizing t used with
| zero => simp [truncate,Valid]
| succ n ih =>
cases t with
| done r => trivial
| ask e l r => exact ⟨h.1, ih l _ h.2.1, ih r _ h.2.2⟩
lemma queryCount_truncate_le (t : SearchTree E R) (n : ℕ) (ω : E → Prop) :
(t.truncate n).queryCount ω ≤ n := by
induction n generalizing t with
| zero => simp [truncate,queryCount]
| succ n ih =>
cases t with
| done r => simp [truncate,queryCount]
| ask e l r =>
simp only [truncate,queryCount]
split_ifs <;> have := ih r <;> have := ih l <;> omega
lemma centeredSum_truncate_eq (t : SearchTree E R) (n : ℕ) (ω : E → Prop)
(h : t.queryCount ω ≤ n) (p : ℝ) :
(t.truncate n).centeredSum p ω = t.centeredSum p ω := by
induction n generalizing t with
| zero =>
cases t with
| done r => rfl
| ask e l r => simp [queryCount] at h
| succ n ih =>
cases t with
| done r => rfl
| ask e l r =>
by_cases he : ω e
· simp only [queryCount,ite_eq_left he] at h
simp only [truncate,centeredSum,ite_eq_left he]
rw [ih r (by omega)]
· simp only [queryCount,ite_eq_right he] at h
simp only [truncate,centeredSum,ite_eq_right he]
rw [ih l (by omega)]
variable [Fintype E] [DecidableEq E]
lemma centeredSum_moments (t : SearchTree E R) (used : Finset E) (ht : t.Valid used)
(p : ℝ) (w : E → Prop → ℝ) (hw : ∀ e, ∑ b, w e b = 1)
(hf : LazyDFS.Fresh w p used) :
expect w (t.centeredSum p) = 0 ∧
expect w (fun ω => (t.centeredSum p ω)^2) =
p*(1-p)*expect w (fun ω => (t.queryCount ω : ℝ)) := by
induction t generalizing used w with
| done r => simp [centeredSum,queryCount,expect_const w hw]
| ask e l r hl hr =>
have h0 := hl (insert e used) ht.2.1 (pin w e False)
(pin_normalized w hw e False) (LazyDFS.fresh_pin w p used hf e False)
have h1 := hr (insert e used) ht.2.2 (pin w e True)
(pin_normalized w hw e True) (LazyDFS.fresh_pin w p used hf e True)
have he := hf e ht.1
constructor
· simp only [centeredSum]
erw [expect_prop_ite]
simp only [expect_add, expect_const _ (pin_normalized w hw e False),
expect_const _ (pin_normalized w hw e True),h0.1,h1.1,he.1,he.2]
ring
· have hx : (fun ω => ((.ask e l r : SearchTree E R).centeredSum p ω)^2) =
(fun ω => if ω e then ((1-p)+r.centeredSum p ω)^2
else (-p+l.centeredSum p ω)^2) := by funext ω; simp only [centeredSum]; split_ifs <;> rfl
rw [hx]
erw [expect_prop_ite,
expect_square_add_const _ (pin_normalized w hw e False),
expect_square_add_const _ (pin_normalized w hw e True),h0.1,h1.1,h0.2,h1.2,he.1,he.2]
have hc : (fun ω => ((.ask e l r : SearchTree E R).queryCount ω : ℝ)) =
(fun ω => 1+(if ω e then (r.queryCount ω : ℝ) else (l.queryCount ω : ℝ))) := by
funext ω; simp only [queryCount]; split_ifs <;> simp
rw [hc,expect_add]
erw [expect_prop_ite,expect_const w hw,he.1,he.2]
ring
lemma truncated_second_le (t : SearchTree E R) (used : Finset E) (ht : t.Valid used)
(p : ℝ) (hp : p ∈ Set.Icc 0 1) (w : E → Prop → ℝ)
(hw : ∀ e, ∑ b, w e b = 1) (hn : ∀ e b, 0 ≤ w e b)
(hf : LazyDFS.Fresh w p used) (n : ℕ) :
expect w (fun ω => ((t.truncate n).centeredSum p ω)^2) ≤ p*(1-p)*n := by
rw [(t.truncate n).centeredSum_moments used (t.truncate_valid n used ht) p w hw hf |>.2]
apply mul_le_mul_of_nonneg_left _ (mul_nonneg hp.1 (sub_nonneg.mpr hp.2))
calc expect w (fun ω => ((t.truncate n).queryCount ω : ℝ)) ≤ expect w (fun _ => (n:ℝ)) :=
expect_mono hn (fun ω => Nat.cast_le.mpr (t.queryCount_truncate_le n ω))
_ = n := expect_const w hw _
end CriticalPercolation.SearchTree
end
end
end
end OAI
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