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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Exploration.QueryBand
import OAI.Probability.CriticalPercolation.Exploration.Score
import OAI.Probability.CriticalPercolation.Criticality.DyadicBound
namespace OAI
section
section
noncomputable section
namespace CriticalPercolation.SearchTree
open scoped Classical
open FiniteProduct
variable {E R : Type*} [Fintype E] [DecidableEq E]
lemma normalized_centered_bound (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) (K : ℕ) :
expect w (fun ω => |t.centeredSum p ω| *
min (1/(4:ℝ)^K) (1/(t.queryCount ω : ℝ))) ≤ 6/(2:ℝ)^K := by
let N := Fintype.card E + 1
let a (k : ℕ) := min (1/(4:ℝ)^K) (1/(4:ℝ)^k)
let b (k : ℕ) (ω : E → Prop) :=
if 4^k ≤ t.queryCount ω ∧ t.queryCount ω < 4^(k+1)
then |t.centeredSum p ω| else 0
have ha (k : ℕ) : 0≤a k := by dsimp [a]; positivity
have hb (k : ℕ) (ω : E → Prop) : 0≤b k ω := by
dsimp [b]; split_ifs <;> positivity
have hpoint (ω : E → Prop) : |t.centeredSum p ω| *
min (1/(4:ℝ)^K) (1/(t.queryCount ω : ℝ)) ≤
∑ k ∈ Finset.range N, b k ω * a k := by
by_cases hT : t.queryCount ω = 0
· rw [t.centeredSum_eq_zero_of_count ω hT p,abs_zero,zero_mul]
exact Finset.sum_nonneg (fun k _ => mul_nonneg (hb k ω) (ha k))
· let k := Nat.log 4 (t.queryCount ω)
have hklo : 4^k ≤ t.queryCount ω := Nat.pow_log_le_self 4 hT
have hkhi : t.queryCount ω < 4^(k+1) := Nat.lt_pow_succ_log_self (by omega) _
have hkN : k ∈ Finset.range N := by
apply Finset.mem_range.mpr
have h1 := Nat.log_le_self 4 (t.queryCount ω)
have h2 := t.queryCount_le_card used ht ω
dsimp [k,N]
omega
calc
_ ≤ b k ω * a k := by
have hband : 4^k ≤ t.queryCount ω ∧ t.queryCount ω < 4^(k+1) := ⟨hklo,hkhi⟩
simp only [b,ite_eq_left hband]
apply mul_le_mul_of_nonneg_left _ (abs_nonneg _)
apply min_le_min le_rfl
apply one_div_le_one_div_of_le (by positivity : (0:ℝ)<(4:ℝ)^k)
exact_mod_cast hklo
_ ≤ _ := Finset.single_le_sum (fun j _ => mul_nonneg (hb j ω) (ha j)) hkN
calc
_ ≤ expect w (fun ω => ∑ k ∈ Finset.range N, b k ω * a k) := expect_mono hn hpoint
_ = ∑ k ∈ Finset.range N, expect w (b k) * a k := by
simp only [expect,Finset.mul_sum]
rw [Finset.sum_comm]
apply Finset.sum_congr rfl
intro k _
rw [Finset.sum_mul]
apply Finset.sum_congr rfl
intro ω _
ring
_ ≤ ∑ k ∈ Finset.range N, (2:ℝ)^(k+1)*a k := by
apply Finset.sum_le_sum
intro k _
exact mul_le_mul_of_nonneg_right (t.band_centered_bound used ht p hp w hw hn hf k) (ha k)
_ = ∑ k ∈ Finset.range N,
min ((2:ℝ)^(k+1)/4^K) ((2:ℝ)^(k+1)/4^k) := by
apply Finset.sum_congr rfl
intro k _
dsimp [a]
rw [mul_min_of_nonneg _ _ (by positivity)]
simp only [mul_one_div]
_ ≤ _ := dyadic_sum_bound N K
end CriticalPercolation.SearchTree
end
end
end
end OAI
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