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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 | import OAI.Probability.CubeShuffle.Basic
namespace OAI
namespace CubeShuffle.QueryTree
open scoped BigOperators
variable {ι : Type*} [DecidableEq ι]
/-- Flipping one bit is a permutation of the uniform coin space. -/
def flipCoin (i : ι) : Equiv.Perm (ι → Bool) :=
(show Function.Involutive (fun ω : ι → Bool => Function.update ω i (!(ω i))) from by
intro ω
funext j
by_cases h : j = i
· subst j; simp
· simp [Function.update_of_ne h]).toPerm
lemma flipCoin_apply_eq (i : ι) (ω : ι → Bool) : flipCoin i ω i = !(ω i) := by
change Function.update ω i (!(ω i)) i = _
exact Function.update_self _ _ _
lemma flipCoin_apply_ne (i j : ι) (h : j ≠ i) (ω : ι → Bool) :
flipCoin i ω j = ω j := by
change Function.update ω i (!(ω i)) j = _
exact Function.update_of_ne h _ _
/-- Half the uniform coin assignments have a specified bit, independently of
any function which does not inspect that bit. This remains valid for adaptive
functions of all the remaining bits. -/
lemma mean_indicator_bit [Fintype ι] (f : (ι → Bool) → ℝ) (i : ι) (b : Bool)
(hf : ∀ ω, f (flipCoin i ω) = f ω) :
finiteMean (fun ω => if ω i = b then f ω else 0) = finiteMean f / 2 := by
let g := fun ω : ι → Bool => if ω i = b then f ω else 0
have hp (ω : ι → Bool) : g ω + g (flipCoin i ω) = f ω := by
dsimp [g]
rw [flipCoin_apply_eq, hf]
cases hω : ω i <;> cases b <;> simp
have hs := Equiv.sum_comp (flipCoin i) g
have hadd : (∑ ω, g ω) + (∑ ω, g (flipCoin i ω)) = ∑ ω, f ω := by
rw [← Finset.sum_add_distrib]
exact Finset.sum_congr rfl (fun ω _ => hp ω)
unfold finiteMean
change (∑ ω, g ω) / _ = _
rw [hs] at hadd
have hh : (∑ ω, g ω) = (∑ ω, f ω) / 2 := by linarith
rw [hh]
ring
end CubeShuffle.QueryTree
end OAI
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