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9dbc64e | 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 | import OAI.Computability.DepthThree.FiniteProbability
import Mathlib.Tactic.Linarith
namespace OAI
universe uDepth1
open scoped Classical
namespace DepthThreeLowerBound
variable {V : Type uDepth1} [Fintype V]
theorem sign_eq_two_indicator_sub_one (b : Bool) :
sign b = 2 * indicator b - 1 := by
cases b <;> norm_num [sign, indicator]
theorem finiteAvg_sign (f : Cube V → Bool) :
finiteAvg (fun x => sign (f x)) =
2 * finiteAvg (fun x => indicator (f x)) - 1 := by
calc
_ = finiteAvg (fun x => 2 * indicator (f x) - 1) :=
finiteAvg_congr (fun x => sign_eq_two_indicator_sub_one (f x))
_ = _ := by rw [finiteAvg_sub, finiteAvg_const_mul, finiteAvg_const]
theorem acceptance_density_lower_of_sign_bound (f : Cube V → Bool) (ε : ℝ)
(h : |finiteAvg (fun x => sign (f x))| ≤ ε) :
(1 - ε) / 2 ≤ finiteAvg (fun x => indicator (f x)) := by
have hl := (abs_le.mp h).1
rw [finiteAvg_sign] at hl
linarith
theorem acceptance_density_lower_of_width_correlation (f : Cube V → Bool)
(k : ℕ) (ε : ℝ)
(h : ∀ H : CNF V, H.WidthAtMost k →
|finiteAvg (fun x : Cube V => sign (f x) * indicator (H.eval x))| ≤ ε) :
(1 - ε) / 2 ≤ finiteAvg (fun x => indicator (f x)) := by
apply acceptance_density_lower_of_sign_bound
have he := h [] (by intro C hC; simp at hC)
simpa using he
end DepthThreeLowerBound
end OAI
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