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8769036 | 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.NumberTheory.TwoPoint.Circuits.CircuitApproximationDegree
namespace OAI
/-! De Morgan duality transfers the sampled AND approximation to OR gates. -/
namespace TwoPointCorrelations
open Finset
open scoped Classical
noncomputable def boolOrValue {k : ℕ} (b : Fin k → Bool) : ℝ :=
if ∃ i, b i = true then 1 else 0
lemma boolAndValue_not {k : ℕ} (b : Fin k → Bool) :
boolAndValue (fun i => !(b i)) = 1 - boolOrValue b := by
have he : (∀ i, Bool.not (b i) = true) ↔ ¬∃ i, b i = true := by simp
simp only [boolAndValue, boolOrValue, he]
split_ifs <;> norm_num
theorem exists_composed_or_approximation {n : ℕ}
(ν : FiniteLaw (BooleanCube n)) (k s d : ℕ)
(b : Fin k → BooleanCube n → Bool) (P : Fin k → BooleanCube n → ℝ)
(hP : ∀ i, WalshDegreeLE (P i) d) (δ : Fin k → ℝ)
(hδ : ∀ i, ν.probability (fun y => P i y ≠ if b i y then 1 else 0) ≤ δ i) :
∃ Q : BooleanCube n → ℝ,
WalshDegreeLE Q (d * s * (Nat.log 2 k + 3)) ∧
ν.probability (fun y => Q y ≠ boolOrValue (fun i => b i y)) ≤
(7 / 8 : ℝ) ^ s + ∑ i, δ i := by
let R := fun i y => 1 - P i y
have hR (i : Fin k) : WalshDegreeLE (R i) d :=
(WalshDegreeLE.const 1 d).sub (hP i)
have hδ' (i : Fin k) :
ν.probability (fun y => R i y ≠ if !(b i y) then 1 else 0) ≤ δ i := by
apply le_trans (ν.probability_mono ?_) (hδ i)
intro y hy heq
apply hy
dsimp only [R]
rw [heq]
cases b i y <;> norm_num
obtain ⟨Q, hQ, he⟩ := exists_composed_and_approximation ν k s d
(fun i y => !(b i y)) R hR δ hδ'
refine ⟨fun y => 1 - Q y, (WalshDegreeLE.const 1 _).sub hQ, ?_⟩
apply le_trans (ν.probability_mono ?_) he
intro y hy heq
apply hy
change 1 - Q y = _
rw [heq, boolAndValue_not]
ring
end TwoPointCorrelations
end OAI
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