| import OAI.NumberTheory.TwoPoint.Circuits.CircuitApproximationDegree |
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| namespace OAI |
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| /-! De Morgan duality transfers the sampled AND approximation to OR gates. -/ |
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| namespace TwoPointCorrelations |
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|
| open Finset |
| open scoped Classical |
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|
| noncomputable def boolOrValue {k : ℕ} (b : Fin k → Bool) : ℝ := |
| if ∃ i, b i = true then 1 else 0 |
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|
| 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 |
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| 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 |
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| end TwoPointCorrelations |
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| end OAI |
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