openAI-math-dataset / lean /OAI /NumberTheory /TwoPoint /Circuits /CircuitBravermanApproximation.lean
| import OAI.NumberTheory.TwoPoint.Circuits.CircuitNormBudget | |
| import OAI.NumberTheory.TwoPoint.Circuits.CircuitDegreeBound | |
| namespace OAI | |
| /-! The combinatorial approximation stage of Braverman's proof, with a | |
| fully explicit error circuit and coarse quantitative bounds. The depth | |
| estimate is sufficient for the fixed-depth comparison. -/ | |
| namespace TwoPointCorrelations | |
| theorem AC0Circuit.polynomial_error_approximation {n : ℕ} | |
| (ν : FiniteLaw (BooleanCube n)) (s : ℕ) (hs : 1 ≤ s) (c : AC0Circuit n) : | |
| ∃ (P : BooleanCube n → ℝ) (E : AC0Circuit n), | |
| WalshDegreeLE P ((s * (Nat.log 2 c.size + 3)) ^ c.depth) ∧ | |
| E.depth ≤ 4 * c.depth + 1 ∧ | |
| E.size ≤ exceptionPolynomialBound c.size s * c.size ∧ | |
| ν.probability (fun x => E.eval x = true) ≤ (c.size : ℝ) * (7 / 8 : ℝ) ^ s ∧ | |
| (∀ x, E.eval x ≠ true → P x = c.indicator x) ∧ | |
| ∀ x, |P x| ≤ (2 * ((c.size : ℝ) + 1) ^ 2) ^ | |
| ((2 * (s * (Nat.log 2 c.size + 3)) + 2) ^ c.depth) := by | |
| let H := c.certificate ν s | |
| refine ⟨H.polynomial, H.error, ?_, H.error_depth, ?_, H.error_probability, | |
| H.exact_off_error, ?_⟩ | |
| · exact H.degree_bound.mono (AC0Circuit.approximationDegree_le s hs c c.size le_rfl) | |
| · exact H.error_size.trans (AC0Circuit.exceptionSizeBound_le s c c.size le_rfl) | |
| · intro x | |
| exact (H.norm_bound x).trans (AC0Circuit.sampleNormBound_le s c c.size le_rfl) | |
| end TwoPointCorrelations | |
| end OAI | |