openAI-math-dataset / lean /OAI /NumberTheory /TwoPoint /Circuits /CircuitApproximationComparison.lean
| import OAI.NumberTheory.TwoPoint.Circuits.CircuitErrorCutoff | |
| import OAI.NumberTheory.TwoPoint.Circuits.CircuitMixedLaw | |
| import OAI.NumberTheory.TwoPoint.Circuits.CircuitApproximationDegree | |
| namespace OAI | |
| /-! Braverman's final finite expectation argument. The combinatorial | |
| approximation and an L² approximation to its small error event imply a | |
| one-sided bounded-independence comparison. -/ | |
| namespace TwoPointCorrelations | |
| open scoped Classical | |
| lemma exceptionalExtension_gap {n : ℕ} {F : BooleanCube n → ℝ} | |
| (hF : ∀ x, F x = 0 ∨ F x = 1) (E : BooleanCube n → Bool) (x : BooleanCube n) : | |
| |F x - exceptionalExtension F E x| ≤ if E x then (1 : ℝ) else 0 := by | |
| cases hE : E x | |
| · simp [exceptionalExtension, hE] | |
| · rcases hF x with h | h <;> simp [exceptionalExtension, hE, h] | |
| theorem TWiseUniformDensity.approximation_lower {n p q t : ℕ} | |
| {g F P T : BooleanCube n → ℝ} {E : BooleanCube n → Bool} | |
| (hwise : TWiseUniformDensity g t) (hg : ∀ x, 0 ≤ g x) (hmean : cubeAverage g = 1) | |
| (hF : ∀ x, F x = 0 ∨ F x = 1) | |
| (hP : WalshDegreeLE P p) (hT : WalshDegreeLE T q) (ht : (p + q) + (p + q) ≤ t) | |
| (houtside : ∀ x, E x = false → P x = F x) | |
| {B δ ε : ℝ} (hB : 1 ≤ B) (hbound : ∀ x, |P x| ≤ B) | |
| (hE : (mixedCubeLaw g hg hmean).probability (fun x => E x = true) ≤ δ) | |
| (herror : cubeAverage (fun x => ((if E x then (1 : ℝ) else 0) - T x) ^ 2) ≤ ε) : | |
| cubeAverage F - (8 * δ + 2 * B ^ 2 * ε) ≤ cubeAverage (fun x => g x * F x) := by | |
| have hmix := mixedCubeLaw_exception_bounds g hg hmean (fun x => E x = true) hE | |
| have hU : cubeAverage (fun x => if E x then (1 : ℝ) else 0) ≤ 2 * δ := by | |
| convert hmix.2 using 1 | |
| congr 1 | |
| funext x | |
| by_cases h : E x = true <;> simp [h] | |
| have hG : cubeAverage (fun x => g x * (if E x then (1 : ℝ) else 0)) ≤ 2 * δ := by | |
| have hm := hmix.1 | |
| rw [FiniteLaw.probability, cubeDensityLaw_average] at hm | |
| convert hm using 1 | |
| congr 1 | |
| funext x | |
| by_cases h : E x = true <;> simp [h] | |
| have hu : cubeAverage (fun x => |F x - exceptionalExtension F E x|) ≤ 2 * δ := | |
| (cubeAverage_mono (exceptionalExtension_gap hF E)).trans hU | |
| have hm : cubeAverage (fun x => g x * |F x - exceptionalExtension F E x|) ≤ 2 * δ := | |
| (cubeAverage_mono (fun x => mul_le_mul_of_nonneg_left | |
| (exceptionalExtension_gap hF E x) (hg x))).trans hG | |
| have hs := errorCutoffPolynomial_mean_sq hF houtside hB hbound hU herror | |
| have hdeg := ((errorCutoffPolynomial_degree hP hT).minorant).mono ht | |
| have hh := hwise.semiExact_lower hg (exceptionalExtension_boolean hF E) | |
| (errorCutoffPolynomial_zero houtside) hdeg hu hm hs | |
| convert hh using 1 | |
| ring | |
| end TwoPointCorrelations | |
| end OAI | |