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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 55 56 57 58 59 60 | 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
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