File size: 3,684 Bytes
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 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 | import OAI.NumberTheory.TwoPoint.Circuits.CircuitSemiExact
namespace OAI
/-! The algebraic error-cutoff step of Braverman's Lemma 10. A Fourier
approximation of the exceptional event suppresses the large polynomial
values on that event while preserving exact zeros. -/
namespace TwoPointCorrelations
open scoped Classical
noncomputable def exceptionalExtension {n : ℕ} (F : BooleanCube n → ℝ)
(E : BooleanCube n → Bool) (x : BooleanCube n) : ℝ :=
if E x then 1 else F x
noncomputable def errorCutoffPolynomial {n : ℕ} (P T : BooleanCube n → ℝ)
(x : BooleanCube n) : ℝ := P x * (1 - T x)
lemma exceptionalExtension_boolean {n : ℕ} {F : BooleanCube n → ℝ}
(hF : ∀ x, F x = 0 ∨ F x = 1) (E : BooleanCube n → Bool) (x : BooleanCube n) :
exceptionalExtension F E x = 0 ∨ exceptionalExtension F E x = 1 := by
unfold exceptionalExtension
split_ifs
· exact Or.inr rfl
· exact hF x
lemma errorCutoffPolynomial_zero {n : ℕ} {F P T : BooleanCube n → ℝ}
{E : BooleanCube n → Bool} (he : ∀ x, E x = false → P x = F x)
(x : BooleanCube n) (hx : exceptionalExtension F E x = 0) :
errorCutoffPolynomial P T x = 0 := by
have hE : E x = false := by
cases h : E x <;> simp_all [exceptionalExtension]
have hF : F x = 0 := by simpa [exceptionalExtension, hE] using hx
simp [errorCutoffPolynomial, he x hE, hF]
lemma errorCutoffPolynomial_degree {n p q : ℕ} {P T : BooleanCube n → ℝ}
(hP : WalshDegreeLE P p) (hT : WalshDegreeLE T q) :
WalshDegreeLE (errorCutoffPolynomial P T) (p + q) :=
hP.mul ((WalshDegreeLE.const 1 q).sub hT)
lemma errorCutoffPolynomial_sq_bound {n : ℕ} {F P T : BooleanCube n → ℝ}
{E : BooleanCube n → Bool} (hF : ∀ x, F x = 0 ∨ F x = 1)
(he : ∀ x, E x = false → P x = F x) {B : ℝ} (hB : 1 ≤ B)
(hP : ∀ x, |P x| ≤ B) (x : BooleanCube n) :
(exceptionalExtension F E x - errorCutoffPolynomial P T x) ^ 2 ≤
2 * (if E x then (1 : ℝ) else 0) +
2 * B ^ 2 * ((if E x then (1 : ℝ) else 0) - T x) ^ 2 := by
have hp : (P x) ^ 2 ≤ B ^ 2 := by
have ha := abs_le.mp (hP x)
nlinarith
have hBsq : 1 ≤ B ^ 2 := by nlinarith
cases hE : E x
· have heq := he x hE
rcases hF x with hf | hf
· simp [exceptionalExtension, errorCutoffPolynomial, hE, heq, hf]
positivity
· simp only [exceptionalExtension, errorCutoffPolynomial, hE,
Bool.false_eq_true, ite_false, heq, hf, one_mul, mul_zero, zero_add]
have hh := mul_le_mul_of_nonneg_right hBsq (sq_nonneg (T x))
nlinarith only [hh, sq_nonneg (T x)]
· simp only [exceptionalExtension, errorCutoffPolynomial, hE, ite_true]
have hh := mul_le_mul_of_nonneg_right hp (sq_nonneg (1 - T x))
nlinarith only [hh, sq_nonneg (1 + P x * (1 - T x))]
theorem errorCutoffPolynomial_mean_sq {n : ℕ} {F P T : BooleanCube n → ℝ}
{E : BooleanCube n → Bool} (hF : ∀ x, F x = 0 ∨ F x = 1)
(he : ∀ x, E x = false → P x = F x) {B δ ε : ℝ} (hB : 1 ≤ B)
(hP : ∀ x, |P x| ≤ B)
(hE : cubeAverage (fun x => if E x then (1 : ℝ) else 0) ≤ δ)
(hT : cubeAverage (fun x => ((if E x then (1 : ℝ) else 0) - T x) ^ 2) ≤ ε) :
cubeAverage (fun x =>
(exceptionalExtension F E x - errorCutoffPolynomial P T x) ^ 2) ≤
2 * δ + 2 * B ^ 2 * ε := by
have hh := cubeAverage_mono (errorCutoffPolynomial_sq_bound (T := T) hF he hB hP)
rw [cubeAverage_add, cubeAverage_mul_const, cubeAverage_mul_const] at hh
exact hh.trans (add_le_add (mul_le_mul_of_nonneg_left hE (by norm_num))
(mul_le_mul_of_nonneg_left hT (by positivity)))
end TwoPointCorrelations
end OAI
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