| 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 |
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|
| open scoped Classical |
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|
| 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 |
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| end OAI |
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