| import OAI.NumberTheory.TwoPoint.Circuits.CircuitDyadicApproximation |
| import OAI.NumberTheory.TwoPoint.Circuits.CircuitDyadicErrors |
| import OAI.NumberTheory.TwoPoint.Circuits.CircuitApproximationComparison |
|
|
| namespace OAI |
|
|
| /-! The fixed-depth bounded-independence theorem with a completely |
| explicit polynomial budget, stated first at dyadic accuracy. -/ |
|
|
| namespace TwoPointCorrelations |
|
|
| open scoped Classical |
|
|
| theorem AC0Circuit.dyadic_comparison_lower {n j t : ℕ} (c : AC0Circuit n) |
| (hc : c.depth ≤ 22) (hm : c.size ≤ 2 ^ j) (ht : bravermanDegree j ≤ t) |
| (g : BooleanCube n → ℝ) (hg : ∀ x, 0 ≤ g x) (hmean : cubeAverage g = 1) |
| (hwise : TWiseUniformDensity g t) : |
| cubeAverage c.indicator - (1 / 2 : ℝ) ^ j ≤ |
| cubeAverage (fun x => g x * c.indicator x) := by |
| obtain ⟨P, E, hP, hdepth, hsize, hprob, hexact, hnorm⟩ := |
| c.dyadic_approximation (mixedCubeLaw g hg hmean) hc hm |
| let D := switchingDegree 89 (bravermanSwitches j) |
| have herror : cubeAverage (fun x => (E.indicator x - walshTruncation E.indicator D x) ^ 2) ≤ |
| 4 * (2 : ℝ) ^ bravermanErrorExponent j * |
| (1 / 2 : ℝ) ^ (bravermanSwitches j + 1) := by |
| apply (E.fourier_tail_polynomial_degree 89 (bravermanSwitches j) hdepth |
| (bravermanSwitches_pos j)).trans |
| have hreal : (E.size : ℝ) ≤ (2 : ℝ) ^ bravermanErrorExponent j := by |
| exact_mod_cast hsize |
| gcongr |
| have hF (x) : c.indicator x = 0 ∨ c.indicator x = 1 := by |
| cases he : c.eval x <;> simp [AC0Circuit.indicator, he] |
| have hbound : 1 ≤ (2 : ℝ) ^ bravermanNormExponent j := one_le_pow₀ (by norm_num) |
| have hdegree : (bravermanBase j ^ 22 + D) + (bravermanBase j ^ 22 + D) ≤ t := by |
| dsimp [bravermanDegree] at ht |
| dsimp [D] |
| omega |
| have hh := hwise.approximation_lower hg hmean hF hP |
| (walshTruncation_degree E.indicator D) hdegree |
| (fun x hx => hexact x (by simp [hx])) hbound hnorm hprob herror |
| have hloss := braverman_total_loss j |
| linarith |
|
|
| theorem AC0Circuit.dyadic_comparison {n j t : ℕ} (c : AC0Circuit n) |
| (hc : c.depth ≤ 22) (hm : c.size ≤ 2 ^ j) (ht : bravermanDegree j ≤ t) |
| (g : BooleanCube n → ℝ) (hg : ∀ x, 0 ≤ g x) (hmean : cubeAverage g = 1) |
| (hwise : TWiseUniformDensity g t) : |
| |cubeAverage (fun x => g x * c.indicator x) - cubeAverage c.indicator| ≤ |
| (1 / 2 : ℝ) ^ j := by |
| have hlo := c.dyadic_comparison_lower hc hm ht g hg hmean hwise |
| have hhi := c.negate.dyadic_comparison_lower (by simpa [AC0Circuit.negate_depth] using hc) |
| (by simpa [AC0Circuit.negate_size] using hm) ht g hg hmean hwise |
| have hu : cubeAverage c.negate.indicator = 1 - cubeAverage c.indicator := by |
| change cubeAverage (fun x => c.negate.indicator x) = _ |
| simp_rw [AC0Circuit.negate_indicator] |
| rw [cubeAverage_sub, cubeAverage_const] |
| have hgneg : cubeAverage (fun x => g x * c.negate.indicator x) = |
| 1 - cubeAverage (fun x => g x * c.indicator x) := by |
| simp_rw [AC0Circuit.negate_indicator, mul_sub, mul_one] |
| rw [cubeAverage_sub, hmean] |
| rw [hu, hgneg] at hhi |
| exact abs_le.mpr ⟨by linarith, by linarith⟩ |
|
|
| end TwoPointCorrelations |
|
|
| end OAI |
|
|
