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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 | import OAI.NumberTheory.TwoPoint.Circuits.CircuitGateComposition
namespace OAI
/-! Degree bookkeeping for the random polynomial construction. The recursive
degree uses the largest child degree, rather than the number of leaves. -/
namespace TwoPointCorrelations
open Finset
open scoped Classical
lemma WalshDegreeLE.mono {n p q : ℕ} {F : BooleanCube n → ℝ}
(hF : WalshDegreeLE F p) (hpq : p ≤ q) : WalshDegreeLE F q := by
obtain ⟨a, ha⟩ := hF
refine ⟨fun S => if S.card ≤ p then a S else 0, ?_⟩
intro x
rw [ha]
rw [sum_filter, sum_filter]
apply sum_congr rfl
intro S _
by_cases hS : S.card ≤ p
· simp [hS, hS.trans hpq]
· simp [hS]
lemma literal_indicator_degree {n : ℕ} (i : Fin n) (b : Bool) :
WalshDegreeLE (AC0Circuit.literal i b).indicator 1 := by
have hw : WalshDegreeLE (walsh {i}) 1 := WalshDegreeLE.of_walsh {i} (by simp)
have hp := (WalshDegreeLE.const (n := n) (1 / 2) 1).sub (hw.smul (1 / 2))
have hn := (WalshDegreeLE.const (n := n) 1 1).sub hp
cases b
· convert hn using 1
funext x
simp only [AC0Circuit.indicator, AC0Circuit.eval, Bool.false_eq_true, ite_false,
walsh, prod_singleton]
cases x i <;> norm_num [booleanSign]
· convert hp using 1
funext x
simp only [AC0Circuit.indicator, AC0Circuit.eval, ite_true, walsh, prod_singleton]
cases x i <;> norm_num [booleanSign]
namespace AC0Circuit
def approximationDegree {n : ℕ} (s : ℕ) : AC0Circuit n → ℕ
| .literal _ _ => 1
| .andGate (k := k) children =>
(univ.sup fun i => approximationDegree s (children i)) * s * (Nat.log 2 k + 3)
| .orGate (k := k) children =>
(univ.sup fun i => approximationDegree s (children i)) * s * (Nat.log 2 k + 3)
end AC0Circuit
end TwoPointCorrelations
end OAI
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