| import OAI.NumberTheory.TwoPoint.Circuits.CircuitOrCertificate |
|
|
| namespace OAI |
|
|
| /-! The recursive error certificate has polynomial size. Only this coarse |
| bound, rather than the detailed shape of its gates, is used by the next stage. -/ |
|
|
| namespace TwoPointCorrelations.AC0Circuit |
|
|
| open Finset |
| open scoped Classical |
|
|
| def exceptionPolynomialBound (m s : ℕ) : ℕ := |
| 5 + m + s * (Nat.log 2 m + 3) * (1 + m * (1 + m)) |
|
|
| theorem exceptionSizeBound_le {n : ℕ} (s : ℕ) (c : AC0Circuit n) |
| (m : ℕ) (hm : c.size ≤ m) : |
| c.exceptionSizeBound s ≤ exceptionPolynomialBound m s * c.size := by |
| induction c generalizing m with |
| | literal i b => |
| simp only [exceptionSizeBound, size, mul_one, exceptionPolynomialBound] |
| omega |
| | @andGate k c ih | @orGate k c ih => |
| have hm' : 1 + ∑ i, (c i).size ≤ m := hm |
| have hsize (i : Fin k) : (c i).size ≤ m := by |
| have hh := single_le_sum (fun j _ => Nat.zero_le (c j).size) (mem_univ i) |
| omega |
| have hcount : k ≤ ∑ i, (c i).size := by |
| calc |
| k = ∑ _i : Fin k, 1 := by simp |
| _ ≤ _ := sum_le_sum (fun i _ => (c i).size_pos) |
| have hk : k ≤ m := by omega |
| have hsum : ∑ i, (c i).exceptionSizeBound s ≤ |
| exceptionPolynomialBound m s * ∑ i, (c i).size := by |
| rw [mul_sum] |
| exact sum_le_sum (fun i _ => ih i m (hsize i)) |
| have hlocal : 4 + (∑ i, (c i).size) + s * (Nat.log 2 k + 3) * |
| (1 + k * (1 + ∑ i, (c i).size)) ≤ exceptionPolynomialBound m s := by |
| unfold exceptionPolynomialBound |
| have hsumsize : ∑ i, (c i).size ≤ m := by omega |
| have hlog := Nat.log_mono_right (b := 2) hk |
| gcongr |
| norm_num |
| change 4 + (∑ i, (c i).size) + s * (Nat.log 2 k + 3) * |
| (1 + k * (1 + ∑ i, (c i).size)) + ∑ i, (c i).exceptionSizeBound s ≤ |
| exceptionPolynomialBound m s * (1 + ∑ i, (c i).size) |
| calc |
| _ ≤ exceptionPolynomialBound m s + |
| exceptionPolynomialBound m s * ∑ i, (c i).size := Nat.add_le_add hlocal hsum |
| _ = _ := by ring |
|
|
| end TwoPointCorrelations.AC0Circuit |
|
|
| end OAI |
|
|
