| import Mathlib.Data.Nat.Log |
|
|
| namespace OAI |
|
|
| /-! # A bounded tail-recursive binary logarithm |
|
|
| This is the integer operation needed to turn measured mixer counts into |
| prime widths. Each iteration divides by two and updates two counters. |
| -/ |
|
|
| namespace LowSpaceSubsetSum.MeasuredLog2 |
|
|
| def loop : ℕ → ℕ → ℕ → ℕ → ℕ × ℕ |
| | 0, _, value, work => (value, work + 1) |
| | fuel + 1, x, value, work => |
| if 2 ≤ x then loop fuel (x / 2) (value + 1) (work + 3) |
| else (value, work + 1) |
|
|
| theorem loop_value (fuel x value work : ℕ) (hx : Nat.log2 x ≤ fuel) : |
| (loop fuel x value work).1 = value + Nat.log2 x := by |
| induction fuel generalizing x value work with |
| | zero => |
| have hz : Nat.log2 x = 0 := by omega |
| simp [loop, hz] |
| | succ fuel ih => |
| rw [loop] |
| split |
| next h => |
| have he := Nat.log2_def x |
| rw [ite_eq_left h] at he |
| rw [ih _ _ _ (by omega), he] |
| omega |
| next h => simp only [Nat.log2_def x, ite_eq_right h, Nat.add_zero] |
|
|
| theorem loop_work (fuel x value work : ℕ) : |
| (loop fuel x value work).2 ≤ work + 3 * fuel + 1 := by |
| induction fuel generalizing x value work with |
| | zero => simp [loop] |
| | succ fuel ih => |
| rw [loop] |
| split |
| · have h := ih (x / 2) (value + 1) (work + 3); omega |
| · simp only; omega |
|
|
| def run (bits x : ℕ) : ℕ × ℕ := loop bits x 0 0 |
|
|
| theorem run_value (bits x : ℕ) (hx : x ≤ 2 ^ bits) : |
| (run bits x).1 = Nat.log2 x := by |
| have hlog : Nat.log2 x ≤ bits := by |
| rw [Nat.log2_eq_log_two] |
| have h := Nat.log_mono_right (b := 2) hx |
| simpa only [Nat.log_pow (by decide : 1 < 2)] using h |
| simpa only [run, Nat.zero_add] using loop_value bits x 0 0 hlog |
|
|
| theorem run_work (bits x : ℕ) : (run bits x).2 ≤ 3 * bits + 1 := by |
| simpa only [run, Nat.zero_add] using loop_work bits x 0 0 |
|
|
| end LowSpaceSubsetSum.MeasuredLog2 |
|
|
| end OAI |
|
|
