| import Mathlib.Data.Nat.Log |
| import Mathlib.Tactic |
|
|
| namespace OAI |
|
|
| /-! # Computing a bounded binary ceiling logarithm by doubling |
|
|
| The executable loop stores the current power of two. It performs no call |
| to `Nat.clog` and never forms a power by repeated recomputation. |
| -/ |
|
|
| namespace LowSpaceSubsetSum.CountedCeilLog |
|
|
| def loop (size : ℕ) : ℕ → ℕ → ℕ → ℕ → ℕ × ℕ |
| | 0, depth, _, work => (depth, work + 2) |
| | fuel + 1, depth, power, work => |
| if size ≤ power then (depth, work + 2) |
| else loop size fuel (depth + 1) (2 * power) (work + 4) |
|
|
| theorem loop_value (size fuel depth power work : ℕ) |
| (hpower : power = 2 ^ depth) (hlo : depth ≤ Nat.clog 2 size) |
| (hhi : Nat.clog 2 size ≤ depth + fuel) : |
| (loop size fuel depth power work).1 = Nat.clog 2 size := by |
| induction fuel generalizing depth power work with |
| | zero => simp only [loop]; omega |
| | succ fuel ih => |
| simp only [loop] |
| split |
| · rename_i h |
| have hc : Nat.clog 2 size ≤ depth := Nat.clog_le_of_le_pow (h.trans_eq hpower) |
| omega |
| · rename_i h |
| apply ih |
| · rw [hpower, pow_succ] |
| omega |
| · have hc : depth < Nat.clog 2 size := |
| (Nat.lt_clog_iff_pow_lt (by decide)).mpr (by omega) |
| omega |
| · omega |
|
|
| theorem loop_work (size fuel depth power work : ℕ) : |
| (loop size fuel depth power work).2 ≤ work + 4 * fuel + 2 := by |
| induction fuel generalizing depth power work with |
| | zero => simp [loop] |
| | succ fuel ih => |
| simp only [loop] |
| split |
| · omega |
| · have h := ih (depth + 1) (2 * power) (work + 4) |
| omega |
|
|
| def run (size fuel : ℕ) : ℕ × ℕ := loop size fuel 0 1 0 |
|
|
| theorem run_value (size fuel : ℕ) (h : size ≤ 2 ^ fuel) : |
| (run size fuel).1 = Nat.clog 2 size := |
| loop_value size fuel 0 1 0 (by simp) (Nat.zero_le _) |
| (by simpa only [Nat.zero_add] using Nat.clog_le_of_le_pow h) |
|
|
| theorem run_work (size fuel : ℕ) : (run size fuel).2 ≤ 4 * fuel + 2 := by |
| simpa only [run, Nat.zero_add] using loop_work size fuel 0 1 0 |
|
|
| end LowSpaceSubsetSum.CountedCeilLog |
|
|
| end OAI |
|
|
