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e5a97c6 | 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 57 58 59 60 61 | 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
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