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
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
|
; Milawa - A Reflective Theorem Prover
; Copyright (C) 2005-2009 Kookamara LLC
;
; Contact:
;
; Kookamara LLC
; 11410 Windermere Meadows
; Austin, TX 78759, USA
; http://www.kookamara.com/
;
; License: (An MIT/X11-style license)
;
; Permission is hereby granted, free of charge, to any person obtaining a
; copy of this software and associated documentation files (the "Software"),
; to deal in the Software without restriction, including without limitation
; the rights to use, copy, modify, merge, publish, distribute, sublicense,
; and/or sell copies of the Software, and to permit persons to whom the
; Software is furnished to do so, subject to the following conditions:
;
; The above copyright notice and this permission notice shall be included in
; all copies or substantial portions of the Software.
;
; THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
; IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
; FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
; AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
; LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
; FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER
; DEALINGS IN THE SOFTWARE.
;
; Original author: Jared Davis <jared@kookamara.com>
(in-package "MILAWA")
(include-book "shift")
(set-verify-guards-eagerness 2)
(set-case-split-limitations nil)
(set-well-founded-relation ord<)
(set-measure-function rank)
;; We say log2(0) = 0, and for all other n we say log2(n) is the maximum k such
;; that 2^k <= n. For example:
;;
;; n Log2(n) 2^Log2(n)
;; ------------------------------------------------
;; 0-1 0 1
;; 2-3 1 2
;; 4-7 2 4
;; 8-15 3 8
;; 16-31 4 16
;; ...
;; ------------------------------------------------
(defun fast-log2 (n acc)
(declare (xargs :guard (and (natp n)
(natp acc))
:measure (nfix n)))
(if (or (zp n)
(equal n 1))
acc
(fast-log2 (bitwise-shr n 1) (+ 1 acc))))
(definlined log2 (n)
(declare (xargs :guard (natp n)))
(fast-log2 n 0))
(encapsulate
()
(local (defthm natp-of-fast-log2
(implies (force (natp acc))
(equal (natp (fast-log2 n acc))
t))
:hints(("Goal" :in-theory (enable fast-log2)))))
(defthm natp-of-log2
(equal (natp (log2 n))
t)
:hints(("Goal" :in-theory (enable log2)))))
|
