/usr/share/acl2-6.3/books/str/arithmetic.lisp is in acl2-books-source 6.3-5.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
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 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 | ; ACL2 String Library
; Copyright (C) 2009-2013 Centaur Technology
;
; Contact:
; Centaur Technology Formal Verification Group
; 7600-C N. Capital of Texas Highway, Suite 300, Austin, TX 78731, USA.
; http://www.centtech.com/
;
; This program is free software; you can redistribute it and/or modify it under
; the terms of the GNU General Public License as published by the Free Software
; Foundation; either version 2 of the License, or (at your option) any later
; version. This program is distributed in the hope that it will be useful but
; WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
; FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for
; more details. You should have received a copy of the GNU General Public
; License along with this program; if not, write to the Free Software
; Foundation, Inc., 51 Franklin Street, Suite 500, Boston, MA 02110-1335, USA.
;
; Original author: Jared Davis <jared@centtech.com>
(in-package "ACL2")
(include-book "coerce")
(include-book "arithmetic/top" :dir :system)
(include-book "std/lists/take" :dir :system)
(include-book "std/lists/len" :dir :system)
(include-book "std/lists/nthcdr" :dir :system)
(include-book "std/lists/append" :dir :system)
(include-book "std/lists/repeat" :dir :system)
;; BOZO fundamental lemmas that should probably be part of other libraries.
(defthm negative-when-natp
(implies (natp x) (equal (< x 0) nil)))
(defthm eqlablep-when-characterp
(implies (characterp x) (eqlablep x)))
(defthm nth-of-len
(equal (nth (len x) x)
nil))
(defthm nth-when-bigger
(implies (<= (len x) (nfix n))
(equal (nth n x)
nil))
:hints(("Goal" :in-theory (enable nth))))
(defthm car-of-repeat
(equal (car (acl2::repeat x n))
(if (zp n)
nil
x))
:hints(("Goal" :in-theory (enable acl2::repeat))))
(defthm len-of-nonempty-string-is-positive
(implies (and (stringp x)
(not (equal x "")))
(< 0 (len (explode x))))
:rule-classes ((:rewrite) (:linear)))
(defthm length-zero-when-stringp
(implies (stringp x)
(equal (equal 0 (length x))
(equal x ""))))
(defthm length-zero-when-stringp-alt
(implies (stringp x)
(equal (equal 0 (len (explode x)))
(equal x ""))))
(defthm subsetp-equal-of-cons-right
(implies (subsetp-equal x y)
(subsetp-equal x (cons b y))))
(defthm subsetp-equal-reflexive
(subsetp-equal x x))
(encapsulate
()
(local (defthm l1
(implies (or (not (natp x))
(<= 256 x))
(equal (code-char x)
(code-char 0)))
:hints(("Goal" :use ((:instance acl2::completion-of-code-char))))))
(local (defthm l2
(implies (natp k)
(equal (char-code (code-char (+ k (char-code a))))
(if (and (integerp k)
(<= 0 (+ k (char-code a)))
(< (+ k (char-code a)) 256))
(+ k (char-code a))
0)))
:hints(("Goal"
:cases ((< (+ k (char-code a)) 256))))))
(local (defthm l0
(implies (and (integerp a)
(not (integerp b)))
(equal (integerp (+ a b))
(not (acl2-numberp b))))))
(defthm char-code-of-code-char-of-sum-with-char-code
(equal (char-code (code-char (+ k (char-code a))))
(cond ((integerp k)
(if (and (<= 0 (+ k (char-code a)))
(< (+ k (char-code a)) 256))
(+ k (char-code a))
0))
((acl2-numberp k)
0)
(t
(char-code a))))
:hints(("Goal" :in-theory (e/d ()
(code-char-char-code-is-identity
str::equal-of-char-codes))))))
(defthm characterp-of-car-when-character-listp
(implies (character-listp x)
(equal (characterp (car x))
(consp x))))
(defthm character-listp-of-cdr-when-character-listp
(implies (character-listp x)
(character-listp (cdr x))))
(defthm character-listp-of-repeat
(implies (characterp x)
(character-listp (acl2::repeat x n)))
:hints(("Goal" :in-theory (enable acl2::repeat))))
(defthm character-listp-of-take
(implies (character-listp x)
(equal (character-listp (take n x))
(<= (nfix n) (len x))))
:hints(("Goal" :in-theory (enable take-redefinition))))
(defthm character-listp-of-rev
(equal (character-listp (rev x))
(character-listp (list-fix x)))
:hints(("Goal" :induct (len x))))
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