/usr/share/acl2-6.3/books/str/iless.lisp is in acl2-books-source 6.3-5.
This file is owned by root:root, with mode 0o644.
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; 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 "STR")
(include-book "ieqv")
(local (include-book "std/lists/nthcdr" :dir :system))
(local (include-book "arithmetic"))
(defsection ichar<
:parents (ordering)
:short "Case-insensitive character less-than test."
:long "<p>@(call ichar<) determines if @('x') is \"smaller\" than @('y'), but
ignoring case. Our approach is basically to downcase upper-case letters and
then compare the resulting character codes.</p>
<p>Something subtle about this is that, in the ASCII character ordering, the
upper-case characters do not immediately preceede the lower-case ones. That
is, upper-case Z is at 90, but lower-case a does not start until 97. So, in
our approach, the 6 intervening characters (the square brackets, backslash,
hat, underscore, and backtick) are considered \"smaller\" than letters, even
though in regular ASCII ordering they are \"larger\" than the upper-case
letters.</p>"
(definlined ichar< (x y)
(declare (type character x)
(type character y)
(xargs :guard-hints(("Goal" :in-theory (enable downcase-char)))))
(mbe :logic
(< (char-code (downcase-char x))
(char-code (downcase-char y)))
:exec
(let* ((xc (the (unsigned-byte 8) (char-code (the character x))))
(yc (the (unsigned-byte 8) (char-code (the character y))))
(xc-fix (if (and (<= (big-a) (the (unsigned-byte 8) xc))
(<= (the (unsigned-byte 8) xc) (big-z)))
(the (unsigned-byte 8) (+ (the (unsigned-byte 8) xc) 32))
(the (unsigned-byte 8) xc)))
(yc-fix (if (and (<= (big-a) (the (unsigned-byte 8) yc))
(<= (the (unsigned-byte 8) yc) (big-z)))
(the (unsigned-byte 8) (+ (the (unsigned-byte 8) yc) 32))
(the (unsigned-byte 8) yc))))
(< (the (unsigned-byte 8) xc-fix)
(the (unsigned-byte 8) yc-fix)))))
(local (in-theory (enable ichar<)))
(defthm ichar<-irreflexive
(not (ichar< x x)))
(defthm ichar<-antisymmetric
(implies (ichar< x y)
(not (ichar< y x))))
(defthm ichar<-transitive
(implies (and (ichar< x y)
(ichar< y z))
(ichar< x z)))
(defthm ichar<-transitive-two
(implies (and (ichar< y z)
(ichar< x y))
(ichar< x z)))
(defthm ichar<-trichotomy-weak
(implies (and (not (ichar< x y))
(not (ichar< y x)))
(equal (ichareqv x y)
t))
:hints(("Goal" :in-theory (enable ichar<
ichareqv
downcase-char
char-fix))))
(defcong ichareqv equal (ichar< x y) 1
:hints(("Goal" :in-theory (enable ichar<
ichareqv
downcase-char
char-fix))))
(defcong ichareqv equal (ichar< x y) 2
:hints(("Goal" :in-theory (enable ichar<
ichareqv
downcase-char
char-fix))))
(defthm ichar<-trichotomy-strong
(equal (ichar< x y)
(and (not (ichareqv x y))
(not (ichar< y x))))
:hints(("Goal"
:in-theory (e/d (ichareqv downcase-char char-fix)
(ichar<-trichotomy-weak))
:use ((:instance ichar<-trichotomy-weak))))
:rule-classes ((:rewrite :loop-stopper ((x y))))))
(defsection icharlist<
:parents (ordering)
:short "Case-insensitive character-list less-than test."
:long "<p>@(call icharlist<) determines whether the character list @('x')
preceeds @('y') in alphabetical order without regards to case. The characters
are compared with @(see ichar<) and shorter strings are considered smaller than
longer strings.</p>"
(defund icharlist< (x y)
(declare (xargs :guard (and (character-listp x)
(character-listp y))
:verify-guards nil))
(mbe :logic (cond ((atom y)
nil)
((atom x)
t)
((ichar< (car x) (car y))
t)
((ichar< (car y) (car x))
nil)
(t
(icharlist< (cdr x) (cdr y))))
:exec
(cond ((atom y)
nil)
((atom x)
t)
(t
(let* ((xc (the (unsigned-byte 8) (char-code (the character (car x)))))
(yc (the (unsigned-byte 8) (char-code (the character (car y)))))
(xc-fix (if (and (<= (big-a) (the (unsigned-byte 8) xc))
(<= (the (unsigned-byte 8) xc) (big-z)))
(the (unsigned-byte 8) (+ (the (unsigned-byte 8) xc) 32))
(the (unsigned-byte 8) xc)))
(yc-fix (if (and (<= (big-a) (the (unsigned-byte 8) yc))
(<= (the (unsigned-byte 8) yc) (big-z)))
(the (unsigned-byte 8) (+ (the (unsigned-byte 8) yc) 32))
(the (unsigned-byte 8) yc))))
(cond ((< (the (unsigned-byte 8) xc-fix) (the (unsigned-byte 8) yc-fix))
t)
((< (the (unsigned-byte 8) yc-fix) (the (unsigned-byte 8) xc-fix))
nil)
(t
(icharlist< (cdr x) (cdr y)))))))))
(local (in-theory (enable icharlist<)))
(verify-guards icharlist<
:hints(("Goal" :in-theory (e/d (ichar< downcase-char)
(ichar<-trichotomy-strong)))))
(defthm icharlist<-irreflexive
(equal (icharlist< x x)
nil))
(defthm icharlist<-antisymmetric
(implies (icharlist< x y)
(not (icharlist< y x))))
(defthm icharlist<-transitive
(implies (and (icharlist< x y)
(icharlist< y z))
(icharlist< x z)))
(defthm icharlist<-trichotomy-weak
(implies (and (not (icharlist< x y))
(not (icharlist< y x)))
(equal (icharlisteqv x y)
t)))
(defcong icharlisteqv equal (icharlist< x y) 1)
(defcong icharlisteqv equal (icharlist< x y) 2)
(defthm icharlist<-trichotomy-strong
(equal (icharlist< x y)
(and (not (icharlisteqv x y))
(not (icharlist< y x))))
:rule-classes ((:rewrite :loop-stopper ((x y))))))
(defsection istr<
:parents (ordering)
:short "Case-insensitive string less-than test."
:long "<p>@(call icharlist<) determines whether the string @('x') preceeds
@('y') in alphabetical order without regards to case. The characters are
compared with @(see ichar<) and shorter strings are considered smaller than
longer strings.</p>
<p>Logically, this is identical to:</p>
@({
(icharlist< (explode x) (explode y))
})
<p>But we use a more efficient implementation which avoids coercing the strings
into lists.</p>
<p>NOTE: for reasoning, we leave this function enabled and prefer to work with
@(see icharlist<) of the explodes as our normal form.</p>"
(defund istr<-aux (x y n xl yl)
(declare (type string x)
(type string y)
(type integer n)
(type integer xl)
(type integer yl)
(xargs :guard (and (stringp x)
(stringp y)
(natp n)
(<= n (length x))
(<= n (length y))
(equal xl (length x))
(equal yl (length y)))
:verify-guards nil
:measure (nfix (- (nfix xl) (nfix n)))))
(mbe :logic
(cond ((zp (- (nfix yl) (nfix n)))
nil)
((zp (- (nfix xl) (nfix n)))
t)
((ichar< (char x n) (char y n))
t)
((ichar< (char y n) (char x n))
nil)
(t
(istr<-aux x y (+ (nfix n) 1) xl yl)))
:exec
(cond ((= (the integer n) (the integer yl))
nil)
((= (the integer n) (the integer xl))
t)
(t
(let* ((xc (the (unsigned-byte 8)
(char-code (the character
(char (the string x) (the integer n))))))
(yc (the (unsigned-byte 8)
(char-code (the character
(char (the string y) (the integer n))))))
(xc-fix (if (and (<= (big-a) (the (unsigned-byte 8) xc))
(<= (the (unsigned-byte 8) xc) (big-z)))
(the (unsigned-byte 8) (+ (the (unsigned-byte 8) xc) 32))
(the (unsigned-byte 8) xc)))
(yc-fix (if (and (<= (big-a) (the (unsigned-byte 8) yc))
(<= (the (unsigned-byte 8) yc) (big-z)))
(the (unsigned-byte 8) (+ (the (unsigned-byte 8) yc) 32))
(the (unsigned-byte 8) yc))))
(cond ((< (the (unsigned-byte 8) xc-fix) (the (unsigned-byte 8) yc-fix))
t)
((< (the (unsigned-byte 8) yc-fix) (the (unsigned-byte 8) xc-fix))
nil)
(t
(istr<-aux (the string x)
(the string y)
(the integer (+ (the integer n) 1))
(the integer xl)
(the integer yl)))))))))
(definline istr< (x y)
(declare (type string x)
(type string y)
(xargs :verify-guards nil))
(mbe :logic
(icharlist< (explode x) (explode y))
:exec
(istr<-aux (the string x)
(the string y)
(the integer 0)
(the integer (length (the string x)))
(the integer (length (the string y))))))
(local (defthm crock
(equal (< a a)
nil)))
(local (defthm crock2
(implies (< a b)
(equal (< b a)
nil))))
(verify-guards istr<-aux
:hints(("Goal" :in-theory (e/d (ichar< downcase-char)))))
(defthm istr<-aux-correct
(implies (and (stringp x)
(stringp y)
(natp n)
(<= n (length x))
(<= n (length y))
(equal xl (length x))
(equal yl (length y)))
(equal (istr<-aux x y n xl yl)
(icharlist< (nthcdr n (coerce x 'list))
(nthcdr n (coerce y 'list)))))
:hints(("Goal" :in-theory (enable istr<-aux icharlist<))))
(verify-guards istr<$inline)
(defcong istreqv equal (istr< x y) 1)
(defcong istreqv equal (istr< x y) 2))
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