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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 "digitp")
(include-book "tools/mv-nth" :dir :system)
(local (include-book "arithmetic"))
(defsection parse-nat-from-charlist
:parents (numbers)
:short "Parse a natural number from the beginning of a character list."
:long "<p>@(call parse-nat-from-charlist) tries to read a natural number from
the beginning of the character list @('x').</p>
<ul>
<li>@('val') is an accumulator for the value of the digits we have read so far,
and typically should be set to 0 to begin with.</li>
<li>@('len') is an accumulator for the number of digits we have read, and
should typically be set to 0 to begin with.</li>
</ul>
<p>We return @('(mv val len rest)'), where @('x') after reading as many digits
as possible.</p>
<p>See also @(call digit-list-value), which is simpler for interpreting strings
where all of the characters are digits.</p>"
(defund parse-nat-from-charlist (x val len)
(declare (type integer val)
(type integer len)
(xargs :guard (and (character-listp x)
(natp val)
(natp len))
:verify-guards nil))
(mbe :logic
(cond ((atom x)
(mv (nfix val) (nfix len) nil))
((digitp (car x))
(let ((digit-val (digit-val (car x))))
(parse-nat-from-charlist (cdr x)
; A silly idea I have (for the purposes of strnatless, at least) would be to
; instead multiply each character by 16, which would mean that the operation
; could be done via ash and logior. I think the values produced by such a
; scheme would be ordered in the same way that the values here are ordered.
; And, rudimentary speed test suggests it could be as much as 50% faster. The
; proof seems difficult, so for now I don't have the patience to attempt it.
(+ digit-val (* 10 (nfix val)))
(+ 1 (nfix len)))))
(t
(mv (nfix val) (nfix len) x)))
:exec
(cond ((atom x)
(mv val len nil))
(t
(let ((code (the (unsigned-byte 8) (char-code (the character (car x))))))
(declare (type (unsigned-byte 8) code))
(if (and (<= (the (unsigned-byte 8) 48) (the (unsigned-byte 8) code))
(<= (the (unsigned-byte 8) code) (the (unsigned-byte 8) 57)))
(let ((digit-val (the (unsigned-byte 8)
(- (the (unsigned-byte 8) code)
(the (unsigned-byte 8) 48)))))
(parse-nat-from-charlist
(cdr x)
(the integer (+ (the (unsigned-byte 8) digit-val)
(the integer (* 10 (the integer val)))))
(the integer (+ 1 (the integer len)))))
(mv val len x)))))))
(local (in-theory (enable parse-nat-from-charlist)))
(verify-guards parse-nat-from-charlist
:hints(("Goal" :in-theory (enable digitp
digit-val
char-fix))))
(encapsulate
()
(local (defund leading-digits-value (x val)
(declare (xargs :verify-guards nil))
(if (consp x)
(if (digitp (car x))
(leading-digits-value (cdr x)
(+ (digit-val (car x)) (* 10 (nfix val))))
(nfix val))
(nfix val))))
(local (defthm lemma-1
(equal (mv-nth 0 (parse-nat-from-charlist x val len))
(leading-digits-value x val))
:hints(("Goal" :in-theory (enable leading-digits-value)))))
(local (defthm lemma-2
(equal (leading-digits-value x val)
(digit-list-value1 (take-leading-digits x) val))
:hints(("Goal" :in-theory (e/d (leading-digits-value
digit-list-value1
take-leading-digits)
(digit-list-value1-removal))))))
(defthm val-of-parse-nat-from-charlist
(equal (mv-nth 0 (parse-nat-from-charlist x val len))
(+ (digit-list-value (take-leading-digits x))
(* (nfix val) (expt 10 (len (take-leading-digits x))))))))
(encapsulate
()
(local (defund count-leading-digits1 (x len)
(declare (xargs :verify-guards nil))
(if (consp x)
(if (digitp (car x))
(count-leading-digits1 (cdr x) (+ 1 (nfix len)))
(nfix len))
(nfix len))))
(local (defthm lemma-1
(equal (mv-nth 1 (parse-nat-from-charlist x val len))
(count-leading-digits1 x len))
:hints(("Goal" :in-theory (enable count-leading-digits1)))))
(local (defthm lemma-2
(equal (count-leading-digits1 x len)
(+ (nfix len)
(len (take-leading-digits x))))
:hints(("Goal" :in-theory (enable count-leading-digits1
take-leading-digits)))))
(defthm len-of-parse-nat-from-charlist
(equal (mv-nth 1 (parse-nat-from-charlist x val len))
(+ (nfix len) (len (take-leading-digits x))))))
(defthm rest-of-parse-nat-from-charlist
(equal (mv-nth 2 (parse-nat-from-charlist x val len))
(skip-leading-digits x))
:hints(("Goal" :in-theory (enable skip-leading-digits)))))
(defsection parse-nat-from-string
:parents (numbers)
:short "Parse a natural number from a string, at some offset."
:long "<p>@(call parse-nat-from-string) is somewhat elaborate:</p>
<ul>
<li>@('x') is a string that we want to parse a value from</li>
<li>@('val') is an accumulator for the value we have parsed, and it should
generally be 0 to begin with.</li>
<li>@('len') is an accumulator for the number of characters we have read so
far, and should generally be 0 to begin with.</li>
<li>@('n') is an offset into @('x') where we should begin parsing. It must be
a valid index into the string, i.e., @('0 <= n < (length x)').</li>
<li>@('xl') must be exactly equal to @('(length x)'), and only serves as a
cache to avoid recomputing the length.</li>
</ul>
<p>We return @('(mv val len)'), the final values of the accumulators, which are
respectively the natural number we have just parsed and the number of digits
that we parsed.</p>
<p>Both @('val') and @('len') are guaranteed to be natural numbers; failure is
indicated by a return @('len') of zero.</p>
<p>Because of leading zeroes, the @('len') may be much larger than you would
expect based on @('val') alone. The @('len') argument is generally useful if
you want to continue parsing through the string, i.e., the @('n') you started
with plus the @('len') you got out will be the next position in the string
after the number.</p>
<p>See also @(see parse-nat-from-charlist) for a simpler function that reads a
number from the start of a character list. This function also serves as part
of our logical definition.</p>"
(local (in-theory (disable acl2::nth-when-bigger
acl2::negative-when-natp
default-+-2
default-+-1
default-<-2
commutativity-of-+
default-<-1
ACL2::|x < y => 0 < y-x|
)))
(defund parse-nat-from-string (x val len n xl)
(declare (type string x)
(type (integer 0 *) val len n xl)
(xargs :guard (and (stringp x)
(natp val)
(natp len)
(natp n)
(equal xl (length x))
(<= n xl))
:measure (nfix (- (nfix xl) (nfix n)))
:verify-guards nil))
(mbe :logic
(cond ((zp (- (nfix xl) (nfix n)))
(mv (nfix val) (nfix len)))
((digitp (char x n))
(let ((digit-val (digit-val (char x n))))
(parse-nat-from-string x
(+ digit-val (* 10 (nfix val)))
(+ 1 (nfix len))
(+ 1 (nfix n))
(nfix xl))))
(t
(mv (nfix val) (nfix len))))
:exec
(cond ((int= n xl)
(mv val len))
(t
(let ((code (the (unsigned-byte 8)
(char-code (the character
(char (the string x)
(the (integer 0 *) n)))))))
(declare (type (unsigned-byte 8) code))
(if (and (<= (the (unsigned-byte 8) 48)
(the (unsigned-byte 8) code))
(<= (the (unsigned-byte 8) code)
(the (unsigned-byte 8) 57)))
(let ((digit-val (the (unsigned-byte 8)
(- (the (unsigned-byte 8) code)
(the (unsigned-byte 8) 48)))))
(parse-nat-from-string
(the string x)
(the (integer 0 *)
(+ (the (unsigned-byte 8) digit-val)
(the (integer 0 *) (* 10 (the (integer 0 *) val)))))
(the (integer 0 *) (+ 1 (the (integer 0 *) len)))
(the (integer 0 *) (+ 1 (the (integer 0 *) n)))
(the (integer 0 *) xl)))
(mv val len)))))))
(local (in-theory (enable parse-nat-from-string)))
(verify-guards parse-nat-from-string
:hints(("Goal" :in-theory (enable digitp digit-val))))
(defthm natp-of-val-of-parse-nat-from-string
(and (integerp (mv-nth 0 (parse-nat-from-string x val len n xl)))
(<= 0 (mv-nth 0 (parse-nat-from-string x val len n xl))))
:rule-classes :type-prescription
:hints(("Goal" :in-theory (disable nth nfix))))
(defthm natp-of-len-of-parse-nat-from-string
(and (integerp (mv-nth 1 (parse-nat-from-string x val len n xl)))
(<= 0 (mv-nth 1 (parse-nat-from-string x val len n xl))))
:rule-classes :type-prescription
:hints(("Goal" :in-theory (disable nth nfix))))
(defthm progress-of-parse-nat-from-string
;; If there's a digit there, we read at least one character.
(implies (and (equal xl (length x))
(or (< 0 (nfix len))
(digitp (char x (nfix n)))))
(< 0 (mv-nth 1 (parse-nat-from-string x val len n xl))))
:rule-classes ((:rewrite) (:linear))
:hints(("Goal"
:induct (parse-nat-from-string x val len n xl))))
(defthm val-of-parse-nat-from-string
(implies (and (natp val)
(natp len)
(natp n)
(equal xl (len (explode x)))
(<= n xl))
(equal (mv-nth 0 (parse-nat-from-string x val len n xl))
(mv-nth 0 (parse-nat-from-charlist (nthcdr n (explode x)) val len))))
:hints(("Goal"
:induct (parse-nat-from-string x val len n xl)
:in-theory (e/d (parse-nat-from-charlist)
(val-of-parse-nat-from-charlist))
:do-not '(generalize fertilize))))
(defthm len-of-parse-nat-from-string
(implies (and (natp val)
(natp len)
(natp n)
(equal xl (len (explode x)))
(<= n xl))
(equal (mv-nth 1 (parse-nat-from-string x val len n xl))
(mv-nth 1 (parse-nat-from-charlist (nthcdr n (explode x)) val len))))
:hints(("Goal"
:induct (parse-nat-from-string x val len n xl)
:in-theory (e/d (parse-nat-from-charlist)
(len-of-parse-nat-from-charlist))
:do-not '(generalize fertilize)))))
(defsection charlistnat<
:parents (ordering)
:short "Mixed alphanumeric character-list less-than test."
:long "<p>@(call charlistnat<) determines if the character list @('x') is
\"smaller\" than the character list @('y'), using an ordering that is nice for
humans.</p>
<p>This is almost an ordinary case-sensitive lexicographic ordering. But,
unlike a simple lexicographic order, we identify sequences of natural number
digits and group them together so that they can be sorted numerically.</p>
<p>Even though this function operates on character lists, let's just talk about
strings instead since they are easier to write down. If you give most string
sorts a list of inputs like \"x0\" through \"x11\", they will end up in a
peculiar order:</p>
@({\"x0\", \"x1\", \"x10\", \"x11\", \"x2\", \"x3\", ... \"x9\"})
<p>But in @('charlistnat<'), we see the adjacent digits as bundles and sort
them as numbers. This leads to a nicer ordering:</p>
@({\"x0\", \"x1\", \"x2\", ..., \"x9\", \"x10\", \"x11\"})
<p>This is almost entirely straightforward. One twist is how to accommodate
leading zeroes. Our approach is: instead of grouping adjacent digits and
treating them as a natural number, treat them as a pair with a value and a
length. We then sort these pairs first by value, and then by length. Hence, a
string such as \"x0\" is considered to be less than \"x00\", etc.</p>
<p>See also @(see strnat<) and @(see icharlist<).</p>"
(local (in-theory (disable acl2::nth-when-bigger
acl2::negative-when-natp
default-+-2
default-+-1
default-<-2
commutativity-of-+
default-<-1
ACL2::|x < y => 0 < y-x|
char<-trichotomy-strong
char<-antisymmetric
char<-trichotomy-weak
char<-transitive
expt
default-car
default-cdr)))
(defund charlistnat< (x y)
(declare (xargs :guard (and (character-listp x)
(character-listp y))
:measure (len x)))
(cond ((atom y)
nil)
((atom x)
t)
((and (digitp (car x))
(digitp (car y)))
(b* (((mv v1 l1 rest-x) (parse-nat-from-charlist x 0 0))
((mv v2 l2 rest-y) (parse-nat-from-charlist y 0 0)))
; The basic idea is to order numbers by their values, and then by their
; lengths. This second part only is necessary to accomodate leading zeroes.
(cond ((or (< v1 v2)
(and (int= v1 v2)
(< l1 l2)))
t)
((or (< v2 v1)
(and (int= v1 v2)
(< l2 l1)))
nil)
(t
(charlistnat< rest-x rest-y)))))
((char< (car x) (car y))
t)
((char< (car y) (car x))
nil)
(t
(charlistnat< (cdr x) (cdr y)))))
(local (in-theory (enable charlistnat<)))
(defcong charlisteqv equal (charlistnat< x y) 1)
(defcong charlisteqv equal (charlistnat< x y) 2)
(defthm charlistnat<-irreflexive
(not (charlistnat< x x)))
(defthm charlistnat<-antisymmetric
(implies (charlistnat< x y)
(not (charlistnat< y x)))
:hints(("goal" :in-theory (enable char<-antisymmetric))))
(encapsulate
()
(local (defthm char<-nonsense-2
(implies (and (char< a y)
(not (digitp a))
(digitp y)
(digitp z))
(char< a z))
:rule-classes ((:rewrite :backchain-limit-lst 0))
:hints(("Goal" :in-theory (enable char< digitp)))))
(local (defthm char<-nonsense-3
(implies (and (char< y a)
(not (digitp a))
(digitp x)
(digitp y))
(char< x a))
:rule-classes ((:rewrite :backchain-limit-lst 0))
:hints(("Goal" :in-theory (enable char< digitp)))))
(local (defthm char<-nonsense-4
(implies (and (char< x y)
(not (digitp y))
(digitp x)
(digitp z))
(not (char< y z)))
:rule-classes ((:rewrite :backchain-limit-lst 0))
:hints(("Goal" :in-theory (enable digitp char<)))))
(defthm charlistnat<-transitive
(implies (and (charlistnat< x y)
(charlistnat< y z))
(charlistnat< x z))
:hints(("Goal" :in-theory (e/d ((:induction charlistnat<)
char<-trichotomy-strong
char<-transitive)
(expt charlistnat<-antisymmetric
take-leading-digits-when-digit-listp
default-+-2 default-+-1
BOUND-OF-LEN-OF-TAKE-LEADING-DIGITS
LEN-OF-PARSE-NAT-FROM-CHARLIST))
:induct t
:expand ((:free (y) (charlistnat< x y))
(:free (z) (charlistnat< y z)))))))
(encapsulate
()
(local
(encapsulate ()
;; A slightly tricky lemma about arithmetic.
(local (defun expr (a x b n)
(+ a (* x (expt b n)))))
(local (include-book "arithmetic-3/floor-mod/floor-mod" :dir :system))
(local (defthm mod-of-expr
(implies (and (natp a1)
(natp x1)
(natp n)
(natp b)
(< a1 (expt b n))
(<= x1 b))
(equal (mod (expr a1 x1 b n)
(expt b n))
a1))))
(local (defthm main-lemma
(implies (and (natp a1)
(natp a2)
(natp x1)
(natp x2)
(natp n)
(natp b)
(< a1 (expt b n))
(< a2 (expt b n))
(<= x1 b)
(<= x2 b)
(not (equal a1 a2)))
(not (equal (expr a1 x1 b n)
(expr a2 x2 b n))))
:hints(("Goal" :in-theory (disable expr mod-of-expr)
:use ((:instance mod-of-expr)
(:instance mod-of-expr (a1 a2) (x1 x2)))))))
(defthmd arith-lemma-1
(implies (and (natp a1)
(natp a2)
(natp x1)
(natp x2)
(natp n)
(natp b)
(< a1 (expt b n))
(< a2 (expt b n))
(<= x1 b)
(<= x2 b)
(not (equal a1 a2)))
(not (equal (+ a1 (* x1 (expt b n)))
(+ a2 (* x2 (expt b n))))))
:hints(("Goal"
:in-theory (enable expr)
:use ((:instance main-lemma)))))))
; The main proof of trichotomy
(local (defthm lemma-1
(IMPLIES (AND (NOT (EQUAL (DIGIT-LIST-VALUE X2)
(DIGIT-LIST-VALUE Y2)))
(NOT (EQUAL X2 Y2))
(CHARACTERP X1)
(CHARACTERP Y1)
(CHARACTER-LISTP X2)
(CHARACTER-LISTP Y2)
(DIGITP X1)
(DIGITP Y1)
(DIGIT-LISTP X2)
(DIGIT-LISTP Y2)
(EQUAL (LEN X2) (LEN Y2)))
(NOT (EQUAL (+ (DIGIT-LIST-VALUE X2)
(* (DIGIT-VAL X1) (EXPT 10 (LEN X2))))
(+ (DIGIT-LIST-VALUE Y2)
(* (DIGIT-VAL Y1) (EXPT 10 (LEN X2)))))))
:hints(("Goal"
:use ((:instance arith-lemma-1
(a1 (digit-list-value x2))
(a2 (digit-list-value y2))
(x1 (digit-val x1))
(x2 (digit-val y1))
(b 10)
(n (len x2))))))))
(local (defun my-induction (x y)
(if (and (consp x)
(consp y))
(my-induction (cdr x) (cdr y))
nil)))
(local (defthm lemma-2
(implies (and (equal (len x) (len y))
(character-listp x)
(character-listp y)
(digit-listp x)
(digit-listp y))
(equal (equal (digit-list-value x)
(digit-list-value y))
(equal x y)))
:hints(("Goal"
:induct (my-induction x y)
:in-theory (enable digit-listp
digit-list-value
commutativity-of-+)))))
(local (defthm lemma-3
(implies (and (equal (len (take-leading-digits y))
(len (take-leading-digits x)))
(equal (digit-list-value (take-leading-digits y))
(digit-list-value (take-leading-digits x)))
(charlisteqv (skip-leading-digits x)
(skip-leading-digits y)))
(equal (charlisteqv x y)
t))
:hints(("Goal" :in-theory (enable take-leading-digits
skip-leading-digits
charlisteqv
digit-list-value)))))
(defthm charlistnat<-trichotomy-weak
(implies (and (not (charlistnat< x y))
(not (charlistnat< y x)))
(equal (charlisteqv x y)
t))
:hints(("Goal" :in-theory (e/d (char<-trichotomy-strong)
(BOUND-OF-LEN-OF-TAKE-LEADING-DIGITS
TAKE-LEADING-DIGITS-WHEN-DIGIT-LISTP
ACL2::RIGHT-CANCELLATION-FOR-+
CHARLISTNAT<-ANTISYMMETRIC
CHARLISTNAT<-IRREFLEXIVE
)))))
(defthm charlistnat<-trichotomy-strong
(equal (charlistnat< x y)
(and (not (charlisteqv x y))
(not (charlistnat< y x))))
:rule-classes ((:rewrite :loop-stopper ((x y)))))))
(defsection strnat<-aux
:parents (strnat<)
:short "Implementation of @(see strnat<)."
:long "<p>@(call strnat<-aux) is basically the adaptation of @(see
charlistnat<) for strings. Here, X and Y are the strings being compared, and
XL and YL are their pre-computed lengths. XN and YN are the indices into the
two strings that are our current positions.</p>
<p>BOZO why do we have XN and YN separately? It seems like we should only need
one index.</p>"
(local (in-theory (disable acl2::nth-when-bigger
acl2::negative-when-natp
default-+-2
default-+-1
default-<-2
commutativity-of-+
default-<-1
ACL2::|x < y => 0 < y-x|
ACL2::|x < y => 0 < -x+y|
char<-trichotomy-strong
char<-antisymmetric
char<-trichotomy-weak
char<-transitive
acl2::negative-when-natp
acl2::natp-rw
expt
default-car
default-cdr
(:rewrite PROGRESS-OF-PARSE-NAT-FROM-STRING)
)))
(defund strnat<-aux (x y xn yn xl yl)
(declare (type string x)
(type string y)
(type integer xn)
(type integer yn)
(type integer xl)
(type integer yl)
(xargs :guard (and (stringp x)
(stringp y)
(natp xn)
(natp yn)
(equal xl (length x))
(equal yl (length y))
(<= xn xl)
(<= yn yl))
:verify-guards nil
:measure
(let* ((x (if (stringp x) x ""))
(y (if (stringp y) y ""))
(xn (nfix xn))
(yn (nfix yn))
(xl (length x))
(yl (length y)))
(nfix (+ (- yl yn) (- xl xn))))
:hints(("Goal" :in-theory (disable val-of-parse-nat-from-string
len-of-parse-nat-from-string))))
(ignorable xl yl))
(mbe :logic
(let* ((x (if (stringp x) x ""))
(y (if (stringp y) y ""))
(xn (nfix xn))
(yn (nfix yn))
(xl (length x))
(yl (length y)))
(cond ((zp (- yl yn))
nil)
((zp (- xl xn))
t)
((and (digitp (char x xn))
(digitp (char y yn)))
(b* (((mv v1 l1)
(parse-nat-from-string x 0 0 xn xl))
((mv v2 l2)
(parse-nat-from-string y 0 0 yn yl)))
(cond ((or (< v1 v2)
(and (int= v1 v2)
(< l1 l2)))
t)
((or (< v2 v1)
(and (int= v1 v2)
(< l2 l1)))
nil)
(t
(strnat<-aux x y (+ xn l1) (+ yn l2) xl yl)))))
((char< (char x xn)
(char y yn))
t)
((char< (char y yn)
(char x xn))
nil)
(t
(strnat<-aux x y (+ 1 xn) (+ 1 yn) xl yl))))
:exec
(cond ((int= yn yl)
nil)
((int= xn xl)
t)
(t
(let* ((char-x (the character (char (the string x) (the integer xn))))
(char-y (the character (char (the string y) (the integer yn))))
(code-x (the (unsigned-byte 8) (char-code (the character char-x))))
(code-y (the (unsigned-byte 8) (char-code (the character char-y)))))
(declare (type character char-x)
(type character char-y)
(type (unsigned-byte 8) code-x)
(type (unsigned-byte 8) code-y))
(cond
((and
;; (digitp (char x xn))
(<= (the (unsigned-byte 8) 48) (the (unsigned-byte 8) code-x))
(<= (the (unsigned-byte 8) code-x) (the (unsigned-byte 8) 57))
;; (digitp (char y yn))
(<= (the (unsigned-byte 8) 48) (the (unsigned-byte 8) code-y))
(<= (the (unsigned-byte 8) code-y) (the (unsigned-byte 8) 57)))
(b* (((mv v1 l1)
(parse-nat-from-string (the string x)
(the integer 0)
(the integer 0)
(the integer xn)
(the integer xl)))
((mv v2 l2)
(parse-nat-from-string (the string y)
(the integer 0)
(the integer 0)
(the integer yn)
(the integer yl))))
(cond ((or (< (the integer v1) (the integer v2))
(and (int= v1 v2)
(< (the integer l1) (the integer l2))))
t)
((or (< (the integer v2) (the integer v1))
(and (int= v1 v2)
(< (the integer l2) (the integer l1))))
nil)
(t
(strnat<-aux (the string x)
(the string y)
(the integer (+ (the integer xn) (the integer l1)))
(the integer (+ (the integer yn) (the integer l2)))
(the integer xl)
(the integer yl))))))
((< (the (unsigned-byte 8) code-x) (the (unsigned-byte 8) code-y))
t)
((< (the (unsigned-byte 8) code-y) (the (unsigned-byte 8) code-x))
nil)
(t
(strnat<-aux (the string x)
(the string y)
(the integer (+ (the integer 1) (the integer xn)))
(the integer (+ (the integer 1) (the integer yn)))
(the integer xl)
(the integer yl)))))))))
(local (in-theory (enable strnat<-aux)))
(set-inhibit-warnings "theory") ;; implicitly local
(encapsulate
nil
(local (in-theory (disable acl2::nth-when-bigger
take-leading-digits-when-digit-listp
digit-listp-when-not-consp
(:type-prescription character-listp)
(:type-prescription eqlable-listp)
(:type-prescription atom-listp)
(:type-prescription digitp$inline)
(:type-prescription strnat<-aux)
(:type-prescription char<)
default-char-code
char<-antisymmetric
char<-trichotomy-strong
default-<-1 default-<-2
default-+-1 default-+-2
acl2::open-small-nthcdr
acl2::nthcdr-when-zp
acl2::nthcdr-when-atom
ACL2::|x < y => 0 < -x+y|
nthcdr len nth not
strnat<-aux
acl2::natp-fc-1
acl2::natp-fc-2
(:FORWARD-CHAINING EQLABLE-LISTP-FORWARD-TO-ATOM-LISTP)
(:FORWARD-CHAINING CHARACTER-LISTP-FORWARD-TO-EQLABLE-LISTP)
(:FORWARD-CHAINING ATOM-LISTP-FORWARD-TO-TRUE-LISTP)
)))
(verify-guards strnat<-aux
:hints((and stable-under-simplificationp
'(:in-theory (enable digitp
digit-val
char-fix
char<))))))
(local (defthm skip-leading-digits-to-nthcdr
(implies (force (true-listp x))
(equal (skip-leading-digits x)
(nthcdr (len (take-leading-digits x)) x)))
:hints(("Goal" :in-theory (enable skip-leading-digits take-leading-digits)))))
(defthm strnat<-aux-correct
(implies (and (stringp x)
(stringp y)
(natp xn)
(natp yn)
(equal xl (length x))
(equal yl (length y))
(<= xn xl)
(<= yn yl))
(equal (strnat<-aux x y xn yn xl yl)
(charlistnat< (nthcdr xn (explode x))
(nthcdr yn (explode y)))))
:hints(("Goal"
:induct (strnat<-aux x y xn yn xl yl)
:expand ((charlistnat< (nthcdr xn (explode x))
(nthcdr yn (explode y)))
(:free (xl yl) (strnat<-aux x y xn yn xl yl)))
:in-theory (e/d (charlistnat<
commutativity-of-+
)
(charlistnat<-antisymmetric
charlistnat<-trichotomy-strong
take-leading-digits-when-digit-listp
digit-listp-when-not-consp
charlistnat<
(:definition strnat<-aux)
default-+-1 default-+-2
acl2::nth-when-bigger))
:do-not '(generalize fertilize)))))
(defsection strnat<
:parents (ordering)
:short "Mixed alphanumeric string less-than test."
:long "<p>@(call strnat<) determines if the string @('x') is \"smaller\"
than the string @('y'), using an ordering that is nice for humans.</p>
<p>See @(see charlistnat<) for a description of this order.</p>
<p>We avoid coercing the strings into character lists, and this is altogether
pretty fast.</p>"
(definlined strnat< (x y)
(declare (type string x y))
(mbe :logic
(charlistnat< (explode x) (explode y))
:exec
(strnat<-aux (the string x)
(the string y)
(the integer 0)
(the integer 0)
(the integer (length (the string x)))
(the integer (length (the string y))))))
(local (in-theory (enable strnat<)))
(defcong streqv equal (strnat< x y) 1)
(defcong streqv equal (strnat< x y) 2)
(defthm strnat<-irreflexive
(not (strnat< x x)))
(defthm strnat<-antisymmetric
(implies (strnat< x y)
(not (strnat< y x))))
(defthm strnat<-transitive
(implies (and (strnat< x y)
(strnat< y z))
(strnat< x z)))
(defthm strnat<-transitive-alt
(implies (and (strnat< y z)
(strnat< x y))
(strnat< x z))))
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