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|
; 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/
;
; 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@centtech.com>
(in-package "STR")
(include-book "decimal")
(include-book "tools/mv-nth" :dir :system)
(local (include-book "arithmetic"))
(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 (dec-digit-char-p (car x))
(dec-digit-char-p (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 (dec-digit-char-p a))
(dec-digit-char-p y)
(dec-digit-char-p z))
(char< a z))
:rule-classes ((:rewrite :backchain-limit-lst 0))
:hints(("Goal" :in-theory (enable char< dec-digit-char-p)))))
(local (defthm char<-nonsense-3
(implies (and (char< y a)
(not (dec-digit-char-p a))
(dec-digit-char-p x)
(dec-digit-char-p y))
(char< x a))
:rule-classes ((:rewrite :backchain-limit-lst 0))
:hints(("Goal" :in-theory (enable char< dec-digit-char-p)))))
(local (defthm char<-nonsense-4
(implies (and (char< x y)
(not (dec-digit-char-p y))
(dec-digit-char-p x)
(dec-digit-char-p z))
(not (char< y z)))
:rule-classes ((:rewrite :backchain-limit-lst 0))
:hints(("Goal" :in-theory (enable dec-digit-char-p 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-dec-digit-chars-when-dec-digit-char-listp
default-+-2 default-+-1
BOUND-OF-LEN-OF-TAKE-LEADING-DEC-DIGIT-CHARS
LEN-OF-PARSE-NAT-FROM-CHARLIST))
:induct t
:expand ((:free (y) (charlistnat< x y))
(:free (z) (charlistnat< y z)))))))
;; (local (defthm crock
;; (implies (and (equal (take-leading-dec-digit-chars y)
;; (take-leading-dec-digit-chars x))
;; (charlisteqv (skip-leading-digits x)
;; (skip-leading-digits y)))
;; (charlisteqv x y))
;; :hints(("Goal"
;; :induct (my-induction x y)
;; :expand ((take-leading-dec-digit-chars x)
;; (take-leading-dec-digit-chars y)
;; (skip-leading-digits x)
;; (skip-leading-digits y))))))
;; (local (defthm lemma-3
;; (implies (and (equal (len (take-leading-dec-digit-chars y))
;; (len (take-leading-dec-digit-chars x)))
;; (equal (dec-digit-chars-value (take-leading-dec-digit-chars y))
;; (dec-digit-chars-value (take-leading-dec-digit-chars x)))
;; (charlisteqv (skip-leading-digits x)
;; (skip-leading-digits y)))
;; (equal (charlisteqv x y)
;; t))
;; :hints(("Goal"
;; :in-theory (enable chareqv
;; take-leading-dec-digit-chars
;; skip-leading-digits
;; dec-digit-chars-value)))))
;; (defthm charlistnat<-trichotomy-weak
;; (implies (and (not (charlistnat< x y))
;; (not (charlistnat< y x)))
;; (equal (charlisteqv x y)
;; t))
;; :hints (("goal" :induct (charlistnat< x y)
;; :expand ((charlisteqv x y)
;; (charlistnat< y x)
;; (charlistnat< x y)))))
(encapsulate
()
(local (defthm dec-digit-chars-value-max
(< (dec-digit-chars-value x) (expt 10 (len x)))
:hints(("Goal" :in-theory (enable dec-digit-chars-value)))
:rule-classes :linear))
(local (defthmd equal-of-sum-digits-lemma1
(implies (and (posp b)
(natp big1)
(natp big2)
(natp little1)
(natp little2)
(< little1 b)
(< little2 b)
(not (equal big1 big2)))
(not (equal (+ (* big1 b) little1)
(+ (* big2 b) little2))))
:hints (("goal" :use ((:instance distributivity (x b) (y big1) (z (- big2)))
(:instance distributivity (x b) (y big2) (z (- big1))))
:in-theory (disable distributivity))
(and stable-under-simplificationp
'(:use ((:instance ACL2::<-*-Y-X-Y
(y b) (x (- big1 big2)))
(:instance ACL2::<-*-Y-X-Y
(y b) (x (- big2 big1))))
:in-theory (disable distributivity acl2::<-*-y-x-y)
:cases ((and (< (+ big1 (- big2)) 1)
(< (+ big2 (- big1)) 1))))))
:otf-flg t))
(local (defthm equal-of-sum-digits
(implies (and (posp b)
(natp big1)
(natp big2)
(natp little1)
(natp little2)
(< little1 b)
(< little2 b))
(iff (equal (+ (* big1 b) little1)
(+ (* big2 b) little2))
(and (equal big1 big2)
(equal little1 little2))))
:hints (("goal" :use ((:instance equal-of-sum-digits-lemma1))))))
(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)
(dec-digit-char-listp x)
(dec-digit-char-listp y))
(equal (equal (dec-digit-chars-value x)
(dec-digit-chars-value y))
(equal x y)))
:hints(("Goal"
:induct (my-induction x y)
:in-theory (enable dec-digit-char-listp
dec-digit-chars-value
commutativity-of-+))
(and stable-under-simplificationp
'(:use ((:instance equal-of-sum-digits
(b (expt 10 (len (cdr x))))
(little1 (dec-digit-chars-value (cdr x)))
(little2 (dec-digit-chars-value (cdr y)))
(big1 (dec-digit-char-value (car x)))
(big2 (dec-digit-char-value (car y))))))))))
(local (defthm crock
(implies (and (equal (take-leading-dec-digit-chars y)
(take-leading-dec-digit-chars x))
(charlisteqv (skip-leading-digits x)
(skip-leading-digits y)))
(charlisteqv x y))
:hints(("Goal"
:induct (my-induction x y)
:expand ((take-leading-dec-digit-chars x)
(take-leading-dec-digit-chars y)
(skip-leading-digits x)
(skip-leading-digits y))))))
(local (defthm lemma-3
(implies (and (equal (len (take-leading-dec-digit-chars y))
(len (take-leading-dec-digit-chars x)))
(equal (dec-digit-chars-value (take-leading-dec-digit-chars y))
(dec-digit-chars-value (take-leading-dec-digit-chars x)))
(charlisteqv (skip-leading-digits x)
(skip-leading-digits y)))
(equal (charlisteqv x y)
t))
:hints(("Goal"
:in-theory (enable chareqv
take-leading-dec-digit-chars
skip-leading-digits
dec-digit-chars-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-DEC-DIGIT-CHARS
TAKE-LEADING-DEC-DIGIT-CHARS-WHEN-DEC-DIGIT-CHAR-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 (dec-digit-char-p (char x xn))
(dec-digit-char-p (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
;; (dec-digit-char-p (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))
;; (dec-digit-char-p (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-dec-digit-chars-when-dec-digit-char-listp
dec-digit-char-listp-when-not-consp
(:type-prescription character-listp)
(:type-prescription eqlable-listp)
(:type-prescription atom-listp)
(:type-prescription dec-digit-char-p$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 dec-digit-char-p
dec-digit-char-value
char-fix
char<))))))
(local (defthm skip-leading-digits-to-nthcdr
(implies (force (true-listp x))
(equal (skip-leading-digits x)
(nthcdr (len (take-leading-dec-digit-chars x)) x)))
:hints(("Goal" :in-theory (enable skip-leading-digits take-leading-dec-digit-chars)))))
(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-dec-digit-chars-when-dec-digit-char-listp
dec-digit-char-listp-when-not-consp
charlistnat<
(:definition strnat<-aux)
default-+-1 default-+-2
acl2::nth-when-bigger))
:do-not '(generalize fertilize)))))
(define strnat< ((x :type string)
(y :type string))
: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>"
:inline t
(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)))))
///
(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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