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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 "strprefixp")
(include-book "std/basic/defs" :dir :system)
(local (include-book "arithmetic"))
; BOZO should probably rewrite this to have a nice listrpos function sort of
; thing.
(defsection strrpos-fast
:parents (strrpos)
:short "Fast implementation of @(see strrpos)."
(defund strrpos-fast (x y n xl yl)
(declare (type string x y)
(type (integer 0 *) n xl yl)
(xargs :guard (and (stringp x)
(stringp y)
(natp xl)
(natp yl)
(natp n)
(<= n (length y))
(= xl (length x))
(= yl (length y)))
:measure (nfix n)))
;; N goes from YL to 0.
(cond ((mbe :logic (prefixp (explode x)
(nthcdr n (explode y)))
:exec (strprefixp-impl (the string x)
(the string y)
(the integer 0)
(the (integer 0 *) n)
(the (integer 0 *) xl)
(the (integer 0 *) yl)))
(lnfix n))
((zp n)
nil)
(t
(strrpos-fast (the string x)
(the string y)
(the (integer 0 *) (+ -1 (lnfix n)))
(the (integer 0 *) xl)
(the (integer 0 *) yl)))))
(local (in-theory (enable strrpos-fast)))
(defthm strrpos-fast-type
(or (and (integerp (strrpos-fast x y n xl yl))
(<= 0 (strrpos-fast x y n xl yl)))
(not (strrpos-fast x y n xl yl)))
:rule-classes :type-prescription)
(defthm strrpos-fast-upper-bound
(implies (force (natp n))
(<= (strrpos-fast x y n xl yl) n))
:rule-classes :linear)
(defthm strrpos-fast-when-empty
(implies (and (not (consp (explode x)))
(equal xl (length x))
(equal yl (length y))
(natp n))
(equal (strrpos-fast x y n xl yl)
n))))
(defsection strrpos
:parents (substrings)
:short "Locate the last occurrence of a substring."
:long "<p>@(call strrpos) searches through the string @('y') for the last
occurrence of the substring @('x'). If @('x') occurs somewhere in @('y'), it
returns the starting index of the last occurrence. Otherwise, it returns
@('nil') to indicate that @('x') never occurs in @('y').</p>
<p>The function is \"efficient\" in the sense that it does not coerce its
arguments into lists, but rather traverses both strings with @(see char). On
the other hand, it is a naive string search which operates by repeatedly
calling @(see strprefixp), rather than some better algorithm.</p>
<p>Corner case: we say that the empty string <b>is</b> an prefix of any other
string. As a consequence, @('(strrpos \"\" x)') is (length x) for all
@('x').</p>"
(definlined strrpos (x y)
(declare (type string x y))
(let ((x (mbe :logic (str-fix x) :exec x))
(y (mbe :logic (str-fix y) :exec y))
(yl (length (the string y))))
(declare (type string x y)
(type (integer 0 *) yl))
(strrpos-fast (the string x)
(the string y)
(the (integer 0 *) yl)
(the (integer 0 *) (length (the string x)))
(the (integer 0 *) yl))))
(local (in-theory (enable strrpos strrpos-fast)))
(defthm strrpos-type
(or (and (integerp (strrpos x y))
(<= 0 (strrpos x y)))
(not (strrpos x y)))
:rule-classes :type-prescription)
(local (in-theory (enable str-fix)))
(encapsulate
()
(local (defthm lemma
(implies (and (stringp x)
(stringp y)
(natp xl)
(natp yl)
(natp n)
(<= n (length y))
(= xl (length x))
(= yl (length y))
(strrpos-fast x y n xl yl))
(prefixp (explode x)
(nthcdr (strrpos-fast x y n xl yl)
(explode y))))
:hints(("Goal" :induct (strrpos-fast x y n xl yl)))))
(defthm prefixp-of-strrpos
(implies (strrpos x y)
(prefixp (explode x)
(nthcdr (strrpos x y) (explode y))))))
(encapsulate
()
(local (defun my-induction (x y n m xl yl)
(declare (xargs :measure (nfix n)))
(cond ((prefixp (explode x)
(nthcdr n (explode y)))
nil)
((zp n)
(list x y n m xl yl))
(t
(my-induction x y
(- (nfix n) 1)
(if (= (nfix n) (nfix m))
(- (nfix m) 1)
m)
xl yl)))))
(local (defthm lemma
(implies (and (stringp x)
(stringp y)
(natp xl)
(natp yl)
(natp n)
(natp m)
(>= n m)
(<= n (length y))
(= xl (length x))
(= yl (length y))
(prefixp (explode x)
(nthcdr m (explode y))))
(and (natp (strrpos-fast x y n xl yl))
(>= (strrpos-fast x y n xl yl) m)))
:hints(("Goal"
:induct (my-induction x y n m xl yl)
:do-not '(generalize fertilize)))))
(defthm completeness-of-strrpos
(implies (and (prefixp (explode x)
(nthcdr m (explode y)))
(<= m (len y))
(force (natp m)))
(and (natp (strrpos x y))
(>= (strrpos x y) m)))))
(defthm strrpos-upper-bound-weak
;; Aah, this force is necessary because explode doesn't properly string-fix
;; things. What a bummer...
(implies (force (stringp y))
(<= (strrpos x y)
(len (explode y))))
:rule-classes ((:rewrite) (:linear)))
(encapsulate
()
(local (defthm lemma
(implies (and (stringp x)
(stringp y)
(posp xl)
(posp yl)
(natp n)
(<= n (length y))
(= xl (length x))
(= yl (length y)))
(< (strrpos-fast x y n xl yl) yl))
:hints(("Goal"
:induct (strrpos-fast x y n xl yl)))))
(defthm strrpos-upper-bound-strong
(implies (and (not (equal y ""))
(not (equal x ""))
(force (stringp x))
(force (stringp y)))
(< (strrpos x y)
(len (explode y))))
:rule-classes ((:rewrite) (:linear))))
(encapsulate
()
(local (defthm lens-same-when-list-equiv
(implies (list-equiv x y)
(equal (len x) (len y)))
:rule-classes :forward-chaining))
(local (defthm len-of-nthcdr
(equal (len (nthcdr n x))
(if (< (nfix n) (len x))
(- (len x) (nfix n))
0))))
(local (defthm len-bounded-by-prefixp
(implies (prefixp x y)
(<= (len x) (len y)))
:rule-classes ((:linear) (:forward-chaining))))
(local (defthm prefixp-of-self-and-cdr
(equal (prefixp x (cdr x))
(atom x))))
(local (defthm prefixp-of-nthcdr-bounds-n
(IMPLIES (AND (PREFIXP X (NTHCDR N Y))
(case-split (consp x)))
(<= (nfix N)
(- (LEN Y) (LEN X))))
:rule-classes ((:rewrite) (:linear))
:hints(("Goal" :in-theory (enable prefixp nthcdr)))))
(local (defthm lemma1
(implies (and (strrpos-fast x y n xl yl)
(stringp x)
(stringp y)
(natp n)
(= xl (length x))
(= yl (length y))
(<= n (length y)))
(<= (strrpos-fast x y n xl yl)
(- yl xl)))
:rule-classes ((:rewrite) (:linear))))
(defthm strrpos-upper-bound-stronger
(implies (and (strrpos x y)
(force (stringp x))
(force (stringp y)))
(<= (strrpos x y)
(- (len (explode y))
(len (explode x)))))
:rule-classes ((:rewrite) (:linear)))))
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