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; Rev function and lemmas
; Copyright (C) 2005-2013 Kookamara LLC
;
; Contact:
;
; Kookamara LLC
; 11410 Windermere Meadows
; Austin, TX 78759, USA
; http://www.kookamara.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@kookamara.com>
;
; rev.lisp
; This file was originally part of the Unicode library.
(in-package "ACL2")
(include-book "list-fix")
(local (include-book "revappend"))
(local (include-book "append"))
(include-book "equiv")
(defun revappend-without-guard (x y)
(declare (xargs :guard t))
(mbe :logic (revappend x y)
:exec (if (consp x)
(revappend-without-guard (cdr x)
(cons (car x) y))
y)))
(defsection rev
:parents (std/lists reverse)
:short "Logically simple alternative to @(see reverse) for lists."
:long "<p>This function is nicer to reason about than ACL2's built-in @(see
reverse) function because it is more limited:</p>
<ul>
<li>@('reverse') can operate on strings or lists, whereas @('rev') can only
operate on lists.</li>
<li>@('reverse') has a tail-recursive definition, which makes it generally
more difficult to induct over than the non tail-recursive @('rev').</li>
</ul>
<p>Despite its simple @(see append)-based logical definition, @('rev') should
perform quite well thanks to @(see mbe).</p>"
(defund rev (x)
(declare (xargs :verify-guards nil
:guard t))
(mbe :logic (if (consp x)
(append (rev (cdr x))
(list (car x)))
nil)
:exec (revappend-without-guard x nil)))
(local (in-theory (enable rev)))
(defthm rev-when-not-consp
(implies (not (consp x))
(equal (rev x)
nil))
:hints(("Goal" :in-theory (enable rev))))
(defthm rev-of-cons
(equal (rev (cons a x))
(append (rev x)
(list a)))
:hints(("Goal" :in-theory (enable rev))))
; Commented out by Matt K., since deduced type-prescription rule for rev at
; definition time already provides this.
; (defthm true-listp-of-rev
; (true-listp (rev x))
; :rule-classes :type-prescription)
(defthm rev-of-list-fix
(equal (rev (list-fix x))
(rev x))
:hints(("Goal" :induct (len x))))
(defthm len-of-rev
(equal (len (rev x))
(len x)))
(defthm rev-of-rev
(equal (rev (rev x))
(list-fix x)))
(defthm consp-of-rev
(equal (consp (rev x))
(consp x))
:hints(("Goal" :induct (len x))))
(defthm rev-under-iff
(iff (rev x) (consp x))
:hints(("Goal" :induct (len x))))
(defthm revappend-removal
(equal (revappend x y)
(append (rev x) y)))
(verify-guards rev)
(defthm reverse-removal
(implies (true-listp x)
(equal (reverse x)
(rev x))))
(defthm rev-of-append
(equal (rev (append x y))
(append (rev y) (rev x))))
(encapsulate
()
(local (defun cdr-cdr-induction (x y)
(if (or (atom x)
(atom y))
nil
(cdr-cdr-induction (cdr x) (cdr y)))))
(local (defthm lemma
(equal (equal (list a) (append y (list b)))
(and (atom y)
(equal a b)))))
(local (defthm lemma2
(implies (and (true-listp x)
(true-listp y))
(equal (equal (append x (list a))
(append y (list b)))
(and (equal x y)
(equal a b))))
:hints(("Goal" :induct (cdr-cdr-induction x y)))))
(defthm equal-of-rev-and-rev
(equal (equal (rev x) (rev y))
(equal (list-fix x) (list-fix y)))
:hints(("Goal" :induct (cdr-cdr-induction x y)))))
(encapsulate
()
(local (defthm make-character-list-of-append
;; Reprove this to avoid including make-character-list
(equal (make-character-list (append x y))
(append (make-character-list x)
(make-character-list y)))))
(defthm make-character-list-of-rev
;; This arguably doesn't belong here, but maybe it makes more sense here
;; than in str/make-character-list, since this way we don't have to include
;; rev just to get lemmas about make-character-list.
(equal (make-character-list (rev x))
(rev (make-character-list x)))
:hints(("Goal" :in-theory (enable make-character-list)))))
(defthm list-equiv-of-rev-and-rev
(equal (list-equiv (rev x) (rev y))
(list-equiv x y))
:hints(("Goal" :in-theory (enable list-equiv))))
(def-listp-rule element-list-p-of-rev
(equal (element-list-p (rev x))
(element-list-p (list-fix x))))
(def-listfix-rule element-list-fix-of-rev
(equal (element-list-fix (rev x))
(rev (element-list-fix x))))
(def-projection-rule elementlist-projection-of-rev
(equal (elementlist-projection (rev x))
(rev (elementlist-projection x)))))
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