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; Butlast 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>
;
; butlast.lisp
(in-package "ACL2")
(include-book "abstract")
(local (include-book "take"))
(local (in-theory (enable take)))
(defsection std/lists/butlast
:parents (std/lists butlast)
:short "Lemmas about @(see butlast) available in the @(see std/lists)
library."
"<p>The usual definition of @('butlast') is in terms of @(see take). But
it's often more convenient to reason about @('butlast') directly. In that
case, its recursive redefinition may be what you want:</p>"
(defthm butlast-redefinition
(equal (butlast x n)
(if (>= (nfix n) (len x))
nil
(cons (car x)
(butlast (cdr x) n))))
:rule-classes ((:definition :clique (butlast)
:controller-alist ((butlast t t)))))
(defun butlast-induction (x n)
(if (>= (nfix n) (len x))
nil
(cons (car x)
(butlast-induction (cdr x) n))))
(defthm use-butlast-induction
t
:rule-classes ((:induction
:pattern (butlast x n)
:scheme (butlast-induction x n))))
"<p>For use with @(see std::deflist):</p>"
(def-listp-rule element-list-p-of-butlast
(implies (element-list-p (double-rewrite x))
(element-list-p (butlast x n)))))
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