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; Resize List Lemmas
; Copyright (C) 2008-2013 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 authors: Sol Swords <sswords@centtech.com>
; Jared Davis <jared@centtech.com>
(in-package "ACL2")
(include-book "list-defuns")
(include-book "xdoc/top" :dir :system)
(local (include-book "repeat"))
(local (include-book "nth"))
(defsection std/lists/resize-list
:parents (std/lists resize-list)
:short "Lemmas about @(see resize-list) available in the @(see std/lists)
library."
"<h4>Trivial reductions</h4>"
(defthm resize-list-when-zp
(implies (zp n)
(equal (resize-list lst n default-value)
nil)))
(defthm resize-list-of-nfix
(equal (resize-list lst (nfix n) default-value)
(resize-list lst n default-value)))
(defthm resize-list-when-atom
(implies (atom lst)
(equal (resize-list lst n default-value)
(repeat n default-value)))
:hints(("Goal" :in-theory (enable repeat))))
"<h4>Relation with other basic list functions</h4>"
(local (defun my-induct (n m lst)
(if (zp n)
(list lst)
(if (zp m)
nil
(my-induct (- n 1) (- m 1)
(if (atom lst)
lst
(cdr lst)))))))
(defthm nth-of-resize-list
(equal (nth n (resize-list lst m default-value))
(let ((n (nfix n))
(m (nfix m)))
(and (< n m)
(if (< n (len lst))
(nth n lst)
default-value))))
:hints(("Goal"
:expand (resize-list lst m default-value)
:induct (my-induct n m lst))))
(defthm len-of-resize-list
(equal (len (resize-list lst n default))
(nfix n)))
(defthm resize-list-of-len-free
(implies (equal (nfix n) (len lst))
(equal (resize-list lst n default-value)
(list-fix lst))))
(defthm equal-of-resize-list-and-self
(equal (equal (resize-list lst n default-value) lst)
(and (true-listp lst)
(equal (len lst) (nfix n))))))
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