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; Copyright (C) 2001 Georgia Institute of Technology
; This program is free software; you can redistribute it and/or modify
; it under the terms of the GNU General Public License as published by
; the Free Software Foundation; either version 2 of the License, or
; (at your option) any later version.
; This program is distributed in the hope that it will be useful,
; but WITHOUT ANY WARRANTY; without even the implied warranty of
; MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
; GNU General Public License for more details.
; You should have received a copy of the GNU General Public License
; along with this program; if not, write to the Free Software
; Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA.
; Written by: Panagiotis Manolios who can be reached as follows.
; Email: manolios@cc.gatech.edu
; Postal Mail:
; College of Computing
; Georgia Institute of Technology
; 801 Atlantic Drive
; Atlanta, Georgia 30332-0280 U.S.A.
(in-package "ACL2")
(defthm nth-cons-open
(implies (equal y (cons a b))
(equal (nth n y)
(if (zp n)
a
(nth (1- n) b)))))
(defthm open-nth
(equal (nth n (cons a b))
(if (zp n)
a
(nth (1- n) b))))
(defthm compare-updates
(implies (and (equal i1 i2)
(not (equal v1 v2)))
(equal (equal (update-nth i1 v1 l1)
(update-nth i2 v2 l2))
nil)))
(defthm update-nth-cons-open
(implies (equal y (cons a b))
(equal (update-nth n x y)
(if (zp n)
(cons x b)
(cons a (update-nth (1- n) x b))))))
(defthm open-update-nth
(equal (update-nth n x (cons a b))
(if (zp n)
(cons x b)
(cons a (update-nth (1- n) x b)))))
(defthm nth-update-nth2
(equal (nth n1 (update-nth n2 v l))
(if (equal (nfix n1) (nfix n2))
v
(nth n1 l))))
(defthm update-nth-update-nth-diff
(implies
(not (equal (nfix i1) (nfix i2)))
(equal (update-nth i1 v1 (update-nth i2 v2 l))
(update-nth i2 v2 (update-nth i1 v1 l))))
:rule-classes ((:rewrite :loop-stopper ((i1 i2)))))
(defthm update-nth-update-nth-same
(equal (update-nth i v1 (update-nth i v2 l))
(update-nth i v1 l)))
(defthm nth-over-if
(equal (nth n (if a b c))
(if a
(nth n b)
(nth n c))))
(in-theory (disable nth nth-over-if update-nth))
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