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; no-duplicates.lisp: Reasoning about no-duplicatesp-equal and
; intersectp-equal. Name-compatible with list-theory.lisp and set-theory.lisp.
; Copyright (C) 2010 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: Sol Swords <sswords@centtech.com>
(in-package "ACL2")
(defthm associativity-of-append
;; Renamed by Jared for compatibility with data-structures/list-defthms
(equal (append (append a b) c)
(append a (append b c))))
(defthm member-equal-append
(iff (member-equal x (append a b))
(or (member-equal x a)
(member-equal x b))))
;; Note: inesrting double-rewrites here because A and B are now in a
;; set-equivalence context within intersectp-equal, whereas they weren't
;; within no-duplicatesp-equal.
(defthm no-duplicatesp-equal-append-iff
(iff (no-duplicatesp-equal (append a b))
(and (no-duplicatesp-equal a)
(no-duplicatesp-equal b)
(not (intersectp-equal (double-rewrite a)
(double-rewrite b))))))
(defthm intersectp-equal-append1
(equal (intersectp-equal (append a b) c)
(or (intersectp-equal a c)
(intersectp-equal b c))))
(defthm intersectp-equal-cons-second
(iff (intersectp-equal a (cons c b))
(or (member-equal c a)
(intersectp-equal a b))))
(defthm intersectp-equal-non-cons-1
(implies (not (consp a))
(not (intersectp-equal a b))))
(defthm intersectp-equal-non-cons
(implies (not (consp b))
(not (intersectp-equal a b))))
(defthm intersectp-equal-append2
(equal (intersectp-equal c (append a b))
(or (intersectp-equal c a)
(intersectp-equal c b))))
(defthm intersectp-equal-commute
(iff (intersectp-equal a b)
(intersectp-equal b a)))
; [Removed by Matt K. to handle changes to member, assoc, etc. after ACL2 4.2.]
; (defthm member-is-member-equal
; (equal (member x y) (member-equal x y)))
; [Removed by Matt K. to handle changes to member, assoc, etc. after ACL2 4.2.]
; (defthm no-duplicatesp-is-no-duplicatesp-equal
; (equal (no-duplicatesp x)
; (no-duplicatesp-equal x)))
(defthm no-duplicatesp-equal-non-cons
(implies (not (consp x))
(no-duplicatesp-equal x)))
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