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; Milawa - A Reflective Theorem Prover
; Copyright (C) 2005-2009 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>
(in-package "MILAWA")
(include-book "utilities")
(set-verify-guards-eagerness 2)
(set-case-split-limitations nil)
(set-well-founded-relation ord<)
(set-measure-function rank)
(defund intersect (x y)
(declare (xargs :guard t))
(if (consp x)
(if (memberp (car x) y)
(cons (car x)
(intersect (cdr x) y))
(intersect (cdr x) y))
nil))
(defthm intersect-when-not-consp-one
(implies (not (consp x))
(equal (intersect x y)
nil))
:hints(("Goal" :in-theory (enable intersect))))
(defthm intersect-of-cons-one
(equal (intersect (cons a x) y)
(if (memberp a y)
(cons a (intersect x y))
(intersect x y)))
:hints(("Goal" :in-theory (enable intersect))))
(defthm intersect-when-not-consp-two
(implies (not (consp y))
(equal (intersect x y)
nil))
:hints(("Goal" :induct (cdr-induction x))))
(defthm intersect-under-iff
(iff (intersect x y)
(not (disjointp x y)))
:hints(("Goal" :induct (cdr-induction x))))
(defthm true-listp-of-intersect
(equal (true-listp (intersect x y))
t)
:hints(("Goal" :induct (cdr-induction x))))
(defthm intersect-of-list-fix-one
(equal (intersect (list-fix x) y)
(intersect x y))
:hints(("Goal" :induct (cdr-induction x))))
(defthm intersect-of-list-fix-two
(equal (intersect x (list-fix y))
(intersect x y))
:hints(("Goal" :induct (cdr-induction x))))
(defthm intersect-of-app-one
(equal (intersect (app x y) z)
(app (intersect x z)
(intersect y z)))
:hints(("Goal" :induct (cdr-induction x))))
(defthm rev-of-intersect
(equal (rev (intersect x y))
(intersect (rev x) y))
:hints(("Goal" :induct (cdr-induction x))))
(defthm intersect-of-rev-two
(equal (intersect x (rev y))
(intersect x y))
:hints(("Goal" :induct (cdr-induction x))))
(defthm subsetp-of-intersect-one
(equal (subsetp (intersect x y) x)
t)
:hints(("Goal" :induct (cdr-induction x))))
(defthm subsetp-of-intersect-two
(equal (subsetp (intersect x y) y)
t)
:hints(("Goal" :induct (cdr-induction x))))
(defthm intersect-when-subsetp
(implies (subsetp x y)
(equal (intersect x y)
(list-fix x)))
:hints(("Goal" :induct (cdr-induction x))))
(defthm intersect-with-self
(equal (intersect x x)
(list-fix x)))
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