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; Fully Ordered Finite Sets
; Copyright (C) 2003-2012 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 "SET")
(include-book "membership")
(set-verify-guards-eagerness 2)
; Fast Difference
;
; As before, we want to show that difference always creates a set and that the
; produced set has the expected membership properties. Also as before, these
; proofs are ugly.
; PATCH (0.91): David Rager noticed that as of v0.9, fast-difference was not
; tail recursive, and submitted an updated version. The original
; fast-difference has been renamed to fast-difference-old, and the new
; fast-difference replaces it.
(defun fast-difference-old (X Y)
(declare (xargs :measure (fast-measure X Y)
:guard (and (setp X) (setp Y))
:verify-guards nil))
(cond ((endp X) nil)
((endp Y) X)
((equal (car X) (car Y))
(fast-difference-old (cdr X) (cdr Y)))
((mbe :logic (<< (car X) (car Y))
:exec (fast-lexorder (car X) (car Y)))
(cons (car X) (fast-difference-old (cdr X) Y)))
(t
(fast-difference-old X (cdr Y)))))
(verify-guards fast-difference-old
:hints(("Goal" :in-theory (enable (:ruleset low-level-rules)))))
(local
(encapsulate ()
(local (defthm l0
(implies (and (consp (fast-difference-old x y))
(or (atom x) (<< a (car x)))
(setp x))
(<< a (car (fast-difference-old x y))))
:hints(("Goal" :in-theory (enable (:ruleset low-level-rules))))))
(defthm fast-difference-old-set
(implies (and (setp X) (setp Y))
(setp (fast-difference-old X Y)))
:hints(("Goal" :in-theory (enable (:ruleset low-level-rules)))))
(local (defthm l1
(implies (and (member a x)
(not (member a y))
(setp x)
(setp y))
(member a (fast-difference-old x y)))
:hints(("Goal" :in-theory (enable (:ruleset low-level-rules))))))
(local (defthm l2
(implies (and (member a (fast-difference-old x y))
(setp x)
(setp y))
(and (member a x)
(not (member a y))))
:hints(("Goal" :in-theory (enable (:ruleset low-level-rules))))))
(local (defthm member-of-fast-difference-old
(implies (and (setp x)
(setp y))
(iff (member a (fast-difference-old x y))
(and (member a x)
(not (member a y)))))))
(defthm fast-difference-old-membership
(implies (and (setp X) (setp Y))
(equal (in a (fast-difference-old X Y))
(and (in a X)
(not (in a Y)))))
:hints(("Goal" :in-theory (enable (:ruleset low-level-rules)))))))
(defun fast-difference (X Y acc)
(declare (xargs :measure (fast-measure X Y)
:guard (and (setp X)
(setp Y)
(true-listp acc))
:verify-guards nil))
(cond ((endp X) (revappend acc nil))
((endp Y) (revappend acc X))
((equal (car X) (car Y))
(fast-difference (cdr X) (cdr Y) acc))
((mbe :logic (<< (car X) (car Y))
:exec (fast-lexorder (car X) (car Y)))
(fast-difference (cdr X) Y (cons (car X) acc)))
(t
(fast-difference X (cdr Y) acc))))
(verify-guards fast-difference
:hints(("Goal" :in-theory (enable (:ruleset low-level-rules)))))
(encapsulate
()
(local (defthm lemma
(implies (true-listp acc)
(equal (fast-difference x y acc)
(revappend acc (fast-difference-old x y))))))
(local (defthm lemma2
(equal (fast-difference x y nil)
(fast-difference-old x y))))
(defthm fast-difference-set
(implies (and (force (setp X))
(force (setp Y)))
(setp (fast-difference X Y nil))))
(defthm fast-difference-membership
(implies (and (setp X) (setp Y))
(equal (in a (fast-difference X Y nil))
(and (in a X)
(not (in a Y))))))
(in-theory (disable fast-difference
fast-difference-set
fast-difference-membership)))
(defsection difference
:parents (std/osets)
:short "@(call difference) removes all members of @('Y') from @('X')."
:long "<p>The logical definition is very simple, and the essential
correctness property is given by @('difference-in').</p>
<p>The execution uses a better, O(n) algorithm to remove the elements by
exploiting the set order.</p>"
(defun difference (X Y)
(declare (xargs :guard (and (setp X) (setp Y))
:verify-guards nil))
(mbe :logic (cond ((empty X) (sfix X))
((in (head X) Y) (difference (tail X) Y))
(t (insert (head X) (difference (tail X) Y))))
:exec (fast-difference X Y nil)))
(defthm difference-set
(setp (difference X Y)))
(defthm difference-sfix-X
(equal (difference (sfix X) Y) (difference X Y)))
(defthm difference-sfix-Y
(equal (difference X (sfix Y)) (difference X Y)))
(defthm difference-empty-X
(implies (empty X)
(equal (difference X Y) (sfix X))))
(defthm difference-empty-Y
(implies (empty Y)
(equal (difference X Y) (sfix X))))
(encapsulate ()
(local (defthm difference-in-X
(implies (in a (difference X Y))
(in a X))))
(local (defthm difference-in-Y
(implies (in a (difference X Y))
(not (in a Y)))))
(defthm difference-in
(equal (in a (difference X Y))
(and (in a X)
(not (in a Y))))))
(encapsulate
()
;; bozo shouldn't really need this
(local (defthm l0
(implies (and (setp y) (setp x) (empty x))
(not (fast-difference x y nil)))
:hints(("Goal" :in-theory (enable fast-difference
(:ruleset low-level-rules))))))
(verify-guards difference
:hints(("Goal" :in-theory (enable fast-difference-set
fast-difference-membership)))))
(defthm difference-subset-X
(subset (difference X Y) X))
(defthm subset-difference
(equal (empty (difference X Y))
(subset X Y))
:hints(("Goal" :in-theory (enable subset))))
(defthm difference-insert-X
(equal (difference (insert a X) Y)
(if (in a Y)
(difference X Y)
(insert a (difference X Y)))))
(defthm difference-preserves-subset
(implies (subset X Y)
(subset (difference X Z)
(difference Y Z)))))
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