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; Lemmas about remove
; Copyright (C) 2013 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 "ACL2")
(include-book "xdoc/top" :dir :system)
(include-book "list-defuns")
(include-book "abstract")
(local (include-book "duplicity"))
(local (include-book "append"))
(local (include-book "sets"))
(defsection std/lists/remove
:parents (std/lists remove)
:short "Lemmas about @(see remove) available in the @(see std/lists)
library."
(defthm remove-when-atom
(implies (atom x)
(equal (remove a x)
nil)))
(defthm remove-of-cons
(equal (remove a (cons b x))
(if (equal a b)
(remove a x)
(cons b (remove a x)))))
(defthm consp-of-remove
;; BOZO consider all-equalp or similar instead?
(equal (consp (remove a x))
(not (subsetp x (list a)))))
(defthm remove-under-iff
(iff (remove a x)
(not (subsetp x (list a)))))
(defthm remove-when-non-member
(implies (not (member a x))
(equal (remove a x)
(list-fix x))))
(defthm upper-bound-of-len-of-remove-weak
(<= (len (remove a x))
(len x))
:rule-classes ((:rewrite) (:linear)))
(defthm upper-bound-of-len-of-remove-strong
(implies (member a x)
(< (len (remove a x))
(len x)))
:rule-classes :linear)
(defthm len-of-remove-exact
;; May not always be desirable, but leave it enabled by default; if someone
;; disables this, they will still have at least the basic upper bound
;; theorems above.
(equal (len (remove a x))
(- (len x) (duplicity a x))))
(defthm remove-is-commutative
(equal (remove b (remove a x))
(remove a (remove b x))))
(defthm remove-is-idempotent
(equal (remove a (remove a x))
(remove a x)))
(defthm duplicity-of-remove
(equal (duplicity a (remove b x))
(if (equal a b)
0
(duplicity a x))))
;; Note: proved elsewhere in std:
;; (defcong list-equiv equal (remove a x) 2)
;; (defcong set-equiv set-equiv (remove a x) 2)
;; (defthm member-of-remove
;; (iff (member a (remove b x))
;; (and (member a x)
;; (not (equal a b)))))
;; (defthm subsetp-of-remove1
;; (equal (subsetp x (remove a y))
;; (and (subsetp x y)
;; (not (member a x)))))
;; (defthm subsetp-of-remove2
;; (implies (subsetp x y)
;; (subsetp (remove a x) y))))
(defthm remove-of-append
(equal (remove a (append x y))
(append (remove a x)
(remove a y))))
(defthm remove-of-revappend
(equal (remove a (revappend x y))
(revappend (remove a x)
(remove a y))))
(defthm remove-of-rev
(equal (remove a (rev x))
(rev (remove a x))))
(defthm remove-of-union-equal
(equal (remove a (union-equal x y))
(union-equal (remove a x)
(remove a y))))
(defthm remove-of-intersection-equal
(equal (remove a (intersection-equal x y))
(intersection-equal (remove a x)
(remove a y))))
(defthm remove-of-set-difference-equal
(equal (remove a (set-difference-equal x y))
(set-difference-equal (remove a x) y)))
(def-listp-rule element-list-p-of-remove
(implies (element-list-p x)
(element-list-p (remove a x)))))
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