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|
; Removing Duplicates
; Copyright (C) 2014 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: Jared Davis <jared@centtech.com>
;
; Additional copyright notice.
;
; Some of the definitions in this book were adapted from the file
; misc/hons-help.lisp, Copyright (C) 2013 the Regents of the University of
; Texas, which was originally written by Bob Boyer and Warren Hunt and is
; available under a 3-clause BSD license.
(in-package "ACL2")
(include-book "xdoc/top" :dir :system)
(include-book "list-defuns")
(local (include-book "equiv"))
(local (include-book "sets"))
(local (include-book "duplicity"))
(local (include-book "no-duplicatesp"))
(defsection std/lists/remove-duplicates-equal
:parents (std/lists remove-duplicates)
:short "Lemmas about @(see remove-duplicates-equal) available in the @(see
std/lists) library."
(defthm remove-duplicates-equal-when-atom
(implies (atom x)
(equal (remove-duplicates-equal x)
nil)))
(defthm remove-duplicates-equal-of-cons
(equal (remove-duplicates-equal (cons a x))
(if (member a x)
(remove-duplicates-equal x)
(cons a (remove-duplicates-equal x)))))
(defthm consp-of-remove-duplicates-equal
(equal (consp (remove-duplicates-equal x))
(consp x)))
(defthm len-of-remove-duplicates-equal
(<= (len (remove-duplicates-equal x))
(len x))
:rule-classes ((:rewrite) (:linear)))
(defthm remove-duplicates-equal-of-list-fix
(equal (remove-duplicates-equal (list-fix x))
(remove-duplicates-equal x)))
(defcong list-equiv equal (remove-duplicates-equal x) 1
:hints(("Goal"
:in-theory (e/d (list-equiv)
(remove-duplicates-equal-of-list-fix))
:use ((:instance remove-duplicates-equal-of-list-fix)
(:instance remove-duplicates-equal-of-list-fix (x x-equiv))))))
(defthm member-of-remove-duplicates-equal
(iff (member a (remove-duplicates-equal x))
(member a x)))
(defthm remove-duplicates-equal-under-set-equiv
(set-equiv (remove-duplicates-equal x)
x)
:hints(("Goal" :in-theory (enable set-equiv))))
(defcong set-equiv set-equiv (remove-duplicates-equal x) 1)
(defthm no-duplicatesp-equal-of-remove-duplicates-equal
(no-duplicatesp-equal (remove-duplicates-equal x)))
(defthm duplicity-in-of-remove-duplicates-equal
(equal (duplicity a (remove-duplicates-equal x))
(if (member a x)
1
0)))
(defthm remove-duplicates-equal-of-remove
(equal (remove-duplicates-equal (remove a x))
(remove a (remove-duplicates-equal x)))))
(defsection hons-remove-duplicates
:parents (std/lists remove-duplicates)
:short "@(call hons-remove-duplicates) removes duplicate elements from a
list, and is implemented using @(see fast-alists)."
:long "<p>This operation is similar to the built-in ACL2 function @(see
remove-duplicates-equal), but it has different performance characteristics and
the answer it produces is in an oddly different order.</p>
<p>ACL2's @(see remove-duplicates) function preserves the order of
non-duplicated elements but has a rather inefficient @('O(n^2)')
implementation: it walks down the list, checking for each element whether there
are any more occurrences of the element in the tail of the list.</p>
<p>In contrast, @('hons-remove-duplicates') is a tail-recursive function that
uses a fast alist to record the elements that have been seen so far. This is
effectively @('O(n)') if we assume that hashing is constant time. Note,
however. that this approach requires us to @(see hons) the elements in the
list, which may be quite expensive if the list contains large structures.</p>
<p>Another reasonably efficient way to remove duplicate elements from a list
include sorting the elements, e.g., via @(see set::mergesort).</p>
<p>In the special case of removing duplicates from lists of natural numbers,
@(see nat-list-remove-duplicates) may be offer superior performance.</p>
<p>Even though @('hons-remove-duplicates') has a reasonably nice logical
definition that is merely in terms of @(see remove-duplicates-equal) and @(see
rev), we leave it disabled by default and go ahead and prove various theorems
about it. These theorems are perhaps best regarded as a back-up plan. In
general, you may often be able to avoid reasoning about
@('hons-remove-duplicates') via the following, strong @(see set-equiv)
rule:</p>
@(def hons-remove-duplicates-under-set-equiv)"
(defund hons-remove-duplicates1 (l tab)
(declare (xargs :guard t))
(cond ((atom l)
;; BOZO a bit unfortunate to free this table here. Better would be
;; to move the freeing to the :exec part of hons-remove-duplicates,
;; except that we might be depending on this freeing somewhere else?
(progn$ (fast-alist-free tab)
nil))
((hons-get (car l) tab)
(hons-remove-duplicates1 (cdr l) tab))
(t
(cons (car l)
(hons-remove-duplicates1 (cdr l)
(hons-acons (car l) t tab))))))
(defund hons-remove-duplicates (l)
(declare (xargs :guard t :verify-guards nil))
(mbe :logic (rev (remove-duplicates-equal (rev l)))
:exec (hons-remove-duplicates1 l (len l))))
(local (in-theory (enable hons-remove-duplicates1
hons-remove-duplicates)))
(defthm hons-remove-duplicates-when-atom
(implies (atom x)
(equal (hons-remove-duplicates x)
nil)))
(defthm hons-remove-duplicates-of-list-fix
(equal (hons-remove-duplicates (list-fix x))
(hons-remove-duplicates x)))
(defcong list-equiv equal (hons-remove-duplicates x) 1
:hints(("Goal"
:in-theory (e/d (list-equiv)
(hons-remove-duplicates
hons-remove-duplicates-of-list-fix))
:use ((:instance hons-remove-duplicates-of-list-fix)
(:instance hons-remove-duplicates-of-list-fix (x x-equiv))))))
(defcong set-equiv set-equiv (hons-remove-duplicates x) 1)
(defthm hons-remove-duplicates-under-set-equiv
(set-equiv (hons-remove-duplicates x)
(remove-duplicates-equal x)))
(defthm no-duplicatesp-equal-of-hons-remove-duplicates
(no-duplicatesp-equal (hons-remove-duplicates x)))
(defthm duplicity-in-of-hons-remove-duplicates
(equal (duplicity a (hons-remove-duplicates x))
(if (member a x)
1
0)))
(encapsulate
()
(local (include-book "remove"))
(local (include-book "../alists/alist-keys"))
;; BOZO we should have these lemmas about set-difference somewhere decent
(local (defthm set-difference-equal-of-cons-right
(equal (set-difference-equal x (cons a y))
(remove a (set-difference-equal x y)))))
(local (defthm set-difference-equal-of-list-fix-left
(equal (set-difference-equal (list-fix x) y)
(set-difference-equal x y))))
(local (defthm set-difference-equal-of-append-left
(equal (set-difference-equal (append a b) c)
(append (set-difference-equal a c)
(set-difference-equal b c)))))
(local (defthm set-difference-equal-of-remove-left
(equal (set-difference-equal (remove a x) y)
(remove a (set-difference-equal x y)))))
(local (defthm rev-of-remove-equal
(equal (rev (remove-equal a x))
(remove-equal a (rev x)))))
(local (in-theory (disable remove-of-rev
remove-of-set-difference-equal)))
(local (defthm l0
(implies (not (hons-assoc-equal x1 tab))
(equal (cons x1 (remove-equal x1 (rev (remove-duplicates-equal x2))))
(rev (remove-duplicates-equal (append x2 (list x1))))))))
(local (defthm hons-remove-duplicates1-redef
(equal (hons-remove-duplicates1 x tab)
(rev (remove-duplicates-equal
(set-difference-equal (rev x)
(alist-keys tab)))))
:hints(("Goal" :induct (hons-remove-duplicates1 x tab)))))
(local (defthm set-difference-equal-of-nil-right
(equal (set-difference-equal a nil)
(list-fix a))))
(defthmd hons-remove-duplicates1-of-nil
(equal (hons-remove-duplicates1 x nil)
(hons-remove-duplicates x)))
(verify-guards hons-remove-duplicates))
(local (defthm hons-remove-duplicates1-acl2-count
(<= (acl2-count (hons-remove-duplicates1 x y))
(acl2-count x))
:hints(("Goal" :in-theory (e/d (hons-remove-duplicates1 acl2-count))))
:rule-classes :linear))
(defthm acl2-count-of-hons-remove-duplicates
(<= (acl2-count (hons-remove-duplicates x))
(acl2-count x))
:hints(("Goal"
:in-theory (disable hons-remove-duplicates1
hons-remove-duplicates)
:use ((:instance hons-remove-duplicates1-of-nil))))
:rule-classes ((:rewrite) (:linear))))
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