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; Duplicity -- Count how many times an element occurs in a list
; Copyright (C) 2008 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>
(in-package "ACL2")
(include-book "xdoc/top" :dir :system)
(include-book "equiv")
(include-book "rev")
(include-book "flatten")
(defsection duplicity
:parents (std/lists count no-duplicatesp)
:short "@(call duplicity) counts how many times the element @('a') occurs
within the list @('x')."
:long "<p>This function is much nicer to reason about than ACL2's built-in
@(see count) function because it is much more limited:</p>
<ul>
<li>@('count') can operate on either strings or lists; @('duplicity') only
works on lists.</li>
<li>@('count') can consider only some particular sub-range of its input;
@('duplicity') always considers the whole list.</li>
</ul>
<p>Reasoning about duplicity is useful when trying to show two lists are
permutations of one another (sometimes called multiset- or bag-equivalence). A
classic exercise for new ACL2 users is to prove that a permutation function is
symmetric. Hint: a duplicity-based argument may compare quite favorably to
induction on the definition of permutation.</p>
<p>This function can also be useful when trying to establish @(see
no-duplicatesp), e.g., see @(see no-duplicatesp-equal-same-by-duplicity).</p>"
(defund duplicity-exec (a x n)
(declare (xargs :guard (natp n)))
(if (atom x)
n
(duplicity-exec a (cdr x)
(if (equal (car x) a)
(+ 1 n)
n))))
(defund duplicity (a x)
(declare (xargs :guard t
:verify-guards nil))
(mbe :logic (cond ((atom x)
0)
((equal (car x) a)
(+ 1 (duplicity a (cdr x))))
(t
(duplicity a (cdr x))))
:exec (duplicity-exec a x 0)))
(defthm duplicity-exec-removal
(implies (natp n)
(equal (duplicity-exec a x n)
(+ (duplicity a x) n)))
:hints(("Goal" :in-theory (enable duplicity duplicity-exec))))
(verify-guards duplicity
:hints(("Goal" :in-theory (enable duplicity))))
(defthm duplicity-when-not-consp
(implies (not (consp x))
(equal (duplicity a x)
0))
:hints(("Goal" :in-theory (enable duplicity))))
(defthm duplicity-of-cons
(equal (duplicity a (cons b x))
(if (equal a b)
(+ 1 (duplicity a x))
(duplicity a x)))
:hints(("Goal" :in-theory (enable duplicity))))
(defthm duplicity-of-list-fix
(equal (duplicity a (list-fix x))
(duplicity a x))
:hints(("Goal" :induct (len x))))
(defcong list-equiv equal (duplicity a x) 2
:hints(("Goal"
:in-theory (e/d (list-equiv)
(duplicity-of-list-fix))
:use ((:instance duplicity-of-list-fix)
(:instance duplicity-of-list-fix (x x-equiv))))))
(defthm duplicity-of-append
(equal (duplicity a (append x y))
(+ (duplicity a x)
(duplicity a y)))
:hints(("Goal" :induct (len x))))
(defthm duplicity-of-rev
(equal (duplicity a (rev x))
(duplicity a x))
:hints(("Goal" :induct (len x))))
(defthm duplicity-of-revappend
(equal (duplicity a (revappend x y))
(+ (duplicity a x)
(duplicity a y)))
:hints(("Goal" :induct (revappend x y))))
(defthm duplicity-of-reverse
(equal (duplicity a (reverse x))
(duplicity a x)))
(defthm duplicity-when-non-member-equal
(implies (not (member-equal a x))
(equal (duplicity a x)
0)))
(defthm duplicity-when-member-equal
;; New lemma by Keshav Kini to complement the nonmember lemma above
(implies (member-equal a x)
(< 0 (duplicity a x)))
:rule-classes ((:rewrite) (:linear)))
(defthm duplicity-zero-to-member-equal
;; This also seems like a solid rule for normalizing nonzero duplicity into
;; membership. It's potentially expensive since it targets equal, but I
;; think we are generally okay with that.
(iff (equal 0 (duplicity a x))
(not (member-equal a x))))
(defthm no-duplicatesp-equal-when-high-duplicity
(implies (> (duplicity a x) 1)
(not (no-duplicatesp-equal x))))
(defthm duplicity-of-flatten-of-rev
(equal (duplicity a (flatten (rev x)))
(duplicity a (flatten x)))
:hints(("Goal" :induct (len x)))))
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