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; List Equivalence
; Copyright (C) 2013 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 authors: Jared Davis <jared@centtech.com>
; Sol Swords <sswords@centtech.com>
(in-package "ACL2")
(include-book "list-fix")
(local (include-book "std/basic/inductions" :dir :system))
(local (include-book "append"))
; [Jared]: I split these out of centaur/misc/lists.lisp. I tried to mainly
; keep the rules about list-equiv and how it relates to built-in functions.
; But this omits things like arith-equivs, list-defthms, and rewrite rules to
; rewrite various functions.
(defsection list-equiv
:parents (std/lists)
:short "@(call list-equiv) is an @(see equivalence) relation that determines
whether @('x') and @('y') are identical except perhaps in their @(see
final-cdr)s."
:long "<p>This is a very common equivalence relation for functions that
process lists. See also @(see list-fix) for more discussion.</p>"
(defund fast-list-equiv (x y)
(declare (xargs :guard t))
(if (consp x)
(and (consp y)
(equal (car x) (car y))
(fast-list-equiv (cdr x) (cdr y)))
(atom y)))
(local (defthm fast-list-equiv-removal
(equal (fast-list-equiv x y)
(equal (list-fix x) (list-fix y)))
:hints(("Goal" :in-theory (enable fast-list-equiv)))))
(defund list-equiv (x y)
(mbe :logic (equal (list-fix x) (list-fix y))
:exec (fast-list-equiv x y)))
(verify-guards list-equiv)
(local (in-theory (enable list-equiv)))
(defequiv list-equiv)
(defthm list-equiv-when-atom-left
(implies (atom x)
(equal (list-equiv x y)
(atom y)))
:hints(("Goal" :in-theory (enable list-equiv))))
(defthm list-equiv-when-atom-right
(implies (atom y)
(equal (list-equiv x y)
(atom x)))
:hints(("Goal" :in-theory (enable list-equiv))))
(defthm list-equiv-of-nil-left
(equal (list-equiv nil y)
(not (consp y))))
(defthm list-equiv-of-nil-right
(equal (list-equiv x nil)
(not (consp x))))
(defthm list-fix-under-list-equiv
(list-equiv (list-fix x) x))
(defthm list-fix-equal-forward-to-list-equiv
(implies (equal (list-fix x) (list-fix y))
(list-equiv x y))
:rule-classes :forward-chaining)
(defthm append-atom-under-list-equiv
(implies (atom y)
(list-equiv (append x y)
x))))
(local (in-theory (enable list-equiv)))
(defsection basic-list-equiv-congruences
:parents (list-equiv)
:short "Basic @(see list-equiv) @(see congruence) theorems for built-in
functions."
(defcong list-equiv equal (list-fix x) 1)
(defcong list-equiv equal (car x) 1)
(defcong list-equiv list-equiv (cdr x) 1)
(defcong list-equiv list-equiv (cons x y) 2)
(defcong list-equiv equal (nth n x) 2)
(defcong list-equiv list-equiv (nthcdr n x) 2)
(defcong list-equiv list-equiv (update-nth n v x) 3)
(defcong list-equiv equal (consp x) 1 :hints (("Goal" :induct (cdr-cdr-induct x x-equiv))))
(defcong list-equiv equal (len x) 1 :hints (("Goal" :induct (cdr-cdr-induct x x-equiv))))
(defcong list-equiv equal (append x y) 1 :hints (("Goal" :induct (cdr-cdr-induct x x-equiv))))
(defcong list-equiv list-equiv (append x y) 2)
(defcong list-equiv list-equiv (member k x) 2 :hints(("Goal" :induct (cdr-cdr-induct x x-equiv))))
(defcong list-equiv iff (member k x) 2 :hints(("Goal" :induct (cdr-cdr-induct x x-equiv))))
(defcong list-equiv equal (subsetp x y) 1 :hints(("Goal" :induct (cdr-cdr-induct x x-equiv))))
(defcong list-equiv equal (subsetp x y) 2 :hints(("Goal" :induct (cdr-cdr-induct x x-equiv))))
(defcong list-equiv equal (remove a x) 2 :hints (("Goal" :induct (cdr-cdr-induct x x-equiv))))
(defcong list-equiv equal (resize-list lst n default) 1)
(defcong list-equiv equal (revappend x y) 1
:hints (("Goal" :induct (and (cdr-cdr-induct x x-equiv)
(revappend x y)))))
(defcong list-equiv list-equiv (revappend x y) 2)
(defcong list-equiv list-equiv (last x) 1
:hints(("Goal" :induct (cdr-cdr-induct x x-equiv))))
(defcong list-equiv list-equiv (make-list-ac n val ac) 3)
(defcong list-equiv equal (no-duplicatesp-equal x) 1
:hints(("Goal" :induct (cdr-cdr-induct x x-equiv))))
(defcong list-equiv equal (string-append-lst x) 1
:hints(("Goal" :induct (cdr-cdr-induct x x-equiv)))))
(defsection list-equiv-reductions
:parents (list-equiv)
:short "Lemmas for reducing @(see list-equiv) terms involving basic functions."
(defthm list-equiv-of-cons-and-cons
(equal (list-equiv (cons a x) (cons a y))
(list-equiv x y)))
(defthm list-equiv-of-append-and-append
(equal (list-equiv (append a x) (append a y))
(list-equiv x y)))
(defthm list-equiv-of-revappend-and-revappend
(equal (list-equiv (revappend a x) (revappend a y))
(list-equiv x y))))
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