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; Standard Utilities Library
; Copyright (C) 2008-2016 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 "STD")
(include-book "xdoc/top" :dir :system)
(defsection prod-consp
:parents (prod-cons)
:short "Special recognizer for conses, except that to save space we require
that (nil . nil) be represented just as nil."
(defund prod-consp (x)
(declare (xargs :guard t))
(if (consp x)
(and (or (car x) (cdr x))
t)
(not x)))
(local (in-theory (enable prod-consp)))
(defthm booleanp-of-prod-consp
(booleanp (prod-consp x))
:rule-classes :type-prescription)
(defthm prod-consp-compound-recognizer
(implies (prod-consp x)
(or (consp x) (not x)))
:rule-classes :compound-recognizer))
(defsection prod-cons
:parents (defaggregate)
:short "Special constructor for products that collapses (nil . nil) into nil."
(defund-inline prod-cons (x y)
(declare (xargs :guard t))
(and (or x y)
(cons x y)))
(local (in-theory (enable prod-cons)))
(defthm prod-consp-of-prod-cons
(prod-consp (prod-cons x y))
:rule-classes (:rewrite :type-prescription)
:hints(("Goal" :in-theory (enable prod-consp))))
(defthm car-of-prod-cons
(equal (car (prod-cons x y)) x))
(defthm cdr-of-prod-cons
(equal (cdr (prod-cons x y)) y))
(defthm prod-cons-of-car/cdr
(implies (prod-consp x)
(equal (prod-cons (car x) (cdr x))
x))
:hints(("Goal" :in-theory (enable prod-consp))))
(defthmd equal-of-prod-cons
(implies (prod-consp x)
(equal (equal x (prod-cons a b))
(and (equal (car x) a)
(equal (cdr x) b))))
:hints(("Goal" :in-theory (enable prod-consp))))
(defthm acl2-count-of-prod-cons
(and (>= (acl2-count (prod-cons x y))
(acl2-count x))
(>= (acl2-count (prod-cons x y))
(acl2-count y)))
:rule-classes :linear)
(defthm prod-cons-equal-cons
(implies (or a b)
(equal (equal (prod-cons a b) (cons c d))
(and (equal a c)
(equal b d))))
:hints(("Goal" :in-theory (enable prod-cons))))
(defthm prod-cons-when-either
(implies (or a b)
(and (prod-cons a b)
(consp (prod-cons a b))))
:rule-classes ((:rewrite)
(:type-prescription :corollary
(implies (or a b)
(consp (prod-cons a b)))))
:hints(("Goal" :in-theory (enable prod-cons))))
(defthm prod-cons-not-consp-forward
(implies (not (consp (prod-cons a b)))
(and (not a) (not b)))
:hints(("Goal" :in-theory (enable prod-cons)))
:rule-classes ((:forward-chaining :trigger-terms ((prod-cons a b))))))
(defsection prod-cons-with-hint
:parents (prod-cons)
:short "Same as @(see prod-cons), but avoids reconsing like @(see cons-with-hint)."
:long "<p>We leave this enabled and expect to reason about @(see prod-cons) instead.</p>"
(defun-inline prod-cons-with-hint (x y hint)
(declare (xargs :guard t))
(mbe :logic (prod-cons x y)
:exec (and (or x y)
(cons-with-hint x y hint)))))
(defsection prod-car
:parents (prod-cons)
:short "Same as @(see car), but guarded with @(see prod-consp)."
:long "<p>We leave this enabled and expect to reason about @(see car) instead.</p>"
(defun-inline prod-car (x)
(declare (xargs :guard (prod-consp x)))
(car x)))
(defsection prod-cdr
:parents (prod-cons)
:short "Same as @(see cdr), but guarded with @(see prod-consp)."
:long "<p>We leave this enabled and expect to reason about @(see car) instead.</p>"
(defun-inline prod-cdr (x)
(declare (xargs :guard (prod-consp x)))
(cdr x)))
(defsection prod-hons
:parents (prod-cons)
:short "Same as @(see prod-cons) but uses @(see hons)."
:long "<p>We leave this enabled and expect to reason about @(see prod-cons) instead.</p>"
(local (in-theory (enable prod-cons)))
(defun-inline prod-hons (x y)
(declare (xargs :guard t))
(mbe :logic (prod-cons x y)
:exec (and (or x y)
(hons x y)))))
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