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; Centaur Bitops Library
; Copyright (C) 2010-2013 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 author: Jared Davis <jared@centtech.com>
(in-package "BITOPS")
(include-book "ihs/basic-definitions" :dir :system)
(include-book "xdoc/top" :dir :system)
(local (include-book "ihsext-basics"))
(local (include-book "integer-length"))
(local (include-book "arithmetic/top-with-meta" :dir :system))
(defsection logbitp-mismatch
:parents (bitops logbitp)
:short "@(call logbitp-mismatch) finds the minimal bit-position for which
@('a') and @('b') differ, or returns @('NIL') if no such bit exists."
:long "<p>This is mainly useful for proving @(see equal-by-logbitp), but it's
also occasionally useful as a witness in other theorems.</p>"
(defund logbitp-mismatch (a b)
(declare (xargs :measure (+ (integer-length a)
(integer-length b))
:guard (and (integerp a)
(integerp b))))
(cond ((not (equal (logcar a) (logcar b)))
0)
((and (zp (integer-length a))
(zp (integer-length b)))
nil)
(t
(let ((tail (logbitp-mismatch (logcdr a) (logcdr b))))
(and tail (+ 1 tail))))))
(local (in-theory (enable logbitp-mismatch integer-length**)))
(local (in-theory (enable* arith-equiv-forwarding)))
(defthm logbitp-mismatch-under-iff
(iff (logbitp-mismatch a b)
(not (equal (ifix a) (ifix b)))))
(local (in-theory (disable logbitp-mismatch-under-iff)))
(defthm logbitp-mismatch-correct
(implies (logbitp-mismatch a b)
(not (equal (logbitp (logbitp-mismatch a b) a)
(logbitp (logbitp-mismatch a b) b))))
:hints(("Goal"
:in-theory (enable logbitp-mismatch logbitp**)
:induct (logbitp-mismatch a b))))
(defthm logbitp-mismatch-upper-bound
(implies (logbitp-mismatch a b)
(<= (logbitp-mismatch a b)
(max (integer-length a)
(integer-length b))))
:rule-classes ((:rewrite) (:linear)))
(defthm logbitp-mismatch-upper-bound-for-nonnegatives
(implies (and (not (and (integerp a) (< a 0)))
(not (and (integerp b) (< b 0)))
(logbitp-mismatch a b))
(< (logbitp-mismatch a b)
(max (integer-length a)
(integer-length b))))
:rule-classes ((:rewrite) (:linear :trigger-terms ((logbitp-mismatch a b)))))
(defthm integerp-of-logbitp-mismatch
;; BOZO why do I have to have this stupid rule when the type prescription
;; for logbitp-mismatch says it's either a nonnegative integer or nil?
(iff (integerp (logbitp-mismatch a b))
(logbitp-mismatch a b))
:hints(("Goal" :in-theory (enable logbitp-mismatch)))))
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