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; Centaur Bitops Library
; Copyright (C) 2010-2011 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: Anna Slobodova <anna@centtech.com>
; Minor modifications by: Shilpi Goel <shilpi@centtech.com>
(in-package "BITOPS")
(include-book "xdoc/top" :dir :system)
(local (include-book "arithmetic/top-with-meta" :dir :system))
(local (include-book "ihs/quotient-remainder-lemmas" :dir :system))
(local (in-theory (e/d () (expt floor))))
(defsection bitops/logbitp-bounds
:parents (bitops)
:short "A book about @(see logbitp) and @(see expt)."
:long "<p>This is a fast-loading book that can be used separately from the rest of Bitops.</p>")
;; ======================================================================
(defsection logbitp-bounds
:parents (bitops/logbitp-bounds logbitp)
:short "Lemmas about inferring @('logbitp') information from bounds
expressed using @('expt') and vice versa"
:long "<ul>
<li>@(' (logbitp n x) => (<= (expt 2 n) x)')</li>
<li>@(' (logbitp n (expt 2 n)) = t')</li>
<li>@(' (<= (expt 2 n) x)') and @('(< x (expt 2 (1+ n))) => (logbitp n
x)') and @(' (not (logbitp (1+ n) x))')</li>
</ul>
"
(defthm logbitp-to-lower-bound
(implies (and (logbitp n x)
(natp n)
(natp x))
(<= (expt 2 n) x))
:hints (("Goal" :in-theory (enable logbitp)))
:rule-classes (:linear :rewrite))
(defthm expt-2-n-to-logbitp
(implies (natp n)
(logbitp n (expt 2 n)))
:hints (("Goal" :in-theory (enable logbitp))))
(local
(defthm lemma-1
(implies (and (< x (expt 2 (+ 1 n)))
(natp n)
(natp x))
(<= (floor x (expt 2 n))
1))
:hints (("Goal" :nonlinearp t))
:rule-classes :linear))
(local
(defthmd lemma-2
(implies (and (rationalp x)
(<= x 1)
(< 0 x))
(not (integerp (* 1/2 x))))
:hints (("Goal" :nonlinearp t))))
(defthm bounds-to-logbitp-lemma
(implies (and (< (expt 2 n) x)
(< x (expt 2 (+ 1 n)))
(natp n)
(natp x))
(logbitp n x))
:hints (("Goal" :in-theory (e/d (logbitp) ())
:use ((:instance lemma-2
(x (floor x (expt 2 n))))))))
(local
(defthm floor-of-expt2
(implies (and (< (expt 2 n) x)
(< x (* 2 (expt 2 n)))
(natp n)
(natp x))
(equal (floor x (expt 2 n)) 1))))
(defthm bounds-to-logbitp
(implies (and (<= (expt 2 n) x)
(< x (expt 2 (+ 1 n)))
(natp n)
(natp x))
(and (logbitp n x)
(not (logbitp (+ 1 n) x))))
:hints (("Goal" :cases ((equal x (expt 2 n)))))))
;; ======================================================================
|
