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|
; RTL - A Formal Theory of Register-Transfer Logic and Computer Arithmetic
; Copyright (C) 1995-2013 Advanced Mirco Devices, Inc.
;
; Contact:
; David Russinoff
; 1106 W 9th St., Austin, TX 78703
; http://www.russsinoff.com/
;
; This program is free software; you can redistribute it and/or modify it under
; the terms of the GNU General Public License as published by the Free Software
; Foundation; either version 2 of the License, or (at your option) any later
; version.
;
; This program is distributed in the hope that it will be useful but WITHOUT ANY
; WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A
; PARTICULAR PURPOSE. See the GNU General Public License for more details.
;
; You should have received a copy of the GNU General Public License along with
; this program; see the file "gpl.txt" in this directory. If not, write to the
; Free Software Foundation, Inc., 51 Franklin Street, Suite 500, Boston, MA
; 02110-1335, USA.
;
; Author: David M. Russinoff (david@russinoff.com)
(in-package "ACL2")
#|
This book is about LIOR0, a nice version of LOGIOR. LIOR0 takes an extra size parameter, N, and always returns
a bit vector of length N.
Todo:
add versions like logand-expt-2 and logand-expt-4
prove (elsewhere) lemmas mixing lior0 with other functions
what should lior0 of non-ints be?
|#
(local ; ACL2 primitive
(defun natp (x)
(declare (xargs :guard t))
(and (integerp x)
(<= 0 x))))
(defund fl (x)
(declare (xargs :guard (real/rationalp x)))
(floor x 1))
(defund bits (x i j)
(declare (xargs :guard (and (natp x)
(natp i)
(natp j))
:verify-guards nil))
(mbe :logic (if (or (not (integerp i))
(not (integerp j)))
0
(fl (/ (mod x (expt 2 (1+ i))) (expt 2 j))))
:exec (if (< i j)
0
(logand (ash x (- j)) (1- (ash 1 (1+ (- i j))))))))
(defund bitn (x n)
(declare (xargs :guard (and (natp x)
(natp n))
:verify-guards nil))
(mbe :logic (bits x n n)
:exec (if (evenp (ash x (- n))) 0 1)))
(defund bvecp (x k)
(declare (xargs :guard (integerp k)))
(and (integerp x)
(<= 0 x)
(< x (expt 2 k))))
(defund all-ones (n)
(declare (xargs :guard (and (integerp n) (<= 0 n))))
(if (zp n)
0 ;degenerate case
(1- (expt 2 n))))
(local (include-book "all-ones"))
(local (include-book "merge"))
(local (include-book "bvecp"))
(local (include-book "logior"))
(local (include-book "bits"))
(local (include-book "bitn"))
(local (include-book "../../arithmetic/top"))
(defund binary-lior0 (x y n)
(declare (xargs :guard (and (natp x)
(natp y)
(integerp n)
(< 0 n))
:verify-guards nil))
(logior (bits x (1- n) 0)
(bits y (1- n) 0)))
(defun formal-+ (x y)
(declare (xargs :guard t))
(if (and (acl2-numberp x) (acl2-numberp y))
(+ x y)
(list '+ x y)))
(defmacro lior0 (&rest x)
(declare (xargs :guard (and (consp x)
(consp (cdr x))
(consp (cddr x)))))
(cond ((endp (cdddr x)) ;(lior0 x y n) -- the base case
`(binary-lior0 ,@x))
(t
`(binary-lior0 ,(car x)
(lior0 ,@(cdr x))
,(car (last x))))))
;Allows things like (in-theory (disable lior0)) to refer to binary-lior0.
(add-macro-alias lior0 binary-lior0)
(defthm lior0-nonnegative-integer-type
(and (integerp (lior0 x y n))
(<= 0 (lior0 x y n)))
:rule-classes (:type-prescription))
;(:type-prescription lior0) is no better than lior0-nonnegative-integer-type and might be worse:
(in-theory (disable (:type-prescription binary-lior0)))
;drop this if we plan to keep natp enabled?
(defthm lior0-natp
(natp (lior0 x y n)))
(defthm lior0-with-n-not-a-natp
(implies (not (natp n))
(equal (lior0 x y n)
0))
:hints (("Goal" :cases ((acl2-numberp n))
:in-theory (enable lior0)))
)
(defthmd lior0-bvecp-simple
(bvecp (lior0 x y n) n)
:hints (("Goal" :cases ((natp n))
:in-theory (enable lior0))))
(defthm lior0-bvecp
(implies (and (<= n k)
(case-split (integerp k)))
(bvecp (lior0 x y n) k))
:hints (("Goal" :in-theory (disable lior0-bvecp-simple)
:use lior0-bvecp-simple)))
;;
;; Rules to normalize lior0 terms (recall that LIOR0 is a macro for BINARY-LIOR0):
;;
;; allow sizes to differ on these?
(defthm lior0-associative
(equal (lior0 (lior0 x y n) z n)
(lior0 x (lior0 y z n) n))
:hints (("Goal" :cases ((natp n))
:in-theory (enable lior0 bits-tail))))
(defthm lior0-commutative
(equal (lior0 y x n)
(lior0 x y n))
:hints (("Goal" :in-theory (enable lior0))))
(defthm lior0-commutative-2
(equal (lior0 y (lior0 x z n) n)
(lior0 x (lior0 y z n) n))
:hints (("Goal" :cases ((natp n))
:in-theory (enable lior0 bits-tail))))
(defthm lior0-combine-constants
(implies (syntaxp (and (quotep x)
(quotep y)
(quotep n)))
(equal (lior0 x (lior0 y z n) n)
(lior0 (lior0 x y n) z n))))
(defthm lior0-0
(implies (case-split (bvecp y n))
(equal (lior0 0 y n)
y))
:hints (("Goal" :in-theory (enable lior0 bits-tail))))
;nicer than the analogous rule for logior?
(defthm lior0-1
(implies (case-split (bvecp y 1))
(equal (lior0 1 y 1)
1))
:hints (("Goal" :in-theory (enable bvecp-1-rewrite))))
(defthm lior0-self
(implies (and (case-split (bvecp x n))
(case-split (integerp n)))
(equal (lior0 x x n)
x))
:hints (("Goal" :in-theory (enable lior0 bits-tail))))
(defthmd bits-lior0-1
(implies (and (< i n)
(case-split (<= 0 j))
(case-split (integerp n))
)
(equal (bits (lior0 x y n) i j)
(lior0 (bits x i j)
(bits y i j)
(+ 1 i (- j)))))
:otf-flg t
:hints (("Goal" :in-theory (enable lior0 bits-logand))))
(defthmd bits-lior0-2
(implies (and (<= n i)
(case-split (<= 0 j))
(case-split (integerp n))
)
(equal (bits (lior0 x y n) i j)
(lior0 (bits x i j)
(bits y i j)
(+ n (- j)))))
:otf-flg t
:hints (("Goal" :in-theory (enable lior0 bits-logand))))
;notice the call to MIN in the conclusion
(defthm bits-lior0
(implies (and (case-split (<= 0 j))
(case-split (integerp n))
(case-split (integerp i))
)
(equal (bits (lior0 x y n) i j)
(lior0 (bits x i j)
(bits y i j)
(+ (min n (+ 1 i)) (- j)))))
:hints (("Goal" :in-theory (enable bits-lior0-1 bits-lior0-2))))
(defthmd bitn-lior0-1
(implies (and (< m n)
(case-split (<= 0 m))
(case-split (integerp n))
)
(equal (bitn (lior0 x y n) m)
(lior0 (bitn x m)
(bitn y m)
1)))
:hints (("Goal" :in-theory (set-difference-theories
(enable bitn)
'(BITS-N-N-REWRITE)))))
(defthmd bitn-lior0-2
(implies (and (<= n m)
(case-split (<= 0 m))
(case-split (integerp n))
)
(equal (bitn (lior0 x y n) m)
0))
:hints (("Goal" :in-theory (enable BVECP-BITN-0))))
;notice the IF in the conclusion
;we expect this to cause case splits only rarely, since m and n will usually be constants
(defthm bitn-lior0
(implies (and (case-split (<= 0 m))
(case-split (integerp n))
)
(equal (bitn (lior0 x y n) m)
(if (< m n)
(lior0 (bitn x m)
(bitn y m)
1)
0)))
:hints (("Goal" :in-theory (enable bitn-lior0-1 bitn-lior0-2))))
;or could wrap bits around conclusion?
(defthm lior0-equal-0
(implies (and (case-split (bvecp x n))
(case-split (bvecp y n))
(case-split (integerp n))
)
(equal (equal 0 (lior0 x y n))
(and (equal x 0)
(equal y 0))))
:hints (("Goal" :in-theory (enable lior0 bits-tail))))
(defthm lior0-of-single-bits-equal-0
(implies (and (case-split (bvecp x 1))
(case-split (bvecp y 1))
)
(equal (equal 0 (lior0 x y 1))
(and (equal x 0)
(equal y 0))))
:hints (("Goal" :in-theory (enable bvecp-1-rewrite))))
(defthm lior0-of-single-bits-equal-1
(implies (and (case-split (bvecp x 1))
(case-split (bvecp y 1))
)
(equal (equal 1 (lior0 x y 1))
(or (equal x 1)
(equal y 1))))
:hints (("Goal" :in-theory (enable bvecp-1-rewrite))))
(defthm lior0-ones
(implies (and (case-split (bvecp x n))
(case-split (natp n)) ;gen
)
(equal (lior0 (1- (expt 2 n)) x n)
(1- (expt 2 n))))
:rule-classes ()
:hints
(("goal" :use logior-ones
:in-theory (enable lior0 bits-tail)
)))
;lior0-with-all-ones will rewrite (lior0 x n) [note there's only one value being ANDed], because (lior0 x n)
;expands to (BINARY-LIOR0 X (ALL-ONES N) N) - now moot???
(defthm lior0-with-all-ones
(implies (case-split (bvecp x n))
(equal (lior0 (all-ones n) x n)
(all-ones n)))
:hints
(("goal" :use lior0-ones
:in-theory (enable all-ones))))
(defthm lior0-ones-rewrite
(implies (and (syntaxp (and (quotep k)
(quotep n)
(equal (cadr k) (1- (expt 2 (cadr n))))))
(force (equal k (1- (expt 2 n))))
(case-split (natp n))
(case-split (bvecp x n)))
(equal (lior0 k x n)
(1- (expt 2 n))))
:hints (("Goal"
:use lior0-ones)))
(local (in-theory (disable MOD-BY-2-REWRITE-TO-EVEN MOD-MULT-OF-N MOD-EQUAL-0 )))
(encapsulate
()
(local
(defthm lior0-def-integerp
(implies (and (integerp x)
(integerp y)
(> n 0)
(integerp n))
(equal (lior0 x y n)
(+ (* 2 (lior0 (fl (/ x 2)) (fl (/ y 2)) (1- n)))
(lior0 (mod x 2) (mod y 2) 1))))
:rule-classes ()
:hints (("Goal" :in-theory (e/d (lior0 bits-fl-by-2)
())
:use ((:instance logior-def (i (bits x (1- n) 0)) (j (bits y (1- n) 0)))
(:instance mod012 (m x))
(:instance mod012 (m y)))))))
; Now we want to eliminate the (integerp x) and (integerp y) hypotheses from
; lior0-def-integerp. First suppose x is not rational.
(local
(defthm lior0-is-0-if-not-rational-1
(implies (not (rationalp x))
(equal (lior0 x y n)
(lior0 0 y n)))
:hints (("Goal" :expand ((lior0 x y n)
(lior0 0 y n))))))
(local
(defthm lior0-is-0-if-not-rational-2
(implies (not (rationalp y))
(equal (lior0 x y n)
(lior0 x 0 n)))
:hints (("Goal" :expand ((lior0 x y n)
(lior0 0 x n))))))
(local
(defthm fl-1/2-is-0-if-not-rational
(implies (not (rationalp x))
(equal (fl (* 1/2 x)) 0))
:hints (("Goal" :cases ((acl2-numberp x))))))
(local
(defthm mod-2-if-not-rational
(implies (not (rationalp x))
(equal (mod x 2)
(fix x)))
:hints (("Goal" :expand ((mod x 2))))))
(local
(defthm lior0-fl-1
(equal (lior0 (fl x) y n)
(lior0 x y n))
:hints (("Goal" :expand ((lior0 y (fl x) n)
(lior0 x y n))))))
(local
(defthm lior0-fl-2
(equal (lior0 y (fl x) n)
(lior0 y x n))
:hints (("Goal" :expand ((lior0 y (fl x) n)
(lior0 x y n))))))
(local
(defthm lior0-def-rational-hack
(implies (and (rationalp x)
(rationalp y)
(>= n 0)
(integerp n))
(equal (lior0 (* 1/2 (fl x)) (* 1/2 (fl y)) n)
(lior0 (* 1/2 x) (* 1/2 y) n)))
:hints (("Goal" :expand ((lior0 (* 1/2 (fl x)) (* 1/2 (fl y)) n)
(lior0 (* 1/2 x) (* 1/2 y) n))))))
(local
(defthm lior0-def-rational
(implies (and (rationalp x)
(rationalp y)
(> n 0)
(integerp n))
(equal (lior0 x y n)
(+ (* 2 (lior0 (fl (/ x 2)) (fl (/ y 2)) (1- n)))
(lior0 (mod x 2) (mod y 2) 1))))
:rule-classes ()
:hints (("Goal"
:use ((:instance lior0-def-integerp (x (fl x)) (y (fl y))))
:in-theory (e/d (mod-fl-eric) (fl-mod))))))
(local
(defthm lior0-def-not-rational-1
(implies (and (not (rationalp x))
(rationalp y)
(> n 0)
(integerp n))
(equal (lior0 x y n)
(+ (* 2 (lior0 (fl (/ x 2)) (fl (/ y 2)) (1- n)))
(lior0 (mod x 2) (mod y 2) 1))))
:hints (("Goal" :use ((:instance lior0-def-rational
(x 0)))))
:rule-classes nil))
(local
(defthm lior0-def-not-rational-2
(implies (and (rationalp x)
(not (rationalp y))
(> n 0)
(integerp n))
(equal (lior0 x y n)
(+ (* 2 (lior0 (fl (/ x 2)) (fl (/ y 2)) (1- n)))
(lior0 (mod x 2) (mod y 2) 1))))
:hints (("Goal" :use ((:instance lior0-def-rational
(y 0)))))
:rule-classes nil))
(defthm lior0-def
(implies (and (> n 0)
(integerp n))
(equal (lior0 x y n)
(+ (* 2 (lior0 (fl (/ x 2)) (fl (/ y 2)) (1- n)))
(lior0 (mod x 2) (mod y 2) 1))))
:rule-classes ()
:hints (("Goal" :use (lior0-def-not-rational-1
lior0-def-not-rational-2
lior0-def-rational)))))
(defthm lior0-mod-2
(implies (and (natp x)
(natp y)
(natp n)
(> n 0))
(equal (mod (lior0 x y n) 2)
(lior0 (mod x 2) (mod y 2) 1)))
:hints (("Goal" :use (lior0-def
(:instance mod012 (m x))
(:instance mod012 (m y))
(:instance quot-mod (m (lior0 x y n)) (n 2))))))
(defthm lior0-fl-2
(implies (and (natp x)
(natp y)
(natp n)
(> n 0))
(equal (fl (/ (lior0 x y n) 2))
(lior0 (fl (/ x 2)) (fl (/ y 2)) (1- n))))
:hints (("Goal" :use (lior0-def
(:instance mod012 (m x))
(:instance mod012 (m y))
(:instance quot-mod (m (lior0 x y n)) (n 2))))))
(in-theory (disable lior0-mod-2 lior0-fl-2))
(defthm lior0-x-y-0
(equal (lior0 x y 0) 0)
:hints (("Goal" :in-theory (enable lior0))))
(defthm lior0-reduce
(implies (and (bvecp x n)
(bvecp y n)
(< n m)
(natp n)
(natp m)
)
(equal (lior0 x y m) (lior0 x y n)))
:hints (("Goal" :in-theory (enable lior0))))
;whoa! this is a *lower* bound !
;make alt version?
(defthm lior0-bnd
(implies (case-split (bvecp x n))
(<= x (lior0 x y n)))
:rule-classes (:rewrite :linear)
:hints (("Goal" :use ((:instance logior-bnd
(x (bits x (1- n) 0))
(y (bits y (1- n) 0))))
:in-theory (enable bits-tail lior0))))
;get rid of the bvecp hyps (do that for many lemmas like this one)
(defthm lior0-with-shifted-arg
(implies (and (bvecp y m)
(bvecp x (- n m))
(<= m n)
(natp m)
(integerp n)
)
(= (lior0 (* (expt 2 m) x) y n)
(+ (* (expt 2 m) x) y)))
:rule-classes ()
:hints (("Goal" :use ((:instance logior-expt (n m)))
:in-theory (enable bvecp-forward bvecp-longer bvecp-shift-up bits-tail lior0))))
(defthm lior0-shift
(implies (and (bvecp x (- n m))
(bvecp y (- n m))
(integerp n) ;(not (zp n))
(natp m)
(<= m n)
)
(= (lior0 (* (expt 2 m) x)
(* (expt 2 m) y)
n)
(* (expt 2 m) (lior0 x y (- n m)))))
:rule-classes ()
:hints (("Goal" :use ((:instance logior-expt-2 (n m)))
:in-theory (enable bvecp-forward bvecp-longer bvecp-shift-up bits-tail lior0))))
(defthm lior0-expt-original
(implies (and (natp n)
(natp k)
(< k n)
(bvecp x n))
(= (lior0 x (expt 2 k) n)
(+ x (* (expt 2 k) (- 1 (bitn x k))))))
:rule-classes ()
:hints (("Goal" :use (logior-expt-3
(:instance expt-strong-monotone (n k) (m n)))
:in-theory (enable bvecp lior0))))
;interesting. not the same as lior0-bvecp (here, m can be smaller than n)
;rename !!
(defthm lior0-bvecp-2
(implies (and (bvecp x m)
(bvecp y m)
)
(bvecp (lior0 x y n) m))
:hints (("Goal" :in-theory (enable lior0))))
(defthm lior0-upper-bound
(< (lior0 x y n) (expt 2 n))
:rule-classes (:rewrite :linear)
:hints (("Goal" :in-theory (enable lior0))))
(defthm lior0-upper-bound-tight
(implies (<= 0 n)
(<= (lior0 x y n) (1- (expt 2 n))))
:rule-classes (:rewrite :linear))
(defthm lior0-fl-1
(equal (lior0 (fl x) y n)
(lior0 x y n))
:hints (("Goal" :in-theory (enable lior0))))
(defthm lior0-fl-2-eric ;BOZO name conflicted...
(equal (lior0 x (fl y) n)
(lior0 x y n))
:hints (("Goal" :in-theory (enable lior0))))
(defthmd lior0-bits-1
(equal (lior0 (bits x (1- n) 0)
y
n)
(lior0 x y n))
:hints (("Goal" :in-theory (e/d (lior0) (logior lior0-commutative)))))
(defthmd lior0-bits-2
(equal (lior0 x
(bits y (1- n) 0)
n)
(lior0 x y n))
:hints (("Goal" :in-theory (e/d (lior0) (logior lior0-commutative)))))
(local
(defthm lior0-base-lemma
(implies (and (bvecp x 1) (bvecp y 1))
(equal (lior0 x y 1)
(if (or (equal (bitn x 0) 1)
(equal (bitn y 0) 1))
1
0)))
:rule-classes nil))
(defthm lior0-base
(equal (lior0 x y 1)
(if (or (equal (bitn x 0) 1)
(equal (bitn y 0) 1))
1
0))
:hints (("Goal" :use ((:instance lior0-base-lemma
(x (bits x 0 0))
(y (bits y 0 0)))
(:instance lior0-bits-1
(x x)
(y (bits y 0 0))
(n 1))
(:instance lior0-bits-2 (n 1)))))
:rule-classes nil)
(defthm lior0-expt
(implies (and (natp n)
(natp k)
(< k n))
(= (lior0 x (expt 2 k) n)
(+ (bits x (1- n) 0)
(* (expt 2 k) (- 1 (bitn x k))))))
:rule-classes ()
:hints (("Goal" :use ((:instance lior0-expt-original
(x (bits x (1- n) 0))))
:in-theory (enable lior0-bits-1 lior0-bits-2))))
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