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|
(in-package "ACL2")
#|
This book is about LXOR, a nice version of LOGXOR. LXOR takes an extra size parameter, N, and always returns
a bit vector of length N.
todo: ;add analogs of the thms in land.lisp past bitn-land
|#
;add macro alias
(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 lnot (x n)
(declare (xargs :guard (and (natp x)
(integerp n)
(< 0 n))
:verify-guards nil))
(if (natp n)
(+ -1 (expt 2 n) (- (bits x (1- n) 0)))
0))
(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 "logand"))
(local (include-book "bits"))
(local (include-book "bitn"))
(local (include-book "../arithmetic/top"))
(defund binary-lxor (x y n)
(declare (xargs :guard (and (natp x)
(natp y)
(integerp n)
(< 0 n))
:verify-guards nil))
(logxor (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 lxor (&rest x)
(declare (xargs :guard (consp x)))
(cond ((endp (cdddr x)) ;(lxor x y n) -- the base case
`(binary-lxor ,@x))
(t
`(binary-lxor ,(car x)
(lxor ,@(cdr x))
,(car (last x))))))
;Allows things like (in-theory (disable lxor)) to refer to binary-lxor.
(add-macro-alias lxor binary-lxor)
(defthm lxor-nonnegative-integer-type
(and (integerp (lxor x y n))
(<= 0 (lxor x y n)))
:rule-classes (:type-prescription))
;(:type-prescription lxor) is no better than lxor-nonnegative-integer-type and might be worse:
(in-theory (disable (:type-prescription binary-lxor)))
;drop this if we plan to keep natp enabled?
(defthm lxor-natp
(natp (lxor x y n)))
(defthm lxor-with-n-not-a-natp
(implies (not (natp n))
(equal (lxor x y n)
0))
:hints (("Goal" :cases ((acl2-numberp n))
:in-theory (enable lxor)))
)
(defthmd lxor-bvecp-simple
(bvecp (lxor x y n) n)
:hints (("Goal" :cases ((natp n))
:in-theory (enable lxor))))
(defthm lxor-bvecp
(implies (and (<= n k)
(case-split (integerp k)))
(bvecp (lxor x y n) k))
:hints (("Goal" :in-theory (disable lxor-bvecp-simple)
:use lxor-bvecp-simple)))
;;
;; Rules to normalize lxor terms (recall that LXOR is a macro for BINARY-LXOR):
;;
;; allow sizes to differ on these?
(defthm lxor-associative
(equal (lxor (lxor x y n) z n)
(lxor x (lxor y z n) n))
:hints (("Goal" :cases ((natp n))
:in-theory (enable lxor bits-tail))))
(defthm lxor-commutative
(equal (lxor y x n)
(lxor x y n))
:hints (("Goal" :in-theory (enable lxor))))
(defthm lxor-commutative-2
(equal (lxor y (lxor x z n) n)
(lxor x (lxor y z n) n))
:hints (("Goal" :cases ((natp n))
:in-theory (enable lxor bits-tail))))
(defthm lxor-combine-constants
(implies (syntaxp (and (quotep x)
(quotep y)
(quotep n)))
(equal (lxor x (lxor y z n) n)
(lxor (lxor x y n) z n))))
(defthm lxor-0
(implies (case-split (bvecp y n))
(equal (lxor 0 y n)
y))
:hints (("Goal" :in-theory (enable lxor bits-tail))))
;nicer than the analogous rule for logand?
(defthm lxor-1
(implies (case-split (bvecp y 1))
(equal (lxor 1 y 1)
(lnot y 1)))
:hints (("Goal" :in-theory (enable bvecp-1-rewrite))))
(defthm lxor-self
(implies (case-split (bvecp x n))
(equal (lxor x x n)
0))
:hints (("Goal" :in-theory (enable lxor bits-tail))))
(defthmd bits-lxor-1
(implies (and (< i n)
(case-split (<= 0 j))
(case-split (integerp n))
)
(equal (bits (lxor x y n) i j)
(lxor (bits x i j)
(bits y i j)
(+ 1 i (- j)))))
:otf-flg t
:hints (("Goal" :in-theory (enable lxor bits-logand))))
(defthmd bits-lxor-2
(implies (and (<= n i)
(case-split (<= 0 j))
(case-split (integerp n))
)
(equal (bits (lxor x y n) i j)
(lxor (bits x i j)
(bits y i j)
(+ n (- j)))))
:otf-flg t
:hints (("Goal" :in-theory (enable lxor bits-logand))))
;notice the call to MIN in the conclusion
(defthm bits-lxor
(implies (and (case-split (<= 0 j))
(case-split (integerp n))
(case-split (integerp i))
)
(equal (bits (lxor x y n) i j)
(lxor (bits x i j)
(bits y i j)
(+ (min n (+ 1 i)) (- j)))))
:hints (("Goal" :in-theory (enable bits-lxor-1 bits-lxor-2))))
(defthmd bitn-lxor-1
(implies (and (< m n)
(case-split (<= 0 m))
(case-split (integerp n))
)
(equal (bitn (lxor x y n) m)
(lxor (bitn x m)
(bitn y m)
1)))
:hints (("Goal" :in-theory (set-difference-theories
(enable bitn)
'(BITS-N-N-REWRITE)))))
(defthmd bitn-lxor-2
(implies (and (<= n m)
(case-split (<= 0 m))
(case-split (integerp n))
)
(equal (bitn (lxor 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-lxor
(implies (and (case-split (<= 0 m))
(case-split (integerp n))
)
(equal (bitn (lxor x y n) m)
(if (< m n)
(lxor (bitn x m)
(bitn y m)
1)
0)))
:hints (("Goal" :in-theory (enable bitn-lxor-1 bitn-lxor-2))))
(defthm lxor-ones
(implies (case-split (bvecp x n))
(equal (lxor (1- (expt 2 n)) x n)
(lnot x n)))
:rule-classes ()
:hints
(("subgoal 1" :use logxor-ones)
("goal" :cases ((natp n))
:in-theory (enable lxor bits-tail)
)))
;lxor-with-all-ones will rewrite (lxor x n) [note there's only one value being ANDed], because (lxor x n)
;expands to (BINARY-LXOR X (ALL-ONES N) N) - now moot???
(defthm lxor-with-all-ones
(implies (case-split (bvecp x n))
(equal (lxor (all-ones n) x n)
(lnot x n)))
:hints
(("goal" :use lxor-ones
:in-theory (enable all-ones))))
(defthm lxor-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 (bvecp x n)))
(equal (lxor k x n)
(lnot x n)))
:hints (("Goal" :use lxor-ones)))
(local (in-theory (disable mod-by-2-rewrite-to-even mod-mult-of-n mod-equal-0)))
(defthm lxor-def
(implies (and (integerp x)
(integerp y)
(< 0 n)
(integerp n)
)
(equal (lxor x y n)
(+ (* 2 (lxor (fl (/ x 2)) (fl (/ y 2)) (1- n)))
(lxor (mod x 2) (mod y 2) 1))))
:rule-classes ()
:hints (("Goal" :in-theory (e/d (lxor bits-fl-by-2) ())
:use ((:instance logxor-def (i (bits x (1- n) 0)) (j (bits y (1- n) 0)))
mod012
(:instance mod012 (x y))))))
(defthm lxor-mod-2
(implies (and (natp x)
(natp y)
(natp n)
(> n 0))
(equal (mod (lxor x y n) 2)
(lxor (mod x 2) (mod y 2) 1)))
:hints (("Goal" :use (lxor-def
mod012
(:instance mod012 (x y))
(:instance quot-mod (m (lxor x y n)) (n 2))))))
(defthm lxor-fl-2
(implies (and (natp x)
(natp y)
(natp n)
(> n 0))
(equal (fl (/ (lxor x y n) 2))
(lxor (fl (/ x 2)) (fl (/ y 2)) (1- n))))
:hints (("Goal" :use (lxor-def
mod012
(:instance mod012 (x y))
(:instance quot-mod (m (lxor x y n)) (n 2))))))
(in-theory (disable lxor-mod-2 lxor-fl-2))
(defthm bitn-lxor-0
(implies (and (integerp x)
(integerp y)
(not (zp n))
)
(= (bitn (lxor x y n) 0)
(bitn (+ x y) 0)))
:rule-classes ()
:hints (("Goal" :use ((:instance bitn-logxor-0 (a (bits x (1- n) 0)) (b (bits y (1- n) 0)))
(:instance mod-mod-sum (n (expt 2 n)) (a x) (b y))
(:instance mod-of-mod (a n) (b 1) (x (+ x y)))
(:instance mod-of-mod (a n) (b 1) (x (+ (mod x (expt 2 n)) (mod y (expt 2 n))))))
:in-theory (enable lxor bitn-rec-0 bits-mod bitn-bits))))
(defthm lxor-x-y-0
(equal (lxor x y 0) 0)
:hints (("Goal" :in-theory (enable lxor))))
;N is a free variable
(defthm lxor-reduce
(implies (and (bvecp x n)
(bvecp y n)
(< n m)
(case-split (integerp m))
)
(equal (lxor x y m)
(lxor x y n)))
:hints (("Goal" :in-theory (enable lxor))))
;move
(defthm lxor-upper-bound
(implies (and (integerp n)
(<= 0 n))
(< (lxor x y n) (expt 2 n)))
:rule-classes (:rewrite :linear)
:hints (("Goal" :in-theory (enable lxor))))
;move
(defthm lxor-upper-bound-tight
(implies (and (integerp n)
(<= 0 n))
(<= (lxor x y n) (1- (expt 2 n)))))
|
