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|
(in-package "ACL2")
#|
This book is about LIOR, a nice version of LOGIOR. LIOR 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 lior with other functions
what should lior 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-lior (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 lior (&rest x)
(declare (xargs :guard (and (consp x)
(consp (cdr x))
(consp (cddr x)))))
(cond ((endp (cdddr x)) ;(lior x y n) -- the base case
`(binary-lior ,@x))
(t
`(binary-lior ,(car x)
(lior ,@(cdr x))
,(car (last x))))))
;Allows things like (in-theory (disable lior)) to refer to binary-lior.
(add-macro-alias lior binary-lior)
(defthm lior-nonnegative-integer-type
(and (integerp (lior x y n))
(<= 0 (lior x y n)))
:rule-classes (:type-prescription))
;(:type-prescription lior) is no better than lior-nonnegative-integer-type and might be worse:
(in-theory (disable (:type-prescription binary-lior)))
;drop this if we plan to keep natp enabled?
(defthm lior-natp
(natp (lior x y n)))
(defthm lior-with-n-not-a-natp
(implies (not (natp n))
(equal (lior x y n)
0))
:hints (("Goal" :cases ((acl2-numberp n))
:in-theory (enable lior)))
)
(defthmd lior-bvecp-simple
(bvecp (lior x y n) n)
:hints (("Goal" :cases ((natp n))
:in-theory (enable lior))))
(defthm lior-bvecp
(implies (and (<= n k)
(case-split (integerp k)))
(bvecp (lior x y n) k))
:hints (("Goal" :in-theory (disable lior-bvecp-simple)
:use lior-bvecp-simple)))
;;
;; Rules to normalize lior terms (recall that LIOR is a macro for BINARY-LIOR):
;;
;; allow sizes to differ on these?
(defthm lior-associative
(equal (lior (lior x y n) z n)
(lior x (lior y z n) n))
:hints (("Goal" :cases ((natp n))
:in-theory (enable lior bits-tail))))
(defthm lior-commutative
(equal (lior y x n)
(lior x y n))
:hints (("Goal" :in-theory (enable lior))))
(defthm lior-commutative-2
(equal (lior y (lior x z n) n)
(lior x (lior y z n) n))
:hints (("Goal" :cases ((natp n))
:in-theory (enable lior bits-tail))))
(defthm lior-combine-constants
(implies (syntaxp (and (quotep x)
(quotep y)
(quotep n)))
(equal (lior x (lior y z n) n)
(lior (lior x y n) z n))))
(defthm lior-0
(implies (case-split (bvecp y n))
(equal (lior 0 y n)
y))
:hints (("Goal" :in-theory (enable lior bits-tail))))
;nicer than the analogous rule for logior?
(defthm lior-1
(implies (case-split (bvecp y 1))
(equal (lior 1 y 1)
1))
:hints (("Goal" :in-theory (enable bvecp-1-rewrite))))
(defthm lior-self
(implies (and (case-split (bvecp x n))
(case-split (integerp n)))
(equal (lior x x n)
x))
:hints (("Goal" :in-theory (enable lior bits-tail))))
(defthmd bits-lior-1
(implies (and (< i n)
(case-split (<= 0 j))
(case-split (integerp n))
)
(equal (bits (lior x y n) i j)
(lior (bits x i j)
(bits y i j)
(+ 1 i (- j)))))
:otf-flg t
:hints (("Goal" :in-theory (enable lior bits-logand))))
(defthmd bits-lior-2
(implies (and (<= n i)
(case-split (<= 0 j))
(case-split (integerp n))
)
(equal (bits (lior x y n) i j)
(lior (bits x i j)
(bits y i j)
(+ n (- j)))))
:otf-flg t
:hints (("Goal" :in-theory (enable lior bits-logand))))
;notice the call to MIN in the conclusion
(defthm bits-lior
(implies (and (case-split (<= 0 j))
(case-split (integerp n))
(case-split (integerp i))
)
(equal (bits (lior x y n) i j)
(lior (bits x i j)
(bits y i j)
(+ (min n (+ 1 i)) (- j)))))
:hints (("Goal" :in-theory (enable bits-lior-1 bits-lior-2))))
(defthmd bitn-lior-1
(implies (and (< m n)
(case-split (<= 0 m))
(case-split (integerp n))
)
(equal (bitn (lior x y n) m)
(lior (bitn x m)
(bitn y m)
1)))
:hints (("Goal" :in-theory (set-difference-theories
(enable bitn)
'(BITS-N-N-REWRITE)))))
(defthmd bitn-lior-2
(implies (and (<= n m)
(case-split (<= 0 m))
(case-split (integerp n))
)
(equal (bitn (lior 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-lior
(implies (and (case-split (<= 0 m))
(case-split (integerp n))
)
(equal (bitn (lior x y n) m)
(if (< m n)
(lior (bitn x m)
(bitn y m)
1)
0)))
:hints (("Goal" :in-theory (enable bitn-lior-1 bitn-lior-2))))
;or could wrap bits around conclusion?
(defthm lior-equal-0
(implies (and (case-split (bvecp x n))
(case-split (bvecp y n))
(case-split (integerp n))
)
(equal (equal 0 (lior x y n))
(and (equal x 0)
(equal y 0))))
:hints (("Goal" :in-theory (enable lior bits-tail))))
(defthm lior-of-single-bits-equal-0
(implies (and (case-split (bvecp x 1))
(case-split (bvecp y 1))
)
(equal (equal 0 (lior x y 1))
(and (equal x 0)
(equal y 0))))
:hints (("Goal" :in-theory (enable bvecp-1-rewrite))))
(defthm lior-of-single-bits-equal-1
(implies (and (case-split (bvecp x 1))
(case-split (bvecp y 1))
)
(equal (equal 1 (lior x y 1))
(or (equal x 1)
(equal y 1))))
:hints (("Goal" :in-theory (enable bvecp-1-rewrite))))
(defthm lior-ones
(implies (and (case-split (bvecp x n))
(case-split (natp n)) ;gen
)
(equal (lior (1- (expt 2 n)) x n)
(1- (expt 2 n))))
:rule-classes ()
:hints
(("goal" :use logior-ones
:in-theory (enable lior bits-tail)
)))
;lior-with-all-ones will rewrite (lior x n) [note there's only one value being ANDed], because (lior x n)
;expands to (BINARY-LIOR X (ALL-ONES N) N) - now moot???
(defthm lior-with-all-ones
(implies (case-split (bvecp x n))
(equal (lior (all-ones n) x n)
(all-ones n)))
:hints
(("goal" :use lior-ones
:in-theory (enable all-ones))))
(defthm lior-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 (lior k x n)
(1- (expt 2 n))))
:hints (("Goal"
:use lior-ones)))
(local (in-theory (disable MOD-BY-2-REWRITE-TO-EVEN MOD-MULT-OF-N MOD-EQUAL-0 )))
(defthm lior-def
(implies (and (integerp x)
(integerp y)
(< 0 n)
(integerp n))
(equal (lior x y n)
(+ (* 2 (lior (fl (/ x 2)) (fl (/ y 2)) (1- n)))
(lior (mod x 2) (mod y 2) 1))))
:rule-classes ()
:hints (("Goal" :in-theory (e/d (lior bits-fl-by-2)
())
:use ((:instance logior-def (i (bits x (1- n) 0)) (j (bits y (1- n) 0)))
mod012
(:instance mod012 (x y))))))
(defthm lior-mod-2
(implies (and (natp x)
(natp y)
(natp n)
(> n 0))
(equal (mod (lior x y n) 2)
(lior (mod x 2) (mod y 2) 1)))
:hints (("Goal" :use (lior-def
mod012
(:instance mod012 (x y))
(:instance quot-mod (m (lior x y n)) (n 2))))))
(defthm lior-fl-2
(implies (and (natp x)
(natp y)
(natp n)
(> n 0))
(equal (fl (/ (lior x y n) 2))
(lior (fl (/ x 2)) (fl (/ y 2)) (1- n))))
:hints (("Goal" :use (lior-def
mod012
(:instance mod012 (x y))
(:instance quot-mod (m (lior x y n)) (n 2))))))
(in-theory (disable lior-mod-2 lior-fl-2))
(defthm lior-x-y-0
(equal (lior x y 0) 0)
:hints (("Goal" :in-theory (enable lior))))
(defthm lior-reduce
(implies (and (bvecp x n)
(bvecp y n)
(< n m)
(natp n)
(natp m)
)
(equal (lior x y m) (lior x y n)))
:hints (("Goal" :in-theory (enable lior))))
;whoa! this is a *lower* bound !
;make alt version?
(defthm lior-bnd
(implies (case-split (bvecp x n))
(<= x (lior 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 lior))))
;get rid of the bvecp hyps (do that for many lemmas like this one)
;BOZO rename to lior-with-shifted-arg
(defthm lior-plus
(implies (and (bvecp y m)
(bvecp x (- n m))
(<= m n)
(natp m)
(integerp n)
)
(= (lior (* (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 lior))))
(defthm lior-shift
(implies (and (bvecp x (- n m))
(bvecp y (- n m))
(integerp n) ;(not (zp n))
(natp m)
(<= m n)
)
(= (lior (* (expt 2 m) x)
(* (expt 2 m) y)
n)
(* (expt 2 m) (lior 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 lior))))
(defthm lior-expt
(implies (and (natp n)
(natp k)
(< k n)
(bvecp x n))
(= (lior 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 lior))))
;interesting. not the same as lior-bvecp (here, m can be smaller than n)
;rename !!
(defthm lior-bvecp-2
(implies (and (bvecp x m)
(bvecp y m)
)
(bvecp (lior x y n) m))
:hints (("Goal" :in-theory (enable lior))))
(defthm lior-upper-bound
(< (lior x y n) (expt 2 n))
:rule-classes (:rewrite :linear)
:hints (("Goal" :in-theory (enable lior))))
(defthm lior-upper-bound-tight
(implies (<= 0 n)
(<= (lior x y n) (1- (expt 2 n))))
:rule-classes (:rewrite :linear))
(defthm lior-fl-1
(equal (lior (fl x) y n)
(lior x y n))
:hints (("Goal" :in-theory (enable lior))))
(defthm lior-fl-2-eric ;BOZO name conflicted...
(equal (lior x (fl y) n)
(lior x y n))
:hints (("Goal" :in-theory (enable lior))))
|
