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|
(in-package "ACL2")
#|
This book is about LAND, a nice version of LOGAND. LAND takes an extra size parameter, N, and always returns
a bit vector of length N.
Todo:
add versions of logand-expt-2 and logand-expt-4
prove (elsewhere) lemmas mixing land with other functions
|#
;;Necessary defuns:
(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 "land-proofs"))
;; New stuff
;We expect n to be a positive integer, and x and y to be bit vectors of length n.
(defund binary-land (x y n)
(declare (xargs :guard (and (natp x)
(natp y)
(integerp n)
(< 0 n))
:verify-guards nil))
(logand (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 land (&rest x)
(declare (xargs :guard (and (consp x)
(consp (cdr x))
(consp (cddr x)))))
(cond ((endp (cdddr x)) ;(land x y n) -- the base case
`(binary-land ,@x))
(t
`(binary-land ,(car x)
(land ,@(cdr x))
,(car (last x))))))
;Allows things like (in-theory (disable land)) to refer to binary-land.
(add-macro-alias land binary-land)
(defthm land-nonnegative-integer-type
(and (integerp (land x y n))
(<= 0 (land x y n)))
:rule-classes (:type-prescription))
;(:type-prescription land) is no better than land-nonnegative-integer-type and might be worse:
(in-theory (disable (:type-prescription binary-land)))
;drop this if we plan to keep natp enabled?
(defthm land-natp
(natp (land x y n)))
;BOZO split into 2 rules?
(defthm land-with-n-not-a-natp
(implies (not (natp n))
(equal (land x y n)
0)))
(defthmd land-bvecp-simple
(bvecp (land x y n) n))
(defthm land-bvecp
(implies (and (<= n k)
(case-split (integerp k)))
(bvecp (land x y n) k)))
;;
;; Rules to normalize land terms (recall that LAND is a macro for BINARY-LAND):
;;
;This guarantees that the n parameters to nested LAND calls match.
;Note the MIN in the conclusion.
;BOZO do we expect MIN to be enabled? Maybe we should use IF instead for this and other rules?
(defthm land-nest-tighten
(implies (and (syntaxp (not (equal m n)))
(case-split (integerp m))
(case-split (integerp n))
)
(equal (land x (land y z m) n)
(land x (land y z (min m n)) (min m n)))))
; allow the n's to differ on this?
(defthm land-associative
(equal (land (land x y n) z n)
(land x (land y z n) n)))
(defthm land-commutative
(equal (land y x n)
(land x y n)))
; allow the n's to differ on this?
(defthm land-commutative-2
(equal (land y (land x z n) n)
(land x (land y z n) n)))
; allow the n's to differ on this?
(defthm land-combine-constants
(implies (syntaxp (and (quotep x)
(quotep y)
(quotep n)))
(equal (land x (land y z n) n)
(land (land x y n) z n))))
(defthm land-0
(equal (land 0 y n)
0))
;nicer than the analogous rule for logand? is it really?
;BOZO gen the second 1 in the lhs?
(defthm land-1
(equal (land 1 y 1)
(bitn y 0)))
(defthm land-self
(equal (land x x n)
(bits x (1- n) 0)))
;perhaps use only the main rule, bits-land?
(defthmd bits-land-1
(implies (and (< i n)
(case-split (<= 0 j))
(case-split (integerp n))
)
(equal (bits (land x y n) i j)
(land (bits x i j)
(bits y i j)
(+ 1 i (- j))))))
;perhaps use only the main rule, bits-land?
(defthmd bits-land-2
(implies (and (<= n i)
(case-split (<= 0 j))
(case-split (integerp n))
)
(equal (bits (land x y n) i j)
(land (bits x i j)
(bits y i j)
(+ n (- j))))))
;Notice the call to MIN in the conclusion.
(defthm bits-land
(implies (and (case-split (<= 0 j))
(case-split (integerp n))
(case-split (integerp i))
)
(equal (bits (land x y n) i j)
(land (bits x i j)
(bits y i j)
(+ (min n (+ 1 i)) (- j))))))
(defthmd bitn-land-1
(implies (and (< m n)
(case-split (<= 0 m))
(case-split (integerp n))
)
(equal (bitn (land x y n) m)
(land (bitn x m)
(bitn y m)
1))))
(defthmd bitn-land-2
(implies (and (<= n m)
(case-split (<= 0 m))
(case-split (integerp n))
)
(equal (bitn (land x y n) m)
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-land
(implies (and (case-split (<= 0 m))
(case-split (integerp n))
)
(equal (bitn (land x y n) m)
(if (< m n)
(land (bitn x m)
(bitn y m)
1)
0))))
;BOZO see land-equal-0
;drop bvecp hyps and put bitn in conclusion?
(defthm land-of-single-bits-equal-0
(implies (and (case-split (bvecp x 1))
(case-split (bvecp y 1))
)
(equal (equal 0 (land x y 1))
(or (equal x 0)
(equal y 0)))))
(defthm land-of-single-bits-equal-1
(implies (and (case-split (bvecp x 1))
(case-split (bvecp y 1))
)
(equal (equal 1 (land x y 1))
(and (equal x 1)
(equal y 1)))))
(defthm land-ones
(equal (land (1- (expt 2 n)) x n)
(bits x (1- n) 0))
:rule-classes ())
;land-with-all-ones will rewrite (land x n) [note there's only one value being ANDed], because (land x n)
;expands to (BINARY-LAND X (ALL-ONES N) N) - now moot???
;BOZO drop bvecp hyp and move to conclusion?
(defthm land-with-all-ones
(implies (case-split (bvecp x n))
(equal (land (all-ones n) x n)
x)))
(defthmd land-ones-rewrite
(implies (and (syntaxp (and (quotep k) (quotep n)))
(equal k (1- (expt 2 n))) ;this computes on constants...
)
(equal (land k x n)
(bits x (1- n) 0))))
(defthm land-def
(implies (and (integerp x)
(integerp y)
(> n 0)
(integerp n)
)
(equal (land x y n)
(+ (* 2 (land (fl (/ x 2)) (fl (/ y 2)) (1- n)))
(land (mod x 2) (mod y 2) 1))))
:rule-classes ())
(defthmd land-mod-2
(implies (and (natp x)
(natp y)
(natp n)
(> n 0))
(equal (mod (land x y n) 2)
(land (mod x 2) (mod y 2) 1))))
;BOZO RHS isn't simplified...
(defthmd land-fl-2
(implies (and (natp x)
(natp y)
(natp n)
(> n 0))
(equal (fl (/ (land x y n) 2))
(land (fl (/ x 2)) (fl (/ y 2)) (1- n)))))
;BOZO rename to land-with-n-0
;what if n is negative? or not an integer?
(defthm land-x-y-0
(equal (land x y 0) 0))
;actually, maybe only either x or y must be a bvecp of length n
;n is a free var
(defthm land-reduce
(implies (and (bvecp x n)
(bvecp y n)
(natp n)
(natp m)
(< n m))
(equal (land x y m)
(land x y n))))
;deceptive name; this only works for single bits!
(defthm land-equal-0
(implies (and (bvecp i 1)
(bvecp j 1))
(equal (equal 0 (land i j 1))
(or (equal i 0)
(equal j 0)))))
;make alt version?
(defthm land-bnd
(implies (case-split (<= 0 x))
(<= (land x y n) x))
:rule-classes (:rewrite :linear))
;enable? make an alt version??
(defthmd land-ignores-bits
(equal (land (bits x (1- n) 0) y n)
(land x y n)))
(defthmd land-with-shifted-arg
(implies (and (integerp x) ;gen?
(rationalp y)
(integerp m)
(integerp n)
(<= 0 m)
)
(equal (land (* (expt 2 m) x) y n)
(* (expt 2 m) (land x (bits y (1- n) m) (+ n (- m)))))))
(defthmd land-expt
(implies (and (natp n)
(natp k)
(< k n))
(equal (land x (expt 2 k) n)
(* (expt 2 k) (bitn x k)))))
(defthm land-slice
(implies (and (< j i) ;drop? or not?
(<= i n)
(integerp n)
(integerp i)
(integerp j)
(<= 0 j)
)
(equal (land x (- (expt 2 i) (expt 2 j)) n)
(* (expt 2 j) (bits x (1- i) j))))
:rule-classes ())
(defthm land-upper-bound
(implies (and (integerp n)
(<= 0 n))
(< (land x y n) (expt 2 n)))
:rule-classes (:rewrite :linear))
(defthm land-upper-bound-tight
(implies (and (integerp n)
(<= 0 n))
(<= (land x y n) (1- (expt 2 n)))))
(defthm land-fl-1
(equal (land (fl x) y n)
(land x y n)))
(defthm land-fl-2-eric ;BOZO name conflicted...
(equal (land x (fl y) n)
(land x y n)))
|
