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|
(in-package "ACL2")
#|
This book includes lemmas about LOGAND. Note that LOGAND is a macro which expands to nested calls to
BINARY-LOGAND. Both LOGAND and BINARY-LOGAND are built into ACL2.
This book contains only results; all the proofs are done in the book logand-proofs.
Todo:
use set-invisible-fns-alist - or find a better way?
rules for logand x with lognot x anywhere in there?
should logand-with-0 be both sides? what about logand-with-minus-one
how order rules for efficiency? perhaps make a separate documentation book?
any other log lemmas?
are the 4 enough for assoc comm functions?
|#
(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))
(include-book "ground-zero")
(local (include-book "logand-proofs"))
(set-inhibit-warnings "theory") ; avoid warning in the next event
(local (in-theory nil))
;(set-inhibit-warnings) ; restore theory warnings (optional)
;;
;; Rules to normalize logand terms (recall that LOGAND is a macro for BINARY-LOGAND):
;;
(defthm logand-associative
(equal (logand (logand i j) k)
(logand i (logand j k))))
(defthm logand-commutative
(equal (logand j i)
(logand i j)))
(defthm logand-commutative-2
(equal (logand j i k)
(logand i j k)))
(defthm logand-combine-constants
(implies (syntaxp (and (quotep i)
(quotep j)))
(equal (logand i j k)
(logand (logand i j) k))))
;;
;; LOGAND with special values
;;
(defthm logand-with-non-integer-arg
(implies (or (not (integerp i))
(not (integerp j)))
(equal (logand i j)
0)))
;0 should always be brought to the front of logand
;should we have a rule with the second arg being 0?
(defthm logand-with-zero
(equal (logand 0 j) 0))
;-1 should always be brought to the front of logand
;should we have both cases or not?
(defthm logand-with-minus-one
(implies (case-split (integerp i))
(equal (logand -1 i) i)))
;;
;; Type facts
;;
;this goes through:
;(thm (integerp (logand i j)))
(defthm logand-integer-type-prescription
(integerp (logand i j))
:rule-classes (:type-prescription)
:hints (("Goal" :in-theory (enable logand))))
;These three go together.
;logand is negative iff either arg is negative
;Didn't make this a rewrite rule to avoid backchaining on (integerp (logand i j)) -- should never happen, but
;just in case.
(defthm logand-non-negative-integer-type-prescription
(implies (or (<= 0 i)
(<= 0 j))
(and (<= 0 (logand i j))
(integerp (logand i j))))
:rule-classes (:type-prescription))
(defthm logand-negative-integer-type-prescription
(implies (and (< i 0)
(< j 0)
(case-split (integerp i))
(case-split (integerp j)))
(and (< (logand i j) 0)
(integerp (logand i j))))
:rule-classes (:type-prescription))
; rewrites (<= 0 (logand i j)) and (< (logand i j) 0)
;could this perhaps not fire (say, during backchaining) because of case-splitting of the conclusion, causing
;us to wish we had a simple rule that natp args imply logand is natp?
;maybe don't want this one?
(defthm logand-negative-rewrite
(implies (and (case-split (integerp i))
(case-split (integerp j)))
(equal (< (logand i j) 0)
(and (< i 0)
(< j 0)))))
(defthm logand-non-negative
(implies (or (<= 0 x)
(<= 0 y)
)
(<= 0 (logand x y))))
;There's no nice logand-positive rule. Nor is there a clear rewrite for (< 0 (logand i j))
;For logand to be positive, the arguments must have bits that overlap, and there's no way to state this.
(defthm logand-non-positive-integer-type-prescription
(implies (and (<= i 0)
(<= j 0))
(and (<= (logand i j) 0)
(integerp (logand i j))))
:rule-classes (:type-prescription))
(defthm logand-non-positive-rewrite
(implies (and (<= i 0)
(<= j 0))
(<= (logand i j) 0)))
#| do we want this?
(defthm logand-negative
(implies (and (< i 0)
(< j 0)
(case-split (integerp i))
(case-split (integerp j))
)
(and (integerp (logand i j))
(< (logand i j) 0)))
:hints (("Goal" :in-theory (enable logand)))
:rule-classes (:rewrite (:type-prescription)))
|#
; If logand is less than -1, then both i and j are <= -1, and at least one of them is strictly < -1.
(defthm logand-less-than-minus-one
(implies (and (case-split (integerp i))
(case-split (integerp j))
)
(equal (< (logand i j) -1)
(or (and (<= i -1) (< j -1))
(and (<= j -1) (< i -1))))))
;BOZO move!
;perhaps put on a backchain limit?
(defthm integer-tighten-bound
(implies (integerp x)
(equal (< -1 x)
(<= 0 x))))
#|
;rewrite < -1 to <= 0?
;simplify the conclusion?
(defthm logand-negative-5
(implies (and (case-split (integerp i))
(case-split (integerp j))
)
(equal (< -1 (logand i j))
(not (and (< i 0)
(< j 0))))))
:hints (("Goal" :cases ((equal j -1) (equal i -1))
:in-theory (enable logand))))
|#
(defthm logand-self
(implies (case-split (integerp i))
(equal (logand i i) i)))
(defthm logand-equal-minus-one
(equal (EQUAL (LOGAND i j) -1)
(and (equal i -1)
(equal j -1))))
(defthm logand-even
(implies (and (case-split (integerp i))
(case-split (integerp j))
)
(equal (INTEGERP (* 1/2 (logand i j)))
(or (INTEGERP (* 1/2 i))
(INTEGERP (* 1/2 j))))))
;weird?
(defthm logand-0-when-one-arg-is-odd
(implies (and (not (integerp (* 1/2 j)))
(case-split (integerp j))
(case-split (integerp i))
)
(and (equal (equal (logand i j) 0)
(and (integerp (* 1/2 i))
(equal (logand (fl (* 1/2 i)) (fl (* 1/2 j))) 0)))
(equal (equal (logand j i) 0)
(and (integerp (* 1/2 i))
(equal (logand (fl (* 1/2 i)) (fl (* 1/2 j))) 0))))))
(defthm logand-simp-1
(implies (and (case-split (integerp i))
(case-split (integerp j)))
(equal (LOGAND (+ 1 (* 2 i))
(+ 1 (* 2 j)))
(+ 1 (* 2 (logand i j))))))
;add to this
;make linear?
(defthm logand-upper-bound-1
(implies (<= 0 i)
(<= (logand i j) i)))
;BOZO same as logand-upper-bound-1
(defthm logand-bnd
(implies (<= 0 x)
(<= (logand x y) x))
:rule-classes :linear
)
;trying disabled...
(defthmd logand-with-1
(implies (case-split (integerp i))
(equal (logand 1 i)
(if (evenp i)
0
1))))
;trying disabled...
;rename
;BOZO make a nice rule for logand with 1?
(defthmd logand-special-value
(implies (case-split (integerp j))
(equal (equal (logand 1 j) j)
(or (equal j 0) (equal j 1)))))
(defthmd logand-def
(implies (and (case-split (integerp i))
(case-split (integerp j))
)
(equal (logand i j)
(+ (* 2 (logand (fl (* 1/2 i)) (fl (* 1/2 j))))
(logand (mod i 2) (mod j 2)))))
:rule-classes ((:definition :controller-alist ((binary-logand t t)))))
(defthm fl-logand-by-2
(implies (and (case-split (integerp i))
(case-split (integerp j))
)
(equal (fl (* 1/2 (logand i j)))
(logand (fl (* 1/2 i)) (fl (* 1/2 j))))))
(defthm floor-logand-by-2
(implies (and (case-split (integerp i))
(case-split (integerp j)))
(equal (floor (logand i j) 2)
(logand (floor i 2) (floor j 2)))))
(defthm mod-logand-by-2
(equal (mod (logand i j) 2)
(logand (mod i 2) (mod j 2))))
;allow them to occur in other orders (perhaps with intervening terms)?
;think about this
;make a version for logior
(defthm logand-i-lognot-i
(implies (case-split (integerp i))
(equal (LOGAND i (LOGNOT i))
0)))
;make a nice recognizer?
;handle negative case?
;rename?
(defthmd logand-ones
(implies (and (< i (expt 2 n)) ;drop and wrap bits around i?
(<= 0 i)
(case-split (integerp i))
)
(equal (logand i (1- (expt 2 n)))
i)))
#|
;change name and param names eventually
(defthm AND-BITS-A
(implies (and (integerp x); (>= x 0)
(integerp k); (>= k 0)
)
(equal (logand x (expt 2 k))
(* (expt 2 k) (bitn x k))))
:rule-classes ())
|#
(defthm AND-DIST-B
(implies (and (integerp x) (>= x 0)
(integerp y) (>= y 0)
(integerp n) (>= n 0))
(= (logand (* (expt 2 n) x) y)
(* (expt 2 n) (logand x (fl (/ y (expt 2 n)))))))
:rule-classes ())
;BOZO also have logand-with-zero
(defthm logand-0
(equal (logand 0 j) 0))
(defthmd logand-rewrite
(implies (and (case-split (integerp x))
(case-split (integerp y))
)
(equal (logand x y)
(+ (* 2 (logand (fl (/ x 2)) (fl (/ y 2))))
(logand (mod x 2) (mod y 2)))))
:rule-classes ((:definition :controller-alist ((binary-logand t t)))))
(defthm logand-even-2
(implies (and (integerp i)
(integerp j))
(equal (or (= (mod i 2) 0)
(= (mod j 2) 0))
(= (mod (logand i j) 2) 0)))
:rule-classes ())
|
