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|
(in-package "ACL2")
(local ; ACL2 primitive
(defun natp (x)
(declare (xargs :guard t))
(and (integerp x)
(<= 0 x))))
(defund bvecp (x k)
(declare (xargs :guard (integerp k)))
(and (integerp x)
(<= 0 x)
(< x (expt 2 k))))
(defund fl (x)
(declare (xargs :guard (real/rationalp x)))
(floor x 1))
(defun 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)))
(local (include-book "bits"))
(local (include-book "bitn"))
;(local (include-book "../arithmetic/top"))
(local (include-book "../arithmetic/expt"))
(local (include-book "../arithmetic/mod"))
(local (include-book "../arithmetic/mod"))
(local (include-book "../arithmetic/arith"))
(local (include-book "../arithmetic/arith2"))
(local (include-book "../arithmetic/integerp"))
(local (include-book "bvecp"))
(local (in-theory (enable expt-minus)))
#|
(defun LNOT (x n)
(1- (- (expt 2 n) x)))
|#
;used to be called COMP1
(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))
;note that this isn't a rewrite rule b/c we believe it will never need to be
(defthm lnot-nonnegative-integer-type
(and (integerp (lnot x n))
(<= 0 (lnot x n)))
:hints (("Goal" :in-theory (enable lnot)))
:rule-classes ((:type-prescription :typed-term (lnot x n))))
;lnot-nonnegative-integer-type is strictly better, and we don't need both
(in-theory (disable (:type-prescription lnot)))
(defthm lnot-natp
(natp (lnot x n)))
(defthm lnot-upper-bound
(< (lnot x n) (expt 2 n))
:hints (("Goal" :in-theory (enable lnot)))
:rule-classes (:rewrite :linear)
)
;why is bvecp enabled here?
(defthm lnot-bvecp-simple
(bvecp (lnot x n) n)
:hints (("Goal" :in-theory (enable bvecp lnot))))
(defthm lnot-bvecp
(implies (and (<= n k)
(case-split (integerp k)))
(bvecp (lnot x n) k))
:hints (("Goal" :in-theory (disable lnot-bvecp-simple)
:use lnot-bvecp-simple)))
(defthm lnot-lnot
(implies (and (case-split (natp n))
(case-split (bvecp x n))
)
(equal (lnot (lnot x n) n)
x))
:hints (("Goal" :in-theory (enable lnot bvecp bits-does-nothing))))
;reorient this rule?
(defthmd lnot-times-2
(implies (and (case-split (natp x))
(case-split (natp n))
)
(equal (+ 1 (* 2 (lnot x n)))
(lnot (* 2 x) (1+ n))))
:hints (("Goal" :in-theory (enable lnot expt-split)
:use (:instance bits-shift (n 1) (i n) (j 0)))))
(defthm lnot-with-n-0
(equal (lnot x 0)
0)
:hints (("Goal" :in-theory (enable lnot)))
)
(encapsulate
()
(local
(defthm fl-lnot-1
(implies (and (integerp n) (>= n k)
(integerp k)
(>= k 0) ;drop? and propagate..
(integerp x) (>= x 0)
(< x (expt 2 n))
)
(equal (/ (lnot x n) (expt 2 k))
(+ (expt 2 (- n k))
(/ (- -1 x) (expt 2 k)))))
:rule-classes ()
:hints (("Goal" :in-theory (set-difference-theories
(enable lnot expt-split)
'( ;a10
))
))))
;this looks fragile
(local (defthm fl=
(implies (equal x y)
(equal (fl x) (fl y)))
:rule-classes ()))
(local (defthm fl-lnot-2
(implies (and (integerp n) (>= n k)
(integerp k) (>= k 0)
(integerp x) (>= x 0) (< x (expt 2 n)))
(equal (fl (/ (lnot x n) (expt 2 k)))
(fl (+ (expt 2 (- n k))
(/ (- -1 x) (expt 2 k))))))
:rule-classes ()
:hints (("Goal" :in-theory (disable ;a10
)
:use ((:instance fl-lnot-1)
(:instance fl=
(x (/ (lnot x n) (expt 2 k)))
(y (+ (expt 2 (- n k))
(/ (- -1 x) (expt 2 k))))))))))
(local (include-book "../arithmetic/fl"))
(local (defthm fl-lnot-3
(implies (and (integerp n) (>= n k)
(integerp k) (>= k 0)
(integerp x) (>= x 0) (< x (expt 2 n)))
(equal (fl (/ (lnot x n) (expt 2 k)))
(+ (expt 2 (- n k))
(fl (/ (- -1 x) (expt 2 k))))))
:rule-classes ()
:hints (("Goal" :in-theory (enable lnot)
:use ((:instance fl-lnot-2)
)))))
;gen?
;make a by-2 version?
;change param name?
;make a better rewrite rule
(defthmd lnot-fl-aux
(implies (and (<= k n)
(bvecp x n)
(<= 0 k)
(integerp n)
(integerp k)
)
(equal (fl (* (/ (expt 2 k)) (lnot x n)))
(lnot (fl (/ x (expt 2 k))) (- n k))))
:hints (("Goal" :in-theory (set-difference-theories
(enable lnot bvecp)
'(bits-fl ;a10
;fl-minus-gen
))
:use ((:instance fl-lnot-3)
(:instance fl-m+1 (m x) (n (expt 2 k)))
))))
)
;disable?
(defthm lnot-ignores-bits
(equal (lnot (bits x (1- n) 0) n)
(lnot x n))
:hints (("Goal" :in-theory (enable lnot))))
(defthmd lnot-ignores-bits-2
(implies (and (integerp i)
(<= (1- n) i))
(equal (lnot (bits x i 0) n)
(lnot x n)))
:hints (("Goal" :in-theory (enable lnot))))
;disable?
(defthm lnot-fl-eric
(equal (lnot (fl x) n)
(lnot x n))
:hints (("Goal" :in-theory (enable lnot))))
;is this okay? dropped the fl...
(local (defthmd lnot-fl-eric-helper
(implies (and (<= k n)
;(bvecp x n)
(<= 0 k)
(integerp n)
(integerp k)
)
(equal (fl (* (/ (expt 2 k)) (lnot x n)))
(lnot (/ (bits x (1- n) 0) (expt 2 k)) (- n k))))
:hints (("Goal" :in-theory (disable lnot-fl-aux)
:use ((:instance lnot-fl-aux (x (bits x (1- n) 0 ))))))
))
;BOZO move!
(DEFTHM BITS-SHIFT-inv
(IMPLIES (AND (CASE-SPLIT (INTEGERP N))
(CASE-SPLIT (INTEGERP I))
(CASE-SPLIT (INTEGERP J)))
(EQUAL (BITS (* (/ (EXPT 2 N)) X) I J)
(BITS X (+ I N) (+ J N))))
:hints (("Goal" :in-theory (disable bits-shift)
:use (:instance bits-shift (n (- n))))))
;why did I have to open up bits??
;perhaps export this?
(local (defthmd lnot-fl-eric-helper-2
(implies (and (<= k n)
(<= 0 k)
(integerp n)
(integerp k)
)
(equal (lnot (/ (bits x (1- n) 0) (expt 2 k)) (- n k))
(lnot (/ x (expt 2 k)) (- n k))))
:hints (("Goal" :in-theory (e/d ( lnot) (
LESS-THAN-MULTIPLY-THROUGH-BY-INVERTED-FACTOR-FROM-LEFT-HAND-SIDE)) ;BOZO
:use ((:instance bits-shift (n (- k)) (i (+ -1 N (* -1 K))) (j 0))
)))))
;lacks the bvecp hyp
(defthmd lnot-fl
(implies (and (<= k n)
(<= 0 k)
(integerp n)
(integerp k)
)
(equal (fl (* (/ (expt 2 k)) (lnot x n)))
(lnot (fl (/ x (expt 2 k))) (- n k))))
:hints (("Goal" :use lnot-fl-eric-helper-2
:in-theory (enable lnot-fl-eric-helper lnot-fl-eric-helper-2))))
(encapsulate
()
(local
(defthm mod-lnot-1
(implies (and (integerp x)
(>= x 0)
(integerp n)
(>= n 0) ;BOZO try dropping
(integerp m)
(>= m n)
(< x (expt 2 m))
(< x (expt 2 n)) ;new
)
(equal (lnot x m)
(+ (lnot x n) (* (expt 2 n) (1- (expt 2 (- m n)))))))
:rule-classes ()
:hints (("goal" :in-theory (enable lnot expt-split)
))))
(local
(defthm mod-lnot-2-thm
(implies (and (integerp x)
(>= x 0)
(integerp n)
(>= n 0)
(integerp m)
(>= m n)
(< x (expt 2 m))
(< x (expt 2 n)) ;new
)
(equal (mod (lnot x m) (expt 2 n))
(mod (+ (lnot x n) (* (expt 2 n) (1- (expt 2 (- m n))))) (expt 2 n))))
:rule-classes ()
:hints (("goal" :use (mod-lnot-1)))))
(local
(defthm mod-lnot-3
(implies (and (integerp x)
(>= x 0)
(integerp n)
(>= n 0)
(integerp m)
(>= m n)
(< x (expt 2 m))
)
(equal (mod (lnot x m) (expt 2 n))
(mod (lnot (mod x (expt 2 n)) m) (expt 2 n))))
:otf-flg t
:rule-classes ()
:hints (("goal" :in-theory (enable lnot ; bits
)
:use ((:instance mod-difference-elim-second (x1 (1- (expt 2 m))) (x2 x) (y (expt 2 n)))
(:instance expt-weak-monotone)
; (:instance lnot-bnds (n m))
; (:instance mod+-thm (m (lnot x n)) (n (expt 2 n)) (a (1- (expt 2 (- m n)))))
)))))
(local
(defthm mod-lnot-4
(implies (and (integerp x)
(>= x 0)
(integerp n)
(>= n 0)
(integerp m)
(>= m n)
(< x (expt 2 n))
)
(equal (mod (lnot x m) (expt 2 n))
(mod (lnot x n) (expt 2 n))))
:rule-classes ()
:hints (("goal"
:use (mod-lnot-2-thm
(:instance expt-weak-monotone)
; (:instance lnot-bnds (n m))
(:instance mod-mult-eric (x (lnot x n)) (y (expt 2 n)) (a (1- (expt 2 (- m n)))))
)))))
(local
(defthm mod-lnot-5
(implies (and (integerp x)
(>= x 0)
(integerp n)
(>= n 0)
(integerp m)
(>= m n)
(< x (expt 2 m)))
(equal (mod (lnot x m) (expt 2 n))
(mod (lnot (mod x (expt 2 n)) n) (expt 2 n))))
:rule-classes ()
:hints (("goal" :use (mod-lnot-3
(:instance mod-lnot-4 (x (mod x (expt 2 n))))
; (:instance mod-bnd-1 (m x) (n (expt 2 n)))
; (:instance mod>=0 (m x) (n (expt 2 n)))
)))))
;gen
;add case-splits
;write in terms of bvecp?
(defthm mod-lnot-aux
(implies (and (< x (expt 2 m)) ;drop!
(<= n m)
(integerp x)
(<= 0 x)
(integerp n)
(<= 0 n) ;gen
(integerp m)
)
(equal (mod (lnot x m) (expt 2 n))
(lnot (mod x (expt 2 n)) n)))
:hints (("goal" :in-theory (enable lnot)
:use (mod-lnot-5
;(:instance mod-equal (m (lnot (mod x (expt 2 n)) n)) (n (expt 2 n)))
;(:instance mod-bnd-1 (m x) (n (expt 2 n)))
; (:instance mod>=0 (m x) (n (expt 2 n)))
))))
)
(local (include-book "../arithmetic/top"))
;BOZO move this!
(defthm bits-ignores-mod-special
(equal (bits (mod x (expt 2 m)) (1- m) 0)
(bits x (1- m) 0)
)
:hints (("goal" :in-theory (enable bits)))
)
;BOZO move this!
(defthm bits-ignores-mod
(implies (and (<= m n)
(case-split (integerp n))
;(integerp m)
)
(equal (bits (mod x (expt 2 n)) (1- m) 0)
(bits x (1- m) 0)
))
:hints (("goal" :in-theory (enable bits)))
)
(defthm lnot-ignores-mod-special
(equal (lnot (mod x (expt 2 m)) m)
(lnot x m))
:hints (("Goal" :in-theory (enable lnot)))
)
(defthm lnot-ignores-mod
(implies (and (<= m n)
(case-split (integerp n)))
(equal (lnot (mod x (expt 2 n)) m)
(lnot x m)))
:hints (("Goal" :in-theory (enable lnot)))
)
;consider enabling?
(defthmd mod-lnot-aux2
(implies (and (<= n m)
(integerp x) ;will be dropped below
(integerp n)
(<= 0 n) ;gen
(integerp m)
)
(equal (mod (lnot x m) (expt 2 n))
(lnot (mod x (expt 2 n)) n))) ;note the lack of m in the conclusion
:hints (("Goal" :in-theory (disable MOD-LNOT-aux)
:use (:instance mod-lnot-aux (x (mod x (expt 2 m)))))))
;no (integerp x) hyp
(defthmd mod-lnot
(implies (and (<= n m) ;handle the other case?
(integerp n)
(<= 0 n) ;gen
(integerp m)
)
(equal (mod (lnot x m) (expt 2 n))
(lnot (mod x (expt 2 n)) n))) ;note the lack of m in the conclusion
:hints (("Goal" :use (:instance mod-lnot-aux2 (x (fl x)))
:in-theory (disable mod-lnot-aux2 ))))
(defthm mod-lnot-by-2
(implies (and (< 0 n)
(integerp x) ;gen?
(integerp n)
)
(equal (mod (lnot x n) 2)
(lnot (mod x 2) 1)))
:hints (("Goal" :in-theory (disable lnot-ignores-mod
LNOT-IGNORES-MOD-SPECIAL
mod-lnot)
:use ((:instance lnot-ignores-mod (n 1) (m n))
(:instance mod-lnot (m n) (n 1))))))
(local (defthmd bits-lnot-aux
(implies (and (< i m)
(case-split (bvecp x m)) ;dropped below..
(case-split (integerp m))
(case-split (integerp i))
(case-split (natp j)) ;gen?
)
(equal (bits (lnot x m) i j)
(lnot (bits x i j) (1+ (- i j)))))
:hints (("Goal" :cases ((>= i j))
:in-theory (e/d (bits bvecp lnot-fl) ( LNOT-IGNORES-MOD MOD-LNOT LNOT-IGNORES-MOD-SPECIAL))))))
;gen?
;BOZO formal m should be n
(defthm bits-lnot
(implies (and (< i m)
(case-split (natp j))
(case-split (integerp m))
(case-split (integerp i))
)
(equal (bits (lnot x m) i j)
(lnot (bits x i j) (1+ (- i j)))))
:hints (("Goal" :use (:instance bits-lnot-aux (x (bits x (1- m) 0)))
:in-theory (e/d (bvecp) ()))))
#|
(defthm bits-lnot-2
(implies (and (< i m)
(case-split (integerp m))
(case-split (integerp i))
(case-split (natp j)) ;gen?
(not (bvecp x m)) ;note!
(integerp x)
)
(equal (bits (lnot x m) i j)
(lnot (bits x i j) (1+ (- i j)))))
:hints (("Goal" :cases ((>= i j))
:in-theory (enable bvecp lnot))))
|#
;gen?
(defthm bitn-lnot
(implies (and (case-split (natp m))
(case-split (natp n))
;(case-split (bvecp x m))
)
(equal (bitn (lnot x m) n)
(if (< n m)
(lnot (bitn x n) 1)
0)))
:hints (("Goal" :in-theory (enable bitn BVECP-BITS-0
))))
;drop?
(defthm bitn-lnot-not-equal
(implies (and (< k n)
(integerp n)
(<= 0 n)
(integerp k)
(<= 0 k)
)
(not (= (bitn (lnot x n) k)
(bitn x k))))
:hints (("Goal" :in-theory (disable BITN-KNOWN-NOT-0-REPLACE-WITH-1) ;why needed?
; :in-theory (enable bvecp)
:use (:instance bitn-0-1 (n k))
))
:rule-classes ())
;could generalize these a lot (when lnot equals a constant, take the lnot of both sides)
;drop bvecp hyp by wrapping bits around conclusion?
(defthm lnot-bvecp-equal-0
(implies (case-split (bvecp x 1))
(equal (equal (lnot x 1) 0)
(not (equal x 0))))
:hints (("goal" :in-theory (enable lnot bvecp))))
(defthm lnot-bvecp-equal-1
(implies (case-split (bvecp x 1))
(equal (equal (lnot x 1) 1)
(equal x 0)))
:hints (("goal" :in-theory (enable lnot bvecp))))
(defthmd lnot-shift
(implies (and (syntaxp (not (quotep x))) ;prevents loops
(case-split (integerp x))
(case-split (< 0 n))
(case-split (integerp n))
)
(equal (lnot (* 2 x) n)
(+ 1 (* 2 (lnot x (1- n))))))
:hints (("Goal" :in-theory (enable lnot))))
#| BOZO get this to work and use in stick-proofs instead of bitn-lnot-not-equal?
;gen!
(defthm lnot-x-not-equal-x
(implies (and (natp n) (natp x))
(not (equal (lnot x n) x)))
:hints (("Goal" :in-theory (enable lnot)))
)
|#
(defthm lnot-with-n-not-an-integer
(implies (not (integerp n))
(equal (lnot x n)
0))
:hints (("Goal" :in-theory (enable lnot))))
(defthm lnot-with-n-not-positive
(implies (<= n 0)
(equal (lnot x n)
0))
:hints (("Goal" :in-theory (enable lnot))))
|
