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|
; RTL - A Formal Theory of Register-Transfer Logic and Computer Arithmetic
;
; Contact:
; David M. Russinoff
; 1106 W 9th St., Austin, TX 78703
; david@russinoff.com
; http://www.russinoff.com/
;
; See license file books/rtl/rel11/license.txt.
;
(in-package "RTL")
(set-enforce-redundancy t) ; for some reason, acl2 4.3 complains about logand-natp
(local (include-book "../support/top"))
(set-inhibit-warnings "theory") ; avoid warning in the next event
(local (in-theory nil))
(include-book "defs")
;;;**********************************************************************
;;; LOGAND, LOGIOR, and LOGXOR
;;;**********************************************************************
(defsection-rtl |Binary Operations| |Logical Operations|
(in-theory (disable logand logior logxor))
(defthmd logand-def
(equal (logand x y)
(if (or (zip x) (zip y))
0
(if (= x y)
x
(+ (* 2 (logand (fl (/ x 2)) (fl (/ y 2))))
(logand (mod x 2) (mod y 2))))))
:rule-classes ((:definition :controller-alist ((binary-logand t t)))))
(defthmd logior-def
(implies (and (integerp x) (integerp y))
(equal (logior x y)
(if (or (zip x) (= x y))
y
(if (zip y)
x
(+ (* 2 (logior (fl (/ x 2)) (fl (/ y 2))))
(logior (mod x 2) (mod y 2)))))))
:rule-classes ((:definition :controller-alist ((binary-logior t t)))))
(defthmd logxor-def
(implies (and (integerp x) (integerp y))
(equal (logxor x y)
(if (zip x)
y
(if (zip y)
x
(if (= x y)
0
(+ (* 2 (logxor (fl (/ x 2)) (fl (/ y 2))))
(logxor (mod x 2) (mod y 2))))))))
:rule-classes ((:definition :controller-alist ((binary-logxor t t)))))
(defun log-induct (x y)
(declare (xargs :measure (abs (* (ifix x) (ifix y)))))
(if (or (not (integerp x)) (not (integerp y)) (= x 0) (= y 0) (= x y))
(+ x y)
(log-induct (fl (/ x 2)) (fl (/ y 2)))))
(defthm logand-bvecp
(implies (and (natp n)
(bvecp x n)
(integerp y))
(bvecp (logand x y) n)))
(defthm logior-bvecp
(implies (and (bvecp x n)
(bvecp y n))
(bvecp (logior x y) n)))
(defthm logxor-bvecp
(implies (and (bvecp x n)
(bvecp y n)
(natp n))
(bvecp (logxor x y) n)))
(defthmd logand-plus-logxor
(implies (and (integerp x)
(integerp y))
(equal (+ (logand x y) (logxor x y))
(logior x y))))
(defthmd logand-mod
(implies (and (integerp x)
(integerp y)
(integerp n))
(equal (mod (logand x y) (expt 2 n))
(logand (mod x (expt 2 n))
(mod y (expt 2 n))))))
(defthmd logior-mod
(implies (and (integerp x)
(integerp y)
(integerp n))
(equal (mod (logior x y) (expt 2 n))
(logior (mod x (expt 2 n))
(mod y (expt 2 n))))))
(defthmd logxor-mod
(implies (and (integerp x)
(integerp y)
(integerp n))
(equal (mod (logxor x y) (expt 2 n))
(logxor (mod x (expt 2 n))
(mod y (expt 2 n))))))
(defthmd fl-logand
(implies (and (integerp x)
(integerp y)
(natp n))
(equal (fl (* (expt 2 (- n)) (logand x y)))
(logand (fl (* (expt 2 (- n)) x)) (fl (* (expt 2 (- n)) y))))))
(defthmd fl-logior
(implies (and (integerp x)
(integerp y)
(natp n))
(equal (fl (* (expt 2 (- n)) (logior x y)))
(logior (fl (* (expt 2 (- n)) x)) (fl (* (expt 2 (- n)) y))))))
(defthmd fl-logxor
(implies (and (integerp x)
(integerp y)
(natp n))
(equal (fl (* (expt 2 (- n)) (logxor x y)))
(logxor (fl (* (expt 2 (- n)) x)) (fl (* (expt 2 (- n)) y))))))
(defthmd logand-cat
(implies (and (case-split (integerp x1))
(case-split (integerp y1))
(case-split (integerp x2))
(case-split (integerp y2))
(case-split (natp n))
(case-split (natp m)))
(equal (logand (cat x1 m y1 n) (cat x2 m y2 n))
(cat (logand x1 x2) m (logand y1 y2) n))))
(defthmd logior-cat
(implies (and (case-split (integerp x1))
(case-split (integerp y1))
(case-split (integerp x2))
(case-split (integerp y2))
(case-split (natp n))
(case-split (natp m)))
(equal (logior (cat x1 m y1 n) (cat x2 m y2 n))
(cat (logior x1 x2) m (logior y1 y2) n))))
(defthmd logxor-cat
(implies (and (case-split (integerp x1))
(case-split (integerp y1))
(case-split (integerp x2))
(case-split (integerp y2))
(case-split (natp n))
(case-split (natp m)))
(equal (logxor (cat x1 m y1 n) (cat x2 m y2 n))
(cat (logxor x1 x2) m (logxor y1 y2) n))))
(defthmd logand-shift
(implies (and (integerp x)
(integerp y)
(natp k))
(equal (logand (* (expt 2 k) x)
(* (expt 2 k) y))
(* (expt 2 k) (logand x y)))))
(defthmd logior-shift
(implies (and (integerp x)
(integerp y)
(natp k))
(equal (logior (* (expt 2 k) x)
(* (expt 2 k) y))
(* (expt 2 k) (logior x y)))))
(defthmd logxor-shift
(implies (and (integerp x)
(integerp y)
(natp k))
(equal (logxor (* (expt 2 k) x)
(* (expt 2 k) y))
(* (expt 2 k) (logxor x y)))))
(defthmd logand-expt
(implies (and (integerp x)
(integerp y)
(natp n))
(equal (logand (* (expt 2 n) x) y)
(* (expt 2 n) (logand x (fl (/ y (expt 2 n))))))))
(defthmd logior-expt
(implies (and (integerp x)
(integerp y)
(natp n))
(equal (logior (* (expt 2 n) x) y)
(+ (* (expt 2 n) (logior x (fl (/ y (expt 2 n)))))
(mod y (expt 2 n))))))
(defthmd logxor-expt
(implies (and (integerp x)
(integerp y)
(natp n))
(equal (logxor (* (expt 2 n) x) y)
(+ (* (expt 2 n) (logxor x (fl (/ y (expt 2 n)))))
(mod y (expt 2 n))))))
(defthmd logior-expt-cor
(implies (and (natp n)
(integerp x)
(bvecp y n))
(equal (logior (* (expt 2 n) x) y)
(+ (* (expt 2 n) x) y))))
(defthmd logior-2**n
(implies (and (natp n)
(integerp x))
(equal (logior (expt 2 n) x)
(if (= (bitn x n) 1)
x
(+ x (expt 2 n))))))
(defthmd logand-bits
(implies (and (integerp x)
(natp n)
(natp k)
(< k n))
(equal (logand x (- (expt 2 n) (expt 2 k)))
(* (expt 2 k) (bits x (1- n) k)))))
(defthmd logand-bit
(implies (and (integerp x)
(natp n))
(equal (logand x (expt 2 n))
(* (expt 2 n) (bitn x n)))))
(defthmd bits-logand
(implies (and (integerp x)
(integerp y)
(integerp i)
(integerp j))
(equal (bits (logand x y) i j)
(logand (bits x i j) (bits y i j)))))
(defthmd bits-logior
(implies (and (integerp x)
(integerp y)
(integerp i)
(integerp j))
(equal (bits (logior x y) i j)
(logior (bits x i j) (bits y i j)))))
(defthmd bits-logxor
(implies (and (integerp x)
(integerp y)
(integerp i)
(integerp j))
(equal (bits (logxor x y) i j)
(logxor (bits x i j) (bits y i j)))))
(defthmd bitn-logand
(implies (and (integerp x)
(integerp y)
(integerp n))
(equal (bitn (logand x y) n)
(logand (bitn x n) (bitn y n)))))
(defthmd bitn-logior
(implies (and (integerp x)
(integerp y)
(integerp n))
(equal (bitn (logior x y) n)
(logior (bitn x n) (bitn y n)))))
(defthmd bitn-logxor
(implies (and (case-split (integerp x))
(case-split (integerp y))
(case-split (integerp n)))
(equal (bitn (logxor x y) n)
(logxor (bitn x n) (bitn y n)))))
)
;;;**********************************************************************
;;; LOGNOT
;;;**********************************************************************
(defsection-rtl |Complementation| |Logical Operations|
(in-theory (disable lognot))
(defthmd lognot-def
(implies (integerp x)
(equal (lognot x)
(1- (- x)))))
(defthmd lognot-shift
(implies (and (integerp x)
(natp k))
(equal (lognot (* (expt 2 k) x))
(+ (* (expt 2 k) (lognot x))
(1- (expt 2 k))))))
(defthmd lognot-fl
(implies (and (integerp x)
(not (zp n)))
(equal (lognot (fl (/ x n)))
(fl (/ (lognot x) n)))))
(defthmd mod-lognot
(implies (and (integerp x)
(natp n))
(equal (mod (lognot x) (expt 2 n))
(1- (- (expt 2 n) (mod x (expt 2 n)))))))
(defthmd bits-lognot
(implies (and (natp i)
(natp j)
(<= j i)
(integerp x))
(equal (bits (lognot x) i j)
(- (1- (expt 2 (- (1+ i) j))) (bits x i j)))))
(defthm bitn-lognot
(implies (and (integerp x)
(natp n))
(not (equal (bitn (lognot x) n)
(bitn x n))))
:rule-classes ())
(defthmd bits-lognot-bits
(implies (and (integerp x)
(natp i)
(natp j)
(natp k)
(natp l)
(<= l k)
(<= k (- i j)))
(equal (bits (lognot (bits x i j)) k l)
(bits (lognot x) (+ k j) (+ l j)))))
(defthmd bits-lognot-bits-lognot
(implies (and (integerp x)
(natp i)
(natp j)
(natp k)
(natp l)
(<= l k)
(<= k (- i j)))
(equal (bits (lognot (bits (lognot x) i j)) k l)
(bits x (+ k j) (+ l j)))))
(defthmd logand-bits-lognot
(implies (and (integerp x)
(integerp n)
(bvecp y n))
(equal (logand y (bits (lognot x) (1- n) 0))
(logand y (lognot (bits x (1- n) 0))))))
)
;;;**********************************************************************
;;; Algebraic Properties
;;;**********************************************************************
(defsection-rtl |Algebraic Properties| |Logical Operations|
(defthm logand-x-0
(equal (logand x 0) 0))
(defthm logand-0-y
(equal (logand 0 y) 0))
(defthm logior-x-0
(implies (integerp x)
(equal (logior x 0) x)))
(defthm logior-0-y
(implies (integerp y)
(equal (logior 0 y) y)))
(defthm logxor-x-0
(implies (integerp x)
(equal (logxor x 0) x)))
(defthm logxor-0-y
(implies (integerp y)
(equal (logxor 0 y) y)))
(defthm logand-self
(implies (case-split (integerp x))
(equal (logand x x) x)))
(defthm logior-self
(implies (case-split (integerp x))
(equal (logior x x) x)))
(defthm logxor-self
(equal (logxor x x) 0))
(defthm lognot-lognot
(implies (case-split (integerp i))
(equal (lognot (lognot i))
i)))
(defthmd logior-not-0
(implies (and (integerp x)
(integerp y))
(iff (equal (logior x y) 0)
(and (= x 0) (= y 0)))))
(defthmd logxor-not-0
(implies (and (integerp x)
(integerp y))
(iff (equal (logxor x y) 0)
(= x y))))
(defthm logand-x-1
(implies (bvecp x 1)
(equal (logand x 1) x)))
(defthm logand-1-x
(implies (bvecp x 1)
(equal (logand 1 x) x)))
(defthm logior-1-x
(implies (bvecp x 1)
(equal (logior 1 x) 1)))
(defthm logior-x-1
(implies (bvecp x 1)
(equal (logior x 1) 1)))
(defthm logand-x-m1
(implies (integerp x)
(equal (logand x -1) x)))
(defthm logand-m1-y
(implies (integerp y)
(equal (logand -1 y) y)))
(defthm logior-x-m1
(implies (integerp x)
(equal (logior x -1) -1)))
(defthm logior-m1-y
(implies (integerp y)
(equal (logior -1 y) -1)))
(defthm logxor-x-m1
(implies (integerp x)
(equal (logxor x -1)
(lognot x))))
(defthm logxor-m1-x
(implies (integerp x)
(equal (logxor -1 x)
(lognot x))))
(defthm logand-commutative
(equal (logand y x) (logand x y)))
(defthm logior-commutative
(equal (logior y x) (logior x y)))
(defthm logxor-commutative
(equal (logxor y x) (logxor x y)))
(defthm logand-commutative-2
(equal (logand y x z)
(logand x y z)))
(defthm logior-commutative-2
(equal (logior y x z)
(logior x y z)))
(defthm logxor-commutative-2
(equal (logxor y x z)
(logxor x y z)))
(defthm logand-associative
(equal (logand (logand x y) z)
(logand x (logand y z))))
(defthm logior-associative
(equal (logior (logior x y) z)
(logior x (logior y z))))
(defthm logxor-associative
(equal (logxor (logxor x y) z)
(logxor x (logxor y z))))
(defthmd logior-logand
(implies (and (integerp x)
(integerp y)
(integerp z))
(equal (logior x (logand y z))
(logand (logior x y) (logior x z)))))
(defthmd logior-logand-1
(implies (and (integerp x)
(integerp y))
(equal (logior x (logand x y))
x)))
(defthmd logand-logior
(implies (and (integerp x)
(integerp y)
(integerp z))
(equal (logand x (logior y z))
(logior (logand x y) (logand x z)))))
(defthmd logior-logand-2
(implies (and (integerp x)
(integerp y)
(integerp z))
(equal (logand (logior y z) x)
(logior (logand y x) (logand z x)))))
(defthmd log3
(implies (and (integerp x)
(integerp y)
(integerp z))
(equal (logior (logand x y) (logior (logand x z) (logand y z)))
(logior (logand x y) (logand (logxor x y) z)))))
(defthmd logxor-rewrite
(implies (and (integerp x)
(integerp y))
(equal (logxor x y)
(logior (logand x (lognot y))
(logand y (lognot x))))))
(defthmd lognot-logxor
(and (equal (logxor (lognot x) y)
(lognot (logxor x y)))
(equal (logxor y (lognot x))
(lognot (logxor x y)))))
)
|
