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(in-package "ACL2")
;;;***************************************************************
;;;an acl2 library of floating point arithmetic
;;;david m. russinoff
;;;advanced micro devices, inc.
;;;february, 1998
;;;***************************************************************
(local (include-book "sticky-proofs"))
;; Necessary functions:
(defund fl (x)
(declare (xargs :guard (real/rationalp x)))
(floor x 1))
(defund cg (x)
(declare (xargs :guard (real/rationalp x)))
(- (fl (- x))))
(defun expo-measure (x)
; (declare (xargs :guard (and (real/rationalp x) (not (equal x 0)))))
(cond ((not (rationalp x)) 0)
((< x 0) '(2 . 0))
((< x 1) (cons 1 (fl (/ x))))
(t (fl x))))
(defund expo (x)
(declare (xargs :guard t
:measure (expo-measure x)))
(cond ((or (not (rationalp x)) (equal x 0)) 0)
((< x 0) (expo (- x)))
((< x 1) (1- (expo (* 2 x))))
((< x 2) 0)
(t (1+ (expo (/ x 2))))))
;could redefine to divide by the power of 2 (instead of making it a negative power of 2)...
(defund sig (x)
(declare (xargs :guard t))
(if (rationalp x)
(if (< x 0)
(- (* x (expt 2 (- (expo x)))))
(* x (expt 2 (- (expo x)))))
0))
;make defund?
(defun sgn (x)
(declare (xargs :guard t))
(if (or (not (rationalp x)) (equal x 0))
0
(if (< x 0)
-1
1)))
(defund exactp (x n)
; (declare (xargs :guard (and (real/rationalp x) (integerp n))))
(integerp (* (sig x) (expt 2 (1- n)))))
(defun fp+ (x n)
(+ x (expt 2 (- (1+ (expo x)) n))))
(defund trunc (x n)
(declare (xargs :guard (integerp n)))
(* (sgn x) (fl (* (expt 2 (1- n)) (sig x))) (expt 2 (- (1+ (expo x)) n))))
(defund away (x n)
(* (sgn x) (cg (* (expt 2 (1- n)) (sig x))) (expt 2 (- (1+ (expo x)) n))))
(defund re (x)
(- x (fl x)))
(defund near (x n)
(let ((z (fl (* (expt 2 (1- n)) (sig x))))
(f (re (* (expt 2 (1- n)) (sig x)))))
(if (< f 1/2)
(trunc x n)
(if (> f 1/2)
(away x n)
(if (evenp z)
(trunc x n)
(away x n))))))
(defund near+ (x n)
(if (< (re (* (expt 2 (1- n)) (sig x)))
1/2)
(trunc x n)
(away x n)))
;;
;; New stuff:
;;
(defund sticky (x n)
(cond ((exactp x (1- n)) x)
(t (+ (trunc x (1- n))
(* (sgn x) (expt 2 (1+ (- (expo x) n))))))))
(defthm sticky-1
(implies (rationalp x)
(equal (sticky x 1)
(* (sgn x) (expt 2 (expo x))))))
;more rule-classes?
(defthm sticky-pos
(implies (and (< 0 x)
(rationalp x)
(integerp n) (> n 0))
(< 0 (sticky x n)))
:rule-classes :linear)
(defthm sticky-shift
(implies (and (rationalp x)
(integerp n) (> n 0)
(integerp k))
(= (sticky (* (expt 2 k) x) n)
(* (expt 2 k) (sticky x n))))
:rule-classes ())
(defthm sticky-minus
(equal (sticky (* -1 x) n)
(* -1 (sticky x n))))
;gen?
(defthm sticky-exactp
(implies (and (rationalp x) (>= x 0)
(integerp n) (> n 0)
)
(exactp (sticky x n) n))
:rule-classes ())
(defthm sticky-exactp-n-1
(implies (and (rationalp x) (> x 0)
(integerp n) (> n 1))
(iff (exactp (sticky x n) (1- n))
(exactp x (1- n))))
:rule-classes ())
(defthm expo-sticky
(implies (and (rationalp x) (> x 0)
(integerp n) (> n 0))
(= (expo (sticky x n))
(expo x)))
:rule-classes ())
(defthm trunc-sticky
(implies (and (rationalp x) (> x 0)
(integerp m) (> m 0)
(integerp n) (> n m))
(= (trunc (sticky x n) m)
(trunc x m)))
:rule-classes ())
(defthm away-sticky
(implies (and (rationalp x) (> x 0)
(integerp m) (> m 0)
(integerp n) (> n m))
(= (away (sticky x n) m)
(away x m)))
:rule-classes ())
(defthm near-sticky
(implies (and (rationalp x) (> x 0)
(integerp m) (> m 0)
(integerp n) (> n (1+ m)))
(= (near (sticky x n) m)
(near x m)))
:rule-classes ())
(defthm near+-sticky
(implies (and (rationalp x) (> x 0)
(integerp m) (> m 0)
(integerp n) (> n (1+ m)))
(= (near+ (sticky x n) m)
(near+ x m)))
:rule-classes ())
;make local?
(defthm minus-trunc-4
(implies (and (rationalp x)
(> x 0)
(rationalp y)
(> y 0)
(< y x)
(integerp k)
(> k 0)
(> (+ k (- (expo (- x y)) (expo y))) 0)
(= n (+ k (- (expo x) (expo y))))
(exactp x n))
(equal (- x (trunc y k))
(away (- x y) (+ k (- (expo (- x y)) (expo y))))))
:rule-classes ())
;make local?
(defthm minus-trunc-5
(implies (and (rationalp x)
(> x 0)
(rationalp y)
(> y 0)
(< x y)
(integerp k)
(> k 0)
(> (+ k (- (expo (- x y)) (expo y))) 0)
(= n (+ k (- (expo x) (expo y))))
(exactp x n))
(equal (- x (trunc y k))
(- (trunc (- y x) (+ k (- (expo (- x y)) (expo y)))))))
:rule-classes ())
(defthm sticky-plus
(implies (and (rationalp x)
(> x 0)
(rationalp y)
(> y 0)
(integerp k)
(= k1 (+ k (- (expo x) (expo y))))
(= k2 (+ k (- (expo (+ x y)) (expo y))))
(> k 1)
(> k1 1)
(> k2 1)
(exactp x (1- k1)))
(= (+ x (sticky y k))
(sticky (+ x y) k2)))
:rule-classes ())
;BOZO move?
(defthm trunc-away
(implies (and (rationalp x) (> x 0)
(integerp n) (> n 0)
(not (exactp x n)))
(= (away x n)
(+ (trunc x n)
(expt 2 (+ (expo x) 1 (- n))))))
:rule-classes ())
(defthm sticky-0
(equal (sticky 0 n)
0))
(defthm minus-sticky
(implies (and (rationalp x)
(> x 0)
(rationalp y)
(> y 0)
(integerp k)
(= k1 (+ k (- (expo x) (expo y))))
(= k2 (+ k (- (expo (- x y)) (expo y))))
(> k 1)
(> k1 1)
(> k2 1)
(exactp x (1- k1)))
(= (- x (sticky y k))
(sticky (- x y) k2)))
:rule-classes ())
(defthm sticky-lemma
(implies (and (rationalp x)
(> x 0)
(rationalp y)
(integerp k)
(= k1 (+ k (- (expo x) (expo y))))
(= k2 (+ k (- (expo (+ x y)) (expo y))))
(> k 1)
(> k1 1)
(> k2 1)
(exactp x (1- k1)))
(= (+ x (sticky y k))
(sticky (+ x y) k2)))
:rule-classes ())
(defthm sticky-sticky
(implies (and (rationalp x)
(integerp m)
(> m 1)
(integerp n)
(>= n m))
(= (sticky (sticky x n) m)
(sticky x m)))
:rule-classes ())
(defthm sticky-exactp-m
(implies (and (rationalp x)
(integerp m)
(integerp n)
(> n m)
(> m 0))
(iff (exactp (sticky x n) m)
(exactp x m)))
:rule-classes ())
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