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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 "oddr-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)))))
(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))))))
;;
;; New stuff:
;;
(defund oddr (x n)
(let ((z (fl (* (expt 2 (1- n)) (sig x)))))
(if (evenp z)
(* (sgn x) (1+ z) (expt 2 (- (1+ (expo x)) n)))
(* (sgn x) z (expt 2 (- (1+ (expo x)) n))))))
(defthm oddr-pos
(implies (and (< 0 x)
(rationalp x)
(integerp n)
(> n 0))
(< 0 (oddr x n)))
:rule-classes ())
(defthm oddr>=trunc
(implies (and (rationalp x)
(> x 0)
(integerp n)
(> n 0))
(>= (oddr x n) (trunc x n)))
:rule-classes ())
;BOZO just opens up ODDR when x is positive
;leave disabled!
(defthmd oddr-rewrite
(implies (and (< 0 x) ;note this hyp
(rationalp x)
(integerp n)
(< 0 n))
(equal (oddr x n)
(let ((z (fl (* (expt 2 (- (1- n) (expo x))) x))))
(if (evenp z)
(* (1+ z) (expt 2 (- (1+ (expo x)) n)))
(* z (expt 2 (- (1+ (expo x)) n))))))))
;move!
(defthm fl/2
(implies (integerp z)
(= (fl (/ z 2))
(if (evenp z)
(/ z 2)
(/ (1- z) 2))))
:rule-classes ())
(defthm oddr-other
(implies (and (rationalp x)
(> x 0)
(integerp n)
(> n 1))
(= (oddr x n)
(+ (trunc x (1- n))
(expt 2 (- (1+ (expo x)) n)))))
:rule-classes ())
(defthm expo-oddr
(implies (and (rationalp x)
(integerp n)
(> x 0)
(> n 1))
(equal (expo (oddr x n)) (expo x))))
(defthm exactp-oddr
(implies (and (rationalp x)
(integerp n)
(> x 0)
(> n 1))
(exactp (oddr x n) n))
:rule-classes ())
(defthm not-exactp-oddr
(implies (and (rationalp x)
(integerp n)
(> x 0)
(> n 1))
(not (exactp (oddr x n) (1- n))))
:rule-classes ())
(defthm trunc-oddr
(implies (and (rationalp x)
(> x 0)
(integerp n)
(integerp m)
(> m 0)
(> n m))
(= (trunc (oddr x n) m)
(trunc x m)))
:rule-classes ())
;disable?
(defun kp (k x y)
(+ k (- (expo (+ x y)) (expo y))))
(defthm oddr-plus
(implies (and (rationalp x)
(rationalp y)
(integerp k)
(> x 0)
(> y 0)
(> k 1)
(> (+ (1- k) (- (expo x) (expo y))) 0)
(exactp x (+ (1- k) (- (expo x) (expo y)))))
(= (+ x (oddr y k))
(oddr (+ x y) (kp k x y))))
:rule-classes ())
(defthm trunc-trunc-oddr
(implies (and (rationalp x)
(rationalp y)
(integerp m)
(integerp k)
(> x y)
(> y 0)
(> k 0)
(>= (- m 2) k))
(>= (trunc x k) (trunc (oddr y m) k)))
:rule-classes ())
(defthm away-away-oddr
(implies (and (rationalp x)
(rationalp y)
(integerp m)
(integerp k)
(> x y)
(> y 0)
(> k 0)
(>= (- m 2) k))
(>= (away x k) (away (oddr y m) k)))
:rule-classes ())
(defthm near-near-oddr
(implies (and (rationalp x)
(rationalp y)
(integerp m)
(integerp k)
(> x y)
(> y 0)
(> k 0)
(>= (- m 2) k))
(>= (near x k) (near (oddr y m) k)))
:rule-classes ())
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