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(in-package "ACL2")
; Eric Smith, David Russinoff, with contributions and suggestions by Matt Kaufmann
; AMD, June 2001
;this file was previously called repsproofs.lisp
(include-book "rtl")
(include-book "float") ;to get the defns...
(local (include-book "ereps-proofs"))
(set-inhibit-warnings "theory") ; avoid warning in the next event
(local (in-theory nil))
;(set-inhibit-warnings) ; restore theory warnings (optional)
; bias of a q bit exponent field is 2^(q-1)-1
(defund bias (q) (- (expt 2 (- q 1)) 1) )
;;Encoding of floating-point numbers with explicit leading one:
;;bit vectors of length p+q+1, consisting of 1-bit sign field,
;;q-bit exponent field (bias = 2**(q-1)-1), and p-bit significand field.
(defund esgnf (x p q) (bitn x (+ p q)))
(defund eexpof (x p q) (bits x (1- (+ p q)) p))
(defund esigf (x p) (bits x (1- p) 0))
;;;**********************************************************************
;;; REPRESENTABLE NUMBERS
;;;**********************************************************************
(defund erepp (x p q)
(and (rationalp x)
(not (= x 0))
(bvecp (+ (expo x) (bias q)) q)
(exactp x p)))
;;;**********************************************************************
;;; VALID ENCODINGS
;;;**********************************************************************
(defund eencodingp (x p q)
(and (bvecp x (+ p q 1))
(= (bitn x (- p 1)) 1)))
;;;**********************************************************************
;;; EENCODE
;;;**********************************************************************
; sig, expo, and sgn are defined in float.lisp
;bozo disable!
(defund eencode (x p q)
(cat (cat (if (= (sgn x) 1) 0 1)
1
(+ (expo x) (bias q))
q)
(1+ q)
(* (sig x) (expt 2 (- p 1)))
p) )
;;;**********************************************************************
;;; EDECODE
;;;**********************************************************************
(defund edecode (x p q)
(* (if (= (esgnf x p q) 0) 1 -1)
(esigf x p)
(expt 2 (+ 1 (- p) (eexpof x p q) (- (bias q))))))
;;;**********************************************************************
;;; Encoding and Decoding are Inverses
;;;**********************************************************************
(defthm erepp-edecode
(implies (and (eencodingp x p q)
(integerp p)
(> p 0)
(integerp q)
(> q 0))
(erepp (edecode x p q) p q)))
(defthm eencodingp-eencode
(implies (and (erepp x p q)
(integerp p)
(> p 0)
(integerp q)
(> q 0))
(eencodingp (eencode x p q) p q) ))
(defthm edecode-eencode
(implies (and (erepp x p q)
(integerp p)
; (> p 0)
(integerp q)
; (> q 0)
)
(equal (edecode (eencode x p q) p q)
x )))
(defthm eencode-edecode
(implies (and (eencodingp x p q)
(integerp p)
(>= p 0)
(integerp q)
(> q 0))
(equal (eencode (edecode x p q) p q)
x )))
(defthm expo-edecode
(implies (and (eencodingp x p q)
(integerp p)
(> p 0)
(integerp q)
(> q 0))
(equal (expo (edecode x p q))
(- (eexpof x p q) (bias q))
)))
(defthm sgn-edecode
(implies (and (eencodingp x p q)
(integerp p)
(> p 0)
(integerp q)
(> q 0))
(equal (sgn (edecode x p q))
(if (= (esgnf x p q) 0) 1 -1))))
(defthm sig-edecode
(implies (and (eencodingp x p q)
(integerp p)
(> p 0)
(integerp q)
(> q 0))
(equal (sig (edecode x p q))
(/ (esigf x p) (expt 2 (- p 1))))))
(defthm eencodingp-not-zero
(implies (and (eencodingp x p q)
(integerp p)
(> p 0)
(integerp q)
(> q 0))
(not (equal (edecode x p q) 0))))
(defund rebias-expo (expo old new)
(+ expo (- (bias new) (bias old))))
;;I actually needed all four of the following lemmas, although I would have thought
;;that the two bvecp lemmas would be enough.
(defthm natp-rebias-up
(implies (and (natp n)
(natp m)
(< 0 m)
(<= m n)
(bvecp x m))
(natp (rebias-expo x m n)))
:hints (("goal" :in-theory (e/d ( expt-split rebias-expo bvecp natp bias
) (expt-compare))
:use (:instance expt-weak-monotone (n m) (m n)))))
(defthm natp-rebias-down
(implies (and (natp n)
(natp m)
(< 0 m)
(<= m n)
(bvecp x n)
(< x (+ (expt 2 (1- n)) (expt 2 (1- m))))
(>= x (- (expt 2 (1- n)) (expt 2 (1- m)))))
(natp (rebias-expo x n m)))
:hints (("goal" :in-theory (enable rebias-expo bvecp natp bias))))
(defthm bvecp-rebias-up
(implies (and (natp n)
(natp m)
(< 0 m)
(<= m n)
(bvecp x m))
(bvecp (rebias-expo x m n) n)))
(defthm bvecp-rebias-down
(implies (and (natp n)
(natp m)
(< 0 m)
(<= m n)
(bvecp x n)
(< x (+ (expt 2 (1- n)) (expt 2 (1- m))))
(>= x (- (expt 2 (1- n)) (expt 2 (1- m)))))
(bvecp (rebias-expo x n m) m)))
(defthm rebias-up
(implies (and (natp n)
(natp m)
(> n m)
(> m 1)
(bvecp x m))
(equal (rebias-expo x m n)
(cat (cat (bitn x (1- m))
1
(mulcat 1 (- n m) (lnot (bitn x (1- m)) 1))
(- n m))
(1+ (- n m))
(bits x (- m 2) 0)
(1- m))))
:rule-classes ())
(defthm rebias-down
(implies (and (natp n)
(natp m)
(> n m)
(> m 1)
(bvecp x n)
(< x (+ (expt 2 (1- n)) (expt 2 (1- m))))
(>= x (- (expt 2 (1- n)) (expt 2 (1- m)))))
(equal (rebias-expo x n m)
(cat (bitn x (1- n))
1
(bits x (- m 2) 0)
(1- m))))
:rule-classes ())
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