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|
(in-package "ACL2")
; eric smith, david russinoff, with suggestions by matt kaufmann
; amd, june 2001
;this file was previously called irepsproofs.lisp
;note: the proofs in this file are messy. i haven't taken much time to clean
;them up by eliminating hacks or increasing elegance. -eric
;todo: add t-p for isigf, etc.
(include-book "rtl") ;BOZO remove!
(include-book "bias")
(include-book "float") ;for the defns
(local (include-book "merge"))
(local (include-book "cat"))
(local (include-book "bvecp"))
(local (include-book "bits"))
(local (include-book "bitn"))
(local (include-book "../arithmetic/top"))
(local (in-theory (enable bvecp-forward)))
;;encoding of floating-point numbers with implicit msb
;;bit vectors of length p+q, consisting of 1-bit sign field,
;;q-bit exponent field (bias = 2**(q-1)-1), and (p-1)-bit
;;significand field:
;; p must be > 1
;;;**********************************************************************
;;; field extractors
;;;**********************************************************************
(defund isgnf (x p q) (bitn x (1- (+ p q))))
(defund iexpof (x p q) (bits x (- (+ p q) 2) (1- p)))
(defund isigf (x p) (bits x (- p 2) 0))
;;;**********************************************************************
;;; representable numbers
;;;**********************************************************************
(defund nrepp (x p q)
(and (rationalp x)
(not (= x 0))
(< 0 (+ (expo x) (bias q)))
(< (+ (expo x) (bias q)) (- (expt 2 q) 1))
(exactp x p)))
(defund drepp (x p q)
(and (rationalp x)
(not (= x 0))
(<= (- 2 p) (+ (expo x) (bias q)))
(<= (+ (expo x) (bias q)) 0)
(exactp x (+ -2 p (expt 2 (- q 1)) (expo x))))) ;use bias here?
;bits available in the sig field = p-1-(-bias-expo(x))
(defund irepp (x p q)
(or (nrepp x p q)
(drepp x p q)))
;;;**********************************************************************
;;; valid encodings
;;;**********************************************************************
(defund nencodingp (x p q)
(and (bvecp x (+ p q))
(< 0 (iexpof x p q))
(< (iexpof x p q) (- (expt 2 q) 1))))
(defund dencodingp (x p q)
(and (bvecp x (+ p q))
(= (iexpof x p q) 0)
(not (= (isigf x p) 0))))
(defund iencodingp (x p q)
(or (nencodingp x p q)
(dencodingp x p q)))
;;;**********************************************************************
;;; encode
;;;**********************************************************************
; sig, expo, and sgn are defined in float.lisp
(defund dencode (x p q)
(cat (cat (if (= (sgn x) 1) 0 1)
1
0
q)
(1+ q)
(* (sig x) (expt 2 (+ -2 p (expo x) (bias q))))
(- p 1)))
(defund nencode (x p q)
(cat (cat (if (= (sgn x) 1) 0 1)
1
(+ (expo x) (bias q))
q)
(1+ q)
(* (- (sig x) 1) (expt 2 (- p 1)))
(- p 1)))
(defund iencode (x p q)
(cond ((nrepp x p q)
(nencode x p q))
((drepp x p q)
(dencode x p q))))
;;;**********************************************************************
;;; decode
;;;**********************************************************************
(defund ndecode (x p q)
(* (if (= (isgnf x p q) 0) 1 -1)
(+ (expt 2 (- (iexpof x p q) (bias q)))
(* (isigf x p)
(expt 2 (+ 1 (iexpof x p q) (- (bias q)) (- p)))))))
(defund ddecode (x p q)
(* (if (= (isgnf x p q) 0) 1 -1)
(isigf x p)
(expt 2 (+ 2 (- (bias q)) (- p)))))
(defund idecode (x p q)
(cond ((nencodingp x p q)
(ndecode x p q))
((dencodingp x p q)
(ddecode x p q))))
;;;**********************************************************************
;;; theorems
;;;**********************************************************************
(defthm nencodingp-means-not-dencodingp
(implies (nencodingp x p q)
(not (dencodingp x p q)))
:rule-classes ((:rewrite :backchain-limit-lst (0)))
:hints (("goal" :in-theory (enable iencodingp nencodingp dencodingp))))
(defthm dencodingp-means-not-nencodingp
(implies (dencodingp x p q)
(not (nencodingp x p q)))
:rule-classes ((:rewrite :backchain-limit-lst (0)))
:hints (("goal" :in-theory (enable iencodingp nencodingp dencodingp))))
(defthm not-both-nencodingp-and-dencodingp
(implies (iencodingp x p q)
(iff (nencodingp x p q) (not (dencodingp x p q))))
:hints (("Goal" :in-theory (enable iencodingp)))
:rule-classes ())
; the field extractors return bit-vectors.
;some of the rules below may be bad because they are put into both the
; forward-chaining and type-prescription rule classes, causing them
; not to always work.
;!! rc
(defthm bvecp-isigf-forward-3
(implies (case-split (integerp p))
(bvecp (isigf x p) (- p 1)))
:hints (("goal" :in-theory (enable isigf)))
:rule-classes (:rewrite
(:forward-chaining :trigger-terms ((isigf x p)))
))
(defthm iexpof-bvecp
(implies (case-split (integerp q))
(bvecp (iexpof x p q) q))
:hints (("goal" :in-theory (enable iexpof))))
(defthm isgnf-bvecp
(bvecp (isgnf x p q) 1)
:hints (("goal" :in-theory (enable isgnf))))
;forward-chaining-rules for encoding types
(defthm nencodingp-forward-to-iencodingp
(implies (nencodingp x p q)
(iencodingp x p q) )
:hints (("goal" :in-theory (enable iencodingp nencodingp)))
:rule-classes (:rewrite :forward-chaining)
)
(defthm dencodingp-forward-to-iencodingp
(implies (dencodingp x p q)
(iencodingp x p q) )
:hints (("goal" :in-theory (enable iencodingp dencodingp)))
:rule-classes (:rewrite :forward-chaining)
)
;needed? t-p?
(defthm not-zero-ddecode
(implies (and (dencodingp x p q)
(integerp p)
(< 1 p)
(integerp q)
(< 0 q))
(not (equal (ddecode x p q) 0)))
:hints (("goal" :in-theory (enable ddecode dencodingp))))
(defthm not-zero-ndecode
(implies (and ;(nencodingp x p q)
(rationalp x)
(>= x 0)
(integerp p)
(< 1 p)
(integerp q)
(< 0 q))
(not (equal (ndecode x p q) 0)))
:hints (("goal" :in-theory (enable ndecode nencodingp))))
;move
;BOZO make a negative-syntaxp version
;or just enable sgn?
(defthm sgn-minus-dist
(implies (and (acl2-numberp x)
(acl2-numberp y))
(equal (sgn (+ (* -1 x) (* -1 y)))
(* -1 (sgn (+ x y)))))
:hints (("Goal" :in-theory (disable sgn-minus)
:use (:instance sgn-minus (x (* -1 (+ x y)))))))
(defthm expo-ndecode
(implies (and (nencodingp x p q)
(integerp p)
(> p 1)
(integerp q)
(> q 0))
(equal (expo (ndecode x p q))
(- (iexpof x p q) (bias q))))
:hints (("goal" :in-theory (e/d (ndecode nencodingp
expt-split
expt-minus
isigf
)
()))))
(defthm sgn-ndecode
(implies (and ; (nencodingp x p q)
(rationalp x)
(>= x 0)
(integerp p)
(> p 1)
(integerp q)
(> q 0))
(equal (sgn (ndecode x p q))
(if (= (isgnf x p q) 0) 1 -1)))
:hints (("goal" :in-theory (enable ndecode sgn))))
;remove from :rewrite?
(defthm nencodingp-forward-to-positive-integerp
(implies (nencodingp x p q)
(and (integerp x)
(<= 0 x)))
:hints (("Goal" :in-theory (enable nencodingp)))
:rule-classes (:forward-chaining))
(defthm dencodingp-forward-to-positive-integerp
(implies (dencodingp x p q)
(and (integerp x)
(<= 0 x)))
:hints (("Goal" :in-theory (enable dencodingp)))
:rule-classes (:forward-chaining))
;BOZO do we even need this?
(defthmd ndecode-rewrite
(implies (and (rationalp x)
(integerp p)
(> p 1)
(integerp q)
(> q 0))
(= (ndecode x p q)
(* (if (= (isgnf x p q) 0) 1 -1)
(* (expt 2 (- (iexpof x p q) (bias q)))
(+ 1 (* (isigf x p)
(expt 2 (+ 1 (- p)))))))))
:hints (("Goal" :in-theory (enable ndecode))))
(defthm isigf-upper-bound-linear
(implies (case-split (integerp p))
(< (isigf x p) (expt 2 (- p 1) )))
:rule-classes (:rewrite :linear))
(defthm sig-ndecode
(implies (and (nencodingp x p q) ;gen?
(integerp p)
(> p 1)
(integerp q)
(> q 0))
(equal (sig (ndecode x p q))
(+ 1 (/ (isigf x p) (expt 2 (- p 1))))
))
:otf-flg t
:hints (("goal" :use (:instance expo-unique (x (+ 1
(* 2 (/ (EXPT 2 P))
(BITS X (+ -2 P) 0))))
(n 0))
:in-theory (set-difference-theories
(enable ndecode
expt-split
expt-minus
isigf iexpof ; isgnf
sig
; nencodingp
)
'( EXPO-UNIQUE-ERIC-2)))))
;instead, just open up sgn?
(defthm sgn-ddecode
(implies (and (dencodingp x p q)
(integerp p)
(> p 1)
(integerp q)
(> q 0))
(equal (sgn (ddecode x p q))
(if (= (isgnf x p q) 0) 1 -1)))
:hints (("Goal" :in-theory (e/d (ddecode dencodingp sgn) ()))))
(defthm sig-ddecode
(implies (and (dencodingp x p q)
(integerp p)
(> p 1)
(integerp q)
(> q 0))
(equal (sig (ddecode x p q))
(sig (isigf x p))))
:hints (("Goal" :in-theory (e/d ( ddecode sgn ) ()))))
(defthm expo-ddecode
(implies (and (dencodingp x p q)
(integerp p)
(> p 1)
(integerp q)
(> q 0))
(equal (expo (ddecode x p q))
(+ 2 (- p) (- (bias q)) (expo (isigf x p)))))
:hints (("Goal" :in-theory (enable ddecode dencodingp ))))
(defthm sgn-idecode
(implies (and (iencodingp x p q)
(integerp p)
(> p 1)
(integerp q)
(> q 0))
(equal (sgn (idecode x p q))
(if (= (isgnf x p q) 0) 1 -1)))
:hints (("Goal" :in-theory (set-difference-theories
(enable idecode iencodingp ISGNF NDECODE)
'()))))
(defthm sig-idecode
(implies (and (iencodingp x p q)
(integerp p)
(> p 1)
(integerp q)
(> q 0))
(equal (sig (idecode x p q))
(cond ((nencodingp x p q)
(+ 1 (/ (isigf x p) (expt 2 (- p 1)))))
((dencodingp x p q)
(sig (isigf x p))))))
:hints (("Goal" :in-theory (set-difference-theories
(enable idecode iencodingp)
'()))))
(defthm expo-idecode
(implies (and (iencodingp x p q)
(integerp p)
(> p 1)
(integerp q)
(> q 0))
(equal (expo (idecode x p q))
(cond ((nencodingp x p q)
(- (iexpof x p q) (bias q)))
((dencodingp x p q)
(+ 2 (- p) (- (bias q)) (expo (isigf x p)))))))
:hints (("Goal" :in-theory (set-difference-theories
(enable idecode iencodingp)
'()))))
;move?
(defthm expo-bvecp-upper-bound
(implies (and (bvecp x n)
(integerp n)
(< 0 n))
(< (expo x) n))
:hints (("Goal" :in-theory (enable bvecp)
:use ((:instance expo<= ( n (- n 1))))))
:rule-classes (:rewrite (:linear :match-free :all)))
(local (in-theory (disable bvecp-exactp))) ;why? efficiency?
;nice lemma?
(local
(defthm drepp-decode-1
(IMPLIES (AND (DENCODINGP X P Q)
(INTEGERP P)
(< 1 P)
(INTEGERP Q)
(< 0 Q))
(EXACTP (DDECODE X P Q)
(+ (* -1 (BIAS Q))
(EXPT 2 (1- Q))
(EXPO (ISIGF X P)))))
:hints (("Goal" :in-theory (enable ddecode dencodingp bias)))))
(defthm drepp-ddecode
(implies (and (dencodingp x p q)
(integerp p)
(> p 1)
(integerp q)
(> q 0))
(drepp (ddecode x p q) p q))
:hints (("Goal" :use drepp-decode-1
:in-theory (set-difference-theories
(enable drepp)
'(drepp-decode-1)))))
;BOZO yuck. better expo-shift rules would help here..
(defthm nrepp-ndecode-hack
(implies (and (nencodingp x p q)
(integerp p)
(> p 1)
(integerp q)
(> q 0))
(exactp (ndecode x p q) p))
:hints (("Goal" :in-theory (set-difference-theories
(enable nrepp nencodingp ndecode exactp
a15
ndecode-rewrite
;expt-split
)
'(a9 distributivity
)))
(and stable-under-simplificationp ;yuck?
'(:in-theory (enable a9 distributivity a15)))))
#|
(defthm nrepp-ndecode-hack
(implies (and (nencodingp x p q)
(integerp p)
(> p 1)
(integerp q)
(> q 0))
(exactp (ndecode x p q) p))
:hints (("Goal" :in-theory (set-difference-theories
(enable nrepp nencodingp ndecode exactp
a15
ndecode-rewrite
expt-split
expt-minus
sig
isigf
isgnf
iexpof
)
'(;a9 distributivity
a15
)))
(and stable-under-simplificationp ;yuck?
'(:in-theory (enable a9 distributivity a15)))))
|#
(defthm nrepp-ndecode
(implies (and (nencodingp x p q)
(integerp p)
(> p 1)
(integerp q)
(> q 0))
(nrepp (ndecode x p q) p q))
:hints (("Goal" :in-theory (set-difference-theories
(enable nrepp nencodingp)
'()))))
(defthm irepp-idecode
(implies (and (iencodingp x p q)
(integerp p)
(> p 1)
(integerp q)
(> q 0))
(irepp (idecode x p q) p q))
:hints (("Goal" :in-theory (set-difference-theories
(enable idecode irepp iencodingp)
'()))))
(local
(defthm nencodingp-nencode-2-1
(IMPLIES (AND (RATIONALP X)
(NOT (EQUAL X 0))
(EXACTP X P)
(INTEGERP P)
(<= 1 P)
)
(integerp (+ (* -1 (EXPT 2 (1- P)))
(* (SIG X) (EXPT 2 (1- P))))
))
:hints (("Goal" :in-theory (enable exactp)
:use ( sig-lower-bound)))))
(local
(defthm nencodingp-nencode-2
(IMPLIES (AND (NREPP X P Q)
(INTEGERP P)
(< 1 P)
(INTEGERP Q)
(< 0 Q))
(<= 0 (NENCODE X P Q)))
:hints (("Goal" :in-theory (enable nencode nrepp)))))
(local
(defthm nencodingp-nencode-3
(IMPLIES (AND (NREPP X P Q)
(INTEGERP P)
(< 1 P)
(INTEGERP Q)
(< 0 Q))
(< (NENCODE X P Q) (EXPT 2 (+ P Q))))
:hints (("Goal" :in-theory (enable nencode)))))
(local
(defthm nencodingp-nencode-4
(IMPLIES (AND (NREPP X P Q)
(INTEGERP P)
(< 1 P)
(INTEGERP Q)
(< 0 Q))
(< 0 (IEXPOF (NENCODE X P Q) P Q)))
:hints (("Goal" :in-theory (e/d ( nencode
iexpof
bvecp
nrepp
bits-tail) (sig-lower-bound sig-upper-bound))
:use (sig-upper-bound sig-lower-bound)))))
(local
(defthm nencodingp-nencode-5
(IMPLIES (AND (NREPP X P Q)
(INTEGERP P)
(< 1 P)
(INTEGERP Q)
(< 0 Q))
(< (IEXPOF (NENCODE X P Q) P Q)
(1- (EXPT 2 Q))))
:hints (("Goal" :in-theory (e/d (nrepp nencode iexpof bvecp
bits-tail) (sig-lower-bound sig-upper-bound))
:use (sig-upper-bound sig-lower-bound)))))
(defthm nencodingp-nencode
(implies (and (nrepp x p q)
(integerp p)
(> p 1)
(integerp q)
(> q 0))
(nencodingp (nencode x p q) p q) )
:hints (("Goal" :in-theory (enable nencodingp bvecp)))
:OTF-FLG T)
(local
(defthm dencodingp-dencode-hack-3
(implies (and (drepp x p q)
(integerp p)
(> p 1)
(integerp q)
(> q 0))
(equal (* (SIG X)
(EXPT 2 (+ -3 P (EXPO X) (EXPT 2 (1- Q)))))
(* (abs x) (EXPT 2 (+ -3 P (EXPT 2 (1- Q)))))))
:hints (("Goal" :in-theory (set-difference-theories
(enable drepp sgn)
'(fp-rep-collapse abs))
:use fp-abs))
:rule-classes nil))
(local (in-theory (disable expt-compare))) ;yuck
(local
(defthmd dencodingp-dencode-hack-4
(implies (and (drepp x p q)
(integerp p)
(> p 1)
(integerp q)
(> q 0))
(bvecp (* (SIG X)
(EXPT 2 (+ -3 P (EXPO X) (EXPT 2 (1- Q)))))
(- p 1)))
:hints (("Goal" :in-theory (set-difference-theories (enable dencodingp drepp dencode
iexpof isigf bias exactp
expt-split
bvecp)
'(abs EXPT-COMPARE sig-upper-bound))
:use (dencodingp-dencode-hack-3
sig-upper-bound
(:instance expt-weak-monotone
(n (+ -3 P (EXPO X) (EXPT 2 (1- Q))))
(m (- p 2))))))))
(defthm dencodingp-dencode
(implies (and (drepp x p q)
(integerp p)
; (> p 1)
(integerp q)
(> q 0)
)
(dencodingp (dencode x p q) p q) )
:hints (("Goal" :in-theory (e/d (exactp
dencodingp
drepp
dencode
iexpof
isigf
bias
bits-tail
bvecp
bvecp-bits-0
)
(sig-upper-bound
BITS-SHIFT
BITS-SPLIT-AROUND-ZERO))
:use (sig-upper-bound
sig-lower-bound
dencodingp-dencode-hack-4
(:instance expt-strong-monotone
(n (- p 1))
(m (+ p q)))
(:instance expt-strong-monotone
(n (- p 1))
(m (+ p q -1)))))))
(defthm drepp-forward-to-rationalp
(implies (drepp x p q)
(rationalp x))
:hints (("Goal" :in-theory (enable drepp)))
:rule-classes ((:forward-chaining :trigger-terms ((drepp x p q)))))
(defthm drepp-zero-false
(not (drepp 0 p q))
:hints (("Goal" :in-theory (enable drepp))))
(defthm iencodingp-iencode
(implies (and (irepp x p q)
(integerp p)
(> p 1)
(integerp q)
(> q 0))
(iencodingp (iencode x p q) p q) )
:hints (("Goal" :in-theory (enable iencodingp irepp iencode))))
;prove a thm about bvecp of sig-1?
(defthm isigf-nencode-0
(implies (and (rationalp x)
(not (equal x 0))
(exactp x p)
(integerp p)
(< 1 p)
)
(bvecp (+ (* -1 (expt 2 (1- p)))
(* (sig x) (expt 2 (1- p))))
(- p 1)))
:hints (("goal" :in-theory (set-difference-theories
(enable bvecp)
'(nencodingp-nencode-2-1
sig-upper-bound))
:use (sig-lower-bound sig-upper-bound nencodingp-nencode-2-1)))
:rule-classes nil)
(defthm isgnf-nencode
(implies (and (nrepp x p q)
(integerp p)
(> p 1)
(integerp q)
(> q 0))
(equal (isgnf (nencode x p q) p q)
(if (equal (sgn x) 1) 0 1)))
:hints (("Goal" :in-theory (set-difference-theories
(enable isgnf nencode nrepp bvecp)
'(nencodingp-nencode-2-1
bitn-bvecp-0
sig-upper-bound
))
:use (nencodingp-nencode-2-1
sig-lower-bound
sig-upper-bound
(:instance bitn-bvecp-0 (m 0)
(x (+ (BIAS Q) (EXPO X)))
(n q))))))
(defthm isgnf-dencode
(implies (and (drepp x p q)
(integerp p)
(> p 1)
(integerp q)
(> q 0))
(equal (isgnf (dencode x p q) p q)
(if (= (sgn x) 1) 0 1)))
:hints (("Goal" :in-theory (set-difference-theories
(enable isgnf dencode drepp bvecp bias)
'(nencodingp-nencode-2-1 bitn-bvecp-0))
:use (dencodingp-dencode-hack-4
(:instance expt-strong-monotone
(n (- p 1))
(m (+ p q -1)))
(:instance bitn-bvecp-0
(m q)
(x (* (SIG X)
(EXPT 2 (+ -3 P (EXPO X) (EXPT 2 (1- Q))))))
(n (- p 1)))))))
(defthm isgnf-iencode
(implies (and (irepp x p q)
(integerp p)
(> p 1)
(integerp q)
(> q 0))
(equal (isgnf (iencode x p q) p q)
(if (equal (sgn x) 1) 0 1)))
:hints (("Goal" :in-theory (enable irepp iencode))))
(defthm isigf-nencode-1
(implies (and (nrepp x p q)
(integerp p)
(> p 1)
(integerp q)
(> q 0))
(bvecp (+ (* -2 (EXPT 2 (+ -2 P)))
(* 2 (SIG X) (EXPT 2 (+ -2 P)))) (- p 1) ))
:hints (("Goal" :in-theory (set-difference-theories
(enable isigf nencode nrepp bvecp expt-split)
'())
:use isigf-nencode-0))
:rule-classes nil)
(defthm isigf-nencode
(implies (and (nrepp x p q)
(integerp p)
(> p 1)
(integerp q)
(> q 0))
(equal (isigf (nencode x p q) p)
(* (- (sig x) 1) (expt 2 (- p 1)))))
:hints (("Goal" :in-theory (e/d (isigf nencode nrepp expt-split bvecp bias bits-tail)
(bits-shift
BITS-SUM-DROP-IRRELEVANT-TERM-1-OF-2
BITS-SHIFT-BY-CONSTANT-POWER-OF-2))
:use (isigf-nencode-1
))))
(defthm isigf-dencode
(implies (and (drepp x p q)
(integerp p)
(> p 1)
(integerp q)
(> q 0))
(equal (isigf (dencode x p q) p)
(* (sig x) (expt 2 (+ -2 p (expo x) (bias q))))))
:hints (("Goal" :in-theory (set-difference-theories
(enable isigf dencode drepp bias bits-tail lead-bit-0)
'(BITS-SHIFT))
:use dencodingp-dencode-hack-4)))
(defthm iexpof-nencode
(implies (and (nrepp x p q)
(integerp p)
(> p 1)
(integerp q)
(> q 0))
(equal (iexpof (nencode x p q) p q)
(+ (expo x) (bias q))))
:hints (("Goal" :in-theory (enable iexpof nencode nrepp bvecp
bits-tail)
:use (isigf-nencode-1))))
(defthm iexpof-dencode
(implies (and (drepp x p q)
(integerp p)
(> p 1)
(integerp q)
(> q 0))
(equal (iexpof (dencode x p q) p q)
0))
:otf-flg t
:hints (("Goal" :in-theory (set-difference-theories
(enable iexpof dencode drepp bias
bvecp-bits-0
)
'())
:use (dencodingp-dencode-hack-4))))
(defthm ndecode-nencode
(implies (and (nrepp x p q)
(integerp p)
(> p 1)
(integerp q)
(> q 0))
(equal (ndecode (nencode x p q) p q)
x))
:hints (("Goal" :in-theory (enable nrepp ndecode sgn expt-split expt-minus))))
(defthm ddecode-dencode
(implies (and (drepp x p q)
(integerp p)
(> p 1)
(integerp q)
(> q 0))
(equal (ddecode (dencode x p q) p q)
x))
:hints (("Goal" :in-theory (enable drepp ddecode sgn expt-split expt-minus))))
(defthm not-both-nrepp-and-drepp
(implies (irepp x p q)
(iff (nrepp x p q)
(not (drepp x p q))))
:rule-classes ()
:hints (("Goal" :in-theory (enable irepp nrepp drepp))))
(defthm idecode-iencode
(implies (and (irepp x p q)
(integerp p)
(> p 1)
(integerp q)
(> q 0))
(equal
(idecode (iencode x p q) p q)
x))
:hints (("Goal" :in-theory (enable irepp idecode iencode idecode)
:use (dencodingp-dencode
not-both-nrepp-and-drepp))))
(defthm nencode-ndecode
(implies (and (nencodingp x p q)
(integerp p)
(> p 1)
(integerp q)
(> q 0))
(equal (nencode (ndecode x p q) p q)
x))
:hints (("Goal" :in-theory (set-difference-theories
(enable nencodingp nencode iexpof isigf isgnf sgn
cat lead-bit-0
expt-split
bits-reduce ;why?
)
; Modification by Matt K. for v2-9, due to the change to rewrite-clause that
; avoids using forward-chaining facts derived from a literal that has been
; rewritten: use sgn-ndecode explicitly (see :use hint below) and hence disable
; it here.
'(sgn-ndecode))
:use (sgn-ndecode
(:instance bits-plus-bits
(m 0)
(p (- p 1))
(n (+ -2 p q)))
(:instance bitn-plus-bits
(n (+ p q -1))
(m 0))
(:instance bitn-0-1 (x x) (n (+ -1 p q)))))))
(defthm dencode-ddecode
(implies (and (dencodingp x p q)
(integerp p)
(> p 1)
(integerp q)
(> q 0))
(equal (dencode (ddecode x p q) p q)
x))
:hints (("Goal" :in-theory (set-difference-theories
(enable dencodingp dencode iexpof isigf isgnf sgn
cat lead-bit-0 expt-split expt-minus)
'())
:use ((:instance bits-plus-bits
(m 0)
(p (- p 1))
(n (+ -2 p q)))
(:instance fp-rep (x (BITS X (+ -2 P) 0)))
(:instance bitn-plus-bits
(n (+ p q -1))
(m 0))
(:instance bitn-0-1 (x x) (n (+ -1 p q)))
))))
(defthm iencode-idecode
(implies (and (iencodingp x p q)
(integerp p)
(> p 1)
(integerp q)
(> q 0))
(equal (iencode (idecode x p q) p q)
x))
:hints (("Goal" :in-theory (set-difference-theories
(enable irepp idecode iencode idecode IENCODINGP)
'())
:use (
(:instance not-both-nrepp-and-drepp (x (ddecode x p q)))))))
(defthm nrepp-bvecp-sig
(implies (and (natp p)
(> p 0)
(nrepp x p q))
(bvecp (* (1- (sig x)) (expt 2 (1- p)))
(1- p)))
:rule-classes ()
:hints (("Goal" :in-theory (e/d (nrepp exactp bvecp) ()))))
; Matt K., after v4-2:
; Commenting out the following rule, which rewrites a term to itself!
#||
(defthm nencode-nencode
(implies (and (nrepp x p q)
(integerp p)
(> p 0)
(integerp q)
(> q 0))
(equal (nencode x p q) (nencode x p q)))
:hints (("Goal" :use (nrepp-bvecp-sig)
:in-theory (enable nencode nencode bits-tail nrepp bvecp))))
||#
(defthm drepp-bvecp-sig
(implies (and (integerp p)
(natp q)
(> q 0)
(drepp x p q))
(bvecp (* (sig x) (expt 2 (+ -2 p (expo x) (bias q))))
(1- p)))
:rule-classes ()
:hints (("Goal" :in-theory (e/d (bias drepp exactp bvecp expt-split) (sig-upper-bound
))
:use (;sig-lower-bound
sig-upper-bound
(:instance expt-weak-monotone (n (+ -3 P (EXPO X) (EXPT 2 (+ -1 Q)))) (m (- p 2)))))))
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