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|
;;;***************************************************************
;;;An ACL2 Library of Floating Point Arithmetic
;;;David M. Russinoff
;;;Advanced Micro Devices, Inc.
;;;February, 1998
;;;***************************************************************
;put all the new defuns in this book together at the t
(in-package "ACL2")
(local (include-book "float"))
(local (include-book "trunc"))
(local (include-book "away"))
(local (include-book "near"))
(local (include-book "near+"))
(local (include-book "sticky"))
(local (include-book "bitn")) ; for roundup
(local (include-book "land")) ; for roundup
(local (include-book "lior")) ; for roundup
;; Necessary functions:
(local ; ACL2 primitive
(defun natp (x)
(declare (xargs :guard t))
(and (integerp x)
(<= 0 x))))
(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)))
(defund sticky (x n)
(cond ((exactp x (1- n)) x)
(t (+ (trunc x (1- n))
(* (sgn x) (expt 2 (1+ (- (expo x) n))))))))
(defund bits (x i j)
(declare (xargs :guard (and (natp x)
(natp i)
(natp j))
:verify-guards nil))
(mbe :logic (if (or (not (integerp i))
(not (integerp j)))
0
(fl (/ (mod x (expt 2 (1+ i))) (expt 2 j))))
:exec (if (< i j)
0
(logand (ash x (- j)) (1- (ash 1 (1+ (- i j))))))))
(defund bitn (x n)
(declare (xargs :guard (and (natp x)
(natp n))
:verify-guards nil))
(mbe :logic (bits x n n)
:exec (if (evenp (ash x (- n))) 0 1)))
;;
;; New stuff:
;;
;Typically, we may plan to have inf and minf enabled, but we have a few lemmas about them anyway..
(defund inf (x n)
(if (>= x 0)
(away x n)
(trunc x n)))
(defund minf (x n)
(if (>= x 0)
(trunc x n)
(away x n)))
(defund IEEE-MODE-P (mode)
(member mode '(trunc inf minf near)))
(defund rounding-mode-p (mode)
(or (IEEE-mode-p mode) (equal mode 'away) (equal mode 'near+)))
(defund rnd (x mode n)
(case mode
(away (away x n))
(near+ (near+ x n))
(trunc (trunc x n))
(inf (inf x n))
(minf (minf x n))
(near (near x n))
(otherwise 0)))
(defund flip (m)
(case m
(inf 'minf)
(minf 'inf)
(t m)))
;rounding constant..
(defun rnd-const (e mode n)
(case mode
((near near+) (expt 2 (- e n)))
((inf away) (1- (expt 2 (1+ (- e n)))))
(otherwise 0)))
(defthmd inf-minus
(equal (inf (* -1 x) n)
(* -1 (minf x n)))
:hints (("Goal" :in-theory (enable inf minf))))
(defthmd minf-minus
(equal (minf (* -1 x) n)
(* -1 (inf x n)))
:hints (("Goal" :in-theory (enable inf minf))))
(defthm inf-shift
(implies (and (rationalp x)
(integerp n)
(integerp k))
(= (inf (* x (expt 2 k)) n)
(* (inf x n) (expt 2 k))))
:hints (("Goal" :in-theory (enable inf)
)))
(defthm minf-shift
(implies (and (rationalp x)
(integerp n)
(integerp k))
(= (minf (* x (expt 2 k)) n)
(* (minf x n) (expt 2 k))))
:hints (("Goal" :in-theory (enable minf))))
(defthm ieee-mode-p-implies-rounding-mode-p
(implies (IEEE-mode-p mode)
(rounding-mode-p mode))
:hints (("Goal" :in-theory (enable rounding-mode-p)))
:rule-classes (:rewrite; :forward-chaining
))
(defthm rationalp-rnd
(rationalp (rnd x mode n))
:hints (("Goal" :in-theory (enable rnd)))
:rule-classes (:type-prescription))
(defthmd rnd-minus
(equal (rnd (* -1 x) mode n)
(* -1 (rnd x (flip mode) n)))
:hints (("Goal" :in-theory (enable rnd flip minf-minus inf-minus near+-minus))))
(local (defthm rnd-const-thm-1
(implies (and (integerp n)
(> n 1)
(integerp x)
(> x 0)
(>= (expo x) n))
(= (near x n)
(if (and (exactp x (1+ n))
(not (exactp x n)))
(trunc (+ x (rnd-const (expo x) 'near n)) (1- n))
(trunc (+ x (rnd-const (expo x) 'near n)) n))))
:rule-classes ()
:hints (("Goal"
:use ((:instance near-trunc))))))
(local (defthm rnd-const-thm-2
(implies (and (integerp n)
(> n 1)
(integerp x)
(> x 0)
(>= (expo x) n))
(= (away x n)
(trunc (+ x (rnd-const (expo x) 'inf n)) n)))
:rule-classes ()
:hints (("Goal" :in-theory (enable exactp2)
:use ((:instance away-imp (m (1+ (expo x)))))))))
(local (defthm rnd-const-thm-3
(implies (and (integerp n)
(> n 1)
(integerp x)
(> x 0)
(>= (expo x) n))
(= (near+ x n)
(trunc (+ x (rnd-const (expo x) 'near+ n)) n)))
:rule-classes ()
:hints (("Goal" :in-theory (enable exactp2)
:use ((:instance near+trunc))))))
(defthm RND-CONST-THM
(implies (and (rounding-mode-p mode)
(integerp n)
(> n 1)
(integerp x)
(> x 0)
(>= (expo x) n))
(= (rnd x mode n)
(if (and (eql mode 'near)
(exactp x (1+ n))
(not (exactp x n)))
(trunc (+ x (rnd-const (expo x) mode n)) (1- n))
(trunc (+ x (rnd-const (expo x) mode n)) n))))
:rule-classes ()
:hints (("Goal" :in-theory (enable inf minf rnd rounding-mode-p ieee-mode-p)
:use (rnd-const-thm-1 rnd-const-thm-2 rnd-const-thm-3))))
(defun roundup (x mode n)
; Returns T when we should add an ulp after truncating x to n digits.
(case mode
(near+ (= (bitn x (- (expo x) n)) 1))
(near (and (= (bitn x (- (expo x) n)) 1)
(or (not (exactp x (1+ n)))
(= (bitn x (- (1+ (expo x)) n)) 1))))
((inf away) (not (exactp x n)))
(otherwise nil)))
(local
(defthm exactp-preserved-up ; could perhaps manage with exactp-<= instead
(implies (and (integerp n)
(< 0 n)
(exactp x n))
(exactp x (1+ n)))
:hints (("Goal" :in-theory (enable exactp)
:expand ((expt 2 n))))))
(local (include-book "merge"))
(local
(defthm roundup-thm-1
(implies (and (rounding-mode-p mode)
(not (eq mode 'near))
(not (eq mode 'near+))
(integerp n)
(> n 1)
(integerp x)
(> x 0)
(>= (expo x) n))
(= (rnd x mode n)
(if (roundup x mode n)
(fp+ (trunc x n) n)
(trunc x n))))
:hints (("Goal" :in-theory (enable inf minf rnd rounding-mode-p
ieee-mode-p)
:use (trunc-away trunc-exactp-a away-exactp-a)))))
(local (include-book "bits-trunc"))
(local
(defthm roundup-thm-2-1-1-1
(implies (and (integerp n)
(< 1 n)
(integerp x)
(< 0 x)
(<= n (expo x))
(equal (bitn x (+ 1 (expo x) (* -1 n)))
0))
(equal (trunc x n)
(trunc x (1- n))))
:hints (("Goal" :use ((:instance bits-trunc-2 (k n)
(n (1+ (expo x))))
(:instance bits-trunc-2 (k (1- n))
(n (1+ (expo x))))
(:instance bits-plus-bitn (n (expo x))
(m (+ 1 (expo x) (* -1 n)))))
:expand ((expt 2 (+ 2 (expo x) (* -1 n))))))
:rule-classes nil))
(local
(defthm roundup-thm-2-1-1
(implies (and (integerp n)
(< 1 n)
(integerp x)
(< 0 x)
(<= n (expo x))
(equal (bitn x (+ 1 (expo x) (* -1 n)))
0))
(exactp (trunc x n) (1- n)))
:hints (("Goal" :use roundup-thm-2-1-1-1))))
(local
(defthm roundup-thm-2-1-2
(implies (and (not (exactp x n))
(integerp n)
(< 1 n)
(integerp x)
(< 0 x)
(<= n (expo x))
(exactp x (+ 1 n)))
(equal (+ x (expt 2 (+ (expo x) (* -1 n))))
(+ (trunc x n)
(expt 2 (1+ (+ (expo x) (* -1 n)))))))
:hints (("Goal" :use trunc-away-a
:expand ((expt 2 (+ 1 (expo x) (* -1 n))))))
:rule-classes nil))
(local
(defthm roundup-thm-2-1
(implies (and (not (exactp x n))
(integerp n)
(< 1 n)
(integerp x)
(< 0 x)
(<= n (expo x))
(exactp x (+ 1 n))
(not (equal (bitn x (+ 1 (expo x) (* -1 n)))
1)))
(equal (trunc (+ x (expt 2 (+ (expo x) (* -1 n))))
(1- n))
(trunc x n)))
:hints (("Goal" :in-theory (enable plus-trunc-corollary
expt-strong-monotone)
:use roundup-thm-2-1-2))))
(local
(defthm roundup-thm-2-2
(implies (and (not (exactp x n))
(exactp x (+ 1 n))
(integerp n)
(< 1 n)
(integerp x)
(< 0 x)
(<= n (expo x)))
(equal (bitn x (+ (expo x) (* -1 n)))
1))
:hints (("Goal" :use ((:instance exact-k+1
(n (1+ (expo x)))
(k (+ (expo x) (* -1 n)))))))))
(local
(encapsulate
()
(local
(defthm roundup-thm-2-3-1-1-1
(implies (and (not (exactp x n))
(exactp x (+ 1 n))
(integerp n)
(< 1 n)
(integerp x)
(< 0 x)
(<= n (expo x))
(equal (bitn x (+ 1 (expo x) (* -1 n)))
1))
(equal (bits x (expo x)
(+ 1 (expo x) (* -1 n)))
(+ 1
(* 2
(bits x (expo x)
(+ 2 (expo x) (* -1 n)))))))
:hints (("Goal" :use ((:instance bits-plus-bitn (n (expo x))
(m (+ 1 (expo x) (* -1 n)))))))))
(defthm roundup-thm-2-3-1-1
(implies (and (not (exactp x n))
(exactp x (+ 1 n))
(integerp n)
(< 1 n)
(integerp x)
(< 0 x)
(<= n (expo x))
(equal (bitn x (+ 1 (expo x) (* -1 n)))
1))
(equal x
(+ (trunc x (1- n))
(expt 2 (+ (expo x) (* -1 n)))
(expt 2 (+ 1 (expo x) (* -1 n))))))
:hints (("Goal"
:use ((:instance bits-trunc-2 (n (1+ (expo x)))
(k (1+ n)))
(:instance bits-plus-bitn (n (expo x))
(m (+ (expo x) (* -1 n))))
(:instance bits-plus-bitn (n (expo x))
(m (+ 1 (expo x) (* -1 n))))
(:instance bits-trunc-2 (n (1+ (expo x)))
(k (1- n))))
:expand
((expt 2 (+ 2 (expo x) (* -1 n)))))
;; BOZO!! We can't put the following as part of Goal's :expand hint.
("Subgoal 4" :expand ((expt 2 (+ 1 (expo x) (* -1 n)))))
("Subgoal 1" :expand ((expt 2 (+ 1 (expo x) (* -1 n))))))
:rule-classes nil)))
(local
(defthm roundup-thm-2-3-1-2
(implies (and (not (exactp x n))
(exactp x (+ 1 n))
(integerp n)
(< 1 n)
(integerp x)
(< 0 x)
(<= n (expo x))
(equal (bitn x (+ 1 (expo x) (* -1 n)))
1))
(equal (+ x (expt 2 (+ (expo x) (* -1 n))))
(+ (trunc x (1- n))
(expt 2 (+ 2 (expo x) (* -1 n))))))
:hints (("Goal" :use roundup-thm-2-3-1-1
:expand ((expt 2 (+ 2 (expo x) (* -1 n)))))
("Subgoal 1" ; !! BOZO: avoid infinite by using separate subgoal hint
:expand ((expt 2 (+ 1 (expo x) (* -1 n))))))
:rule-classes nil))
(local
(defthm roundup-thm-2-3-1
(implies (and (not (exactp x n))
(exactp x (+ 1 n))
(integerp n)
(< 1 n)
(integerp x)
(< 0 x)
(<= n (expo x))
(equal (bitn x (+ 1 (expo x) (* -1 n)))
1))
(exactp (+ x (expt 2 (+ (expo x) (* -1 n))))
(1- n)))
:hints (("Goal" :use (roundup-thm-2-3-1-1
roundup-thm-2-3-1-2
(:instance fp+2
(x (trunc x (1- n)))
(n (1- n))))))
:rule-classes nil))
(local
(defthm roundup-thm-2-3-2
(implies (and (not (exactp x n))
(exactp x (+ 1 n))
(integerp n)
(< 1 n)
(integerp x)
(< 0 x)
(<= n (expo x))
(equal (bitn x (+ 1 (expo x) (* -1 n)))
1))
(equal (+ (trunc x n)
(expt 2 (+ 1 (expo x) (* -1 n))))
(+ x (expt 2 (+ (expo x) (* -1 n))))))
:hints (("Goal" :use trunc-away-a
:expand ((expt 2 (+ 1 (expo x) (* -1 n))))))
:rule-classes nil))
(local
(defthm roundup-thm-2-3
(implies (and (not (exactp x n))
(exactp x (+ 1 n))
(integerp n)
(< 1 n)
(integerp x)
(< 0 x)
(<= n (expo x))
(equal (bitn x (+ 1 (expo x) (* -1 n)))
1))
(equal (trunc (+ x (expt 2 (+ (expo x) (* -1 n))))
(1- n))
(+ (trunc x n)
(expt 2 (+ 1 (expo x) (* -1 n))))))
:hints (("Goal" :use (roundup-thm-2-3-1
roundup-thm-2-3-2
(:instance trunc-exactp-a
(x (+ x (expt 2 (+ (expo x) (* -1 n)))))
(n (1- n))))))))
; We need a special case of the following lemma for roundup-thm-2-4, so let's
; prove a nice version to include in the library.
(defthmd trunc-split
(implies (and (= n (1+ (expo x)))
(>= x 0)
(integerp m)
(> m k)
(integerp k)
(> k 0))
(equal (trunc x m)
(+ (trunc x k)
(* (expt 2 (- n m))
(bits x (1- (- n k)) (- n m))))))
:hints (("Goal" :use ((:instance expt-split (r 2)
(i (+ m (* -1 k)))
(j (+ 1 (expo x) (* -1 m))))
bits-trunc-2
(:instance bits-trunc-2 (k m)
(n (1+ (expo x))))
(:instance bits-plus-bits (n (expo x))
(m (- (1+ (expo x)) m))
(p (- (1+ (expo x)) k)))))))
(defund trunc-rem (x n)
(- x (trunc x n)))
(defthm trunc-rem-small
(implies (and (integerp n)
(<= 0 n)
(rationalp x)
(<= 0 x))
(< (trunc-rem x n)
(expt 2 (+ 1 (expo x) (* -1 n)))))
:hints (("Goal" :use (trunc-diff-pos trunc-rem)))
:rule-classes :linear)
(defthm trunc-rem-nonnegative
(implies (and (integerp n)
(<= 0 n)
(rationalp x)
(<= 0 x))
(and (rationalp (trunc-rem x n))
(<= 0 (trunc-rem x n))))
:hints (("Goal" :use (trunc-lower-pos trunc-rem)))
:rule-classes :type-prescription)
; First, break x into the high n bits, the next bit, and the rest.
(local
(defthm roundup-thm-2-4-1
(implies (and (integerp n)
(< 1 n)
(integerp x)
(< 0 x)
(<= n (expo x))
(equal (bitn x (+ (expo x) (* -1 n)))
1))
(equal x
(+ (trunc x n)
(expt 2 (- (expo x) n))
(trunc-rem x (1+ n)))))
:hints (("Goal" :use ((:instance trunc-split
(n (1+ (expo x)))
(m (1+ n))
(k n)))
:in-theory (enable trunc-rem bitn)))
:rule-classes nil))
; Next, trivially introduce fp+. A key fact is that (exactp (fp+ (trunc x n)
; n) n), by fp+2 and trunc-exact-b. We need that fact in order to apply
; plus-trunc-corollary.
(local
(defthm roundup-thm-2-4-2
(implies (and (integerp n)
(< 1 n)
(integerp x)
(< 0 x)
(<= n (expo x))
(equal (bitn x (+ (expo x) (* -1 n)))
1))
(equal (+ x (expt 2 (+ (expo x) (* -1 n))))
(+ (fp+ (trunc x n) n)
(trunc-rem x (1+ n)))))
:hints (("Goal" :use (roundup-thm-2-4-1)
:expand ((expt 2 (+ 1 (expo x) (* -1 n))))))
:rule-classes nil))
(local
(defthm expt-2-+-constant
(implies (and (syntaxp (quotep k))
(integerp n)
(<= 0 n)
(integerp k)
(<= 0 k))
(equal (expt 2 (+ k n))
(* 2 (expt 2 (+ (1- k) n)))))
:hints (("Goal" :expand ((expt 2 (+ k n)))))))
; Note: fp+-positive was first discovered at about this point.
(local
(defthm roundup-thm-2-4-3
(implies (and (integerp n)
(< 1 n)
(integerp x)
(< 0 x)
(<= n (expo x))
(equal (bitn x (+ (expo x) (* -1 n)))
1))
(equal (trunc (+ (fp+ (trunc x n) n)
(trunc-rem x (1+ n)))
n)
(fp+ (trunc x n) n)))
:hints (("Goal" :use ((:instance fp+2 (x (trunc x n)))
(:instance plus-trunc-corollary
(x (fp+ (trunc x n) n))
(y (trunc-rem x (1+ n))))
(:instance trunc-rem-small (n (1+ n)))
(:instance fp+2-2 (x (trunc x n))))
:in-theory (disable fp+ plus-trunc-corollary)))
:rule-classes nil))
(local
(defthm roundup-thm-2-4
(implies (and (integerp n)
(< 1 n)
(integerp x)
(< 0 x)
(<= n (expo x))
(equal (bitn x (+ (expo x) (* -1 n)))
1))
(equal (trunc (+ x (expt 2 (+ (expo x) (* -1 n))))
n)
(+ (trunc x n)
(expt 2 (+ 1 (expo x) (* -1 n))))))
:hints (("Goal" :use (roundup-thm-2-4-2 roundup-thm-2-4-3)))))
(local
(defthm roundup-thm-2-5-1
(implies (and (integerp n)
(< 1 n)
(integerp x)
(< 0 x)
(<= n (expo x))
(equal (bitn x (+ (expo x) (* -1 n)))
0))
(equal x
(+ (trunc x n)
(trunc-rem x (1+ n)))))
:hints (("Goal" :use ((:instance trunc-split
(n (1+ (expo x)))
(m (1+ n))
(k n)))
:in-theory (enable trunc-rem bitn)))
:rule-classes nil))
(local
(defthm roundup-thm-2-5-2
(implies (and (integerp n)
(< 1 n)
(integerp x)
(< 0 x)
(<= n (expo x))
(equal (bitn x (+ (expo x) (* -1 n)))
0))
(equal (+ x (expt 2 (+ (expo x) (* -1 n))))
(+ (trunc x n)
(expt 2 (+ (expo x) (* -1 n)))
(trunc-rem x (1+ n)))))
:hints (("Goal" :use (roundup-thm-2-5-1)
:expand ((expt 2 (+ 1 (expo x) (* -1 n))))))
:rule-classes nil))
(local
(defthm roundup-thm-2-5-3
(implies (and (integerp n)
(< 1 n)
(integerp x)
(< 0 x)
(<= n (expo x)))
(< (+ (expt 2 (+ (expo x) (* -1 n)))
(trunc-rem x (1+ n)))
(expt 2 (+ 1 (expo x) (* -1 n)))))
:hints (("Goal" :expand ((expt 2 (+ 1 (expo x) (* -1 n))))
:use ((:instance trunc-rem-small (n (1+ n))))))
:rule-classes nil))
(local
(defthm roundup-thm-2-5
(implies (and (integerp n)
(< 1 n)
(integerp x)
(< 0 x)
(<= n (expo x))
(not (equal (bitn x (+ (expo x) (* -1 n)))
1)))
(equal (trunc (+ x (expt 2 (+ (expo x) (* -1 n))))
n)
(trunc x n)))
:hints (("Goal" :use ((:instance plus-trunc-corollary
(x (trunc x n))
(y (+ (trunc-rem x (+ 1 n))
(expt 2 (+ (expo x) (* -1 n))))))
roundup-thm-2-5-2
roundup-thm-2-5-3)
:in-theory (disable fp+ plus-trunc-corollary)))))
(local
(defthm roundup-thm-2-6
(implies (and (exactp x n)
(integerp n)
(< 1 n)
(integerp x)
(< 0 x)
(<= n (expo x)))
(equal (bitn x (+ (expo x) (* -1 n)))
0))
:hints (("Goal" :use ((:instance exact-bits-1
(n (1+ (expo x)))
(k (- (1+ (expo x)) n)))
(:instance exact-bits-3
(k (- (1+ (expo x)) n))))))))
(local
(defthm roundup-thm-2
(implies (and (eq mode 'near)
(integerp n)
(> n 1)
(integerp x)
(> x 0)
(>= (expo x) n))
(= (rnd x mode n)
(if (roundup x mode n)
(fp+ (trunc x n) n)
(trunc x n))))
:hints (("Goal" :in-theory (enable rnd)
:use (near-exactp-a rnd-const-thm-1)))
:rule-classes ()))
(local
(defthm roundup-thm-3
(implies (and (eq mode 'near+)
(integerp n)
(> n 1)
(integerp x)
(> x 0)
(>= (expo x) n))
(= (rnd x mode n)
(if (roundup x mode n)
(fp+ (trunc x n) n)
(trunc x n))))
:hints (("Goal" :in-theory (enable rnd)
:use near+trunc))
:rule-classes ()))
(defthm roundup-thm
(implies (and (rounding-mode-p mode)
(integerp n)
(> n 1)
(integerp x)
(> x 0)
(>= (expo x) n))
(= (rnd x mode n)
(if (roundup x mode n)
(fp+ (trunc x n) n)
(trunc x n))))
:hints (("Goal" :in-theory (enable rounding-mode-p)
:use (roundup-thm-1
roundup-thm-2
roundup-thm-3)))
:rule-classes ())
;rephrase?
(defthmd rnd-sticky
(implies (and (> n (1+ k))
(rounding-mode-p mode)
(rationalp x) (> x 0)
(integerp k) (> k 0)
(integerp n) )
(equal (rnd (sticky x n) mode k)
(rnd x mode k)))
:hints (("Goal" :in-theory (enable rnd minf inf)
:use (sticky-pos
(:instance trunc-sticky (m k))
(:instance away-sticky (m k))
(:instance near-sticky (m k))
(:instance near+-sticky (m k))))))
(defthm rnd-shift
(implies (and (rationalp x)
(integerp n)
(rounding-mode-p mode)
(integerp k))
(= (rnd (* x (expt 2 k)) mode n)
(* (rnd x mode n) (expt 2 k))))
:rule-classes ()
:hints (("goal" :in-theory (enable rnd IEEE-MODE-P
rounding-mode-p)
:use (trunc-shift
away-shift
near-shift
near+-shift
inf-shift
minf-shift
))))
;elim <-- why?
(defthm expo-rnd
(implies (and (rationalp x)
(not (= x 0))
(integerp n)
(> n 0)
(rounding-mode-p mode)
(not (= (abs (rnd x mode n))
(expt 2 (1+ (expo x))))))
(= (expo (rnd x mode n))
(expo x)))
:rule-classes ()
:hints (("goal" :in-theory (enable rounding-mode-p
ieee-mode-p near near+ rnd minf inf)
:use (expo-trunc expo-away))))
;better rule-classes?
(defthm rnd-pos
(implies (and (< 0 x)
(rationalp x)
(integerp n)
(> n 0)
(rounding-mode-p mode))
(> (rnd x mode n) 0))
:rule-classes (:type-prescription)
:hints (("goal" :in-theory (enable rounding-mode-p ieee-mode-p near rnd inf minf)
:use ())))
(defthm rnd-0
(equal (rnd 0 mode n)
0)
:hints (("Goal" :in-theory (enable rnd rounding-mode-p ieee-mode-p inf minf)
:use (trunc-0 away-0))))
;better rule-classes?
(defthm rnd-neg
(implies (and (< x 0)
(rationalp x)
(integerp n)
(> n 0)
(rounding-mode-p mode))
(< (rnd x mode n) 0))
:rule-classes (:type-prescription)
:hints (("Goal" :in-theory (enable rnd rounding-mode-p ieee-mode-p inf minf)
:use (
near-neg))))
;would like to not open minf, inf here?
(defthm rnd-non-pos
(implies (<= x 0)
(<= (rnd x mode n) 0))
:hints (("goal" :in-theory (enable rnd near+ inf minf)))
:rule-classes (:rewrite :type-prescription :linear))
;would like to not open minf, inf here?
;add to lib?
(defthm rnd-non-neg
(implies (<= 0 x)
(<= 0 (rnd x mode n)))
:hints (("goal" :in-theory (enable rnd near+ inf minf)))
:rule-classes (:rewrite :type-prescription :linear))
(defthm sgn-rnd
(implies (and; (rationalp x)
(rounding-mode-p mode)
(integerp n)
(> n 0)
)
(equal (sgn (rnd x mode n))
(sgn x)))
:hints (("Goal" :in-theory (enable ieee-mode-p rounding-mode-p rnd near+ inf minf)
:use (sgn-trunc
sgn-away
sgn-near-2))))
;enable?
(defthmd rnd-exactp-a
(implies (and (rationalp x)
(rounding-mode-p mode)
(integerp n)
(> n 0))
(equal (equal x (rnd x mode n))
(exactp x n)))
:hints (("Goal" :in-theory (enable ieee-mode-p rounding-mode-p rnd near+ minf inf)
:use (near-exactp-a
trunc-exactp-a
away-exactp-a))))
(defthm rnd-exactp-b
(implies (< 0 n)
(exactp (rnd x mode n) n))
:hints (("Goal" :in-theory (enable rnd near+ minf inf))))
(defthm rnd-exactp-c
(implies (and (rationalp x)
(> x 0)
(rounding-mode-p mode)
(integerp n)
(> n 0)
(rationalp a)
(exactp a n)
(>= a x))
(>= a (rnd x mode n)))
:hints (("Goal" :in-theory (enable ieee-mode-p rnd minf inf)
:use (near-exactp-c
away-exactp-c
trunc-upper-pos))))
(defthm rnd-exactp-d
(implies (and (rationalp x)
(> x 0)
(rounding-mode-p mode)
(integerp n)
(> n 0)
(rationalp a)
(exactp a n)
(<= a x))
(<= a (rnd x mode n)))
:hints (("Goal" :in-theory (enable ieee-mode-p rounding-mode-p rnd minf inf)
:use (near-exactp-c
trunc-exactp-c
away-lower-pos))))
(defthm rnd<=away
(implies (and (rationalp x)
(> x 0)
(rounding-mode-p mode)
(integerp n)
(> n 0))
(<= (rnd x mode n) (away x n)))
:rule-classes ()
:hints (("Goal" :in-theory (enable ieee-mode-p rnd minf inf)
:use (trunc-upper-pos
away-lower-pos
near-choice))))
(defthm rnd>=trunc
(implies (and (rationalp x)
(> x 0)
(rounding-mode-p mode)
(integerp n)
(> n 0))
(>= (rnd x mode n) (trunc x n)))
:rule-classes ()
:hints (("Goal" :in-theory (enable ieee-mode-p rounding-mode-p rnd minf inf)
:use (trunc-upper-pos
away-lower-pos
near-choice))))
(defthm rnd-monotone
(implies (and (<= x y)
(rationalp x)
(rationalp y)
(< 0 x)
(rounding-mode-p mode)
(integerp n)
(> n 0))
(<= (rnd x mode n) (rnd y mode n)))
:hints (("Goal" :in-theory (enable ieee-mode-p rnd minf inf)
:use (trunc-monotone
away-monotone
near-monotone))))
(defthm exactp-rnd
(implies (and (rationalp x)
(rounding-mode-p mode)
(integerp n)
(> n 0))
(exactp (rnd x mode n) n))
:hints (("Goal" :in-theory (enable ieee-mode-p rounding-mode-p rnd inf minf))))
(defthm rnd-choice
(implies (rounding-mode-p mode)
(or (equal (rnd x mode n) (rnd x 'away n))
(equal (rnd x mode n) (rnd x 'trunc n))))
:hints (("Goal" :in-theory (set-difference-theories
(enable rnd near+ inf minf near rounding-mode-p ieee-mode-p)
'(re evenp))))
:rule-classes nil)
(defthm ieee-mode-p-flip
(implies (ieee-mode-p m)
(ieee-mode-p (flip m)))
:hints (("Goal" :in-theory (enable ieee-mode-p flip))))
(defthm rounding-mode-p-flip
(implies (rounding-mode-p m)
(rounding-mode-p (flip m)))
:hints (("Goal" :in-theory (enable ieee-mode-p flip))))
(defthm expo-rnd-bnd
(implies (and (rationalp x)
(integerp n)
(> n 0)
(rounding-mode-p mode))
(>= (expo (rnd x mode n))
(expo x)))
:rule-classes ()
:hints (("Goal" :in-theory (disable expo-minus)
:use (expo-rnd
(:instance expo-minus (x (rnd x mode n)))))))
(defthm plus-inf
(implies (and (rationalp x)
(>= x 0)
(rationalp y)
(>= y 0)
(integerp k)
(exactp x (+ k (- (expo x) (expo y)))))
(= (+ x (inf y k))
(inf (+ x y) (+ k (- (expo (+ x y)) (expo y))))))
:rule-classes ()
:hints (("goal" :in-theory (enable inf)
:use plus-away)))
(defthm plus-minf
(implies (and (rationalp x)
(>= x 0)
(rationalp y)
(>= y 0)
(integerp k)
(exactp x (+ k (- (expo x) (expo y)))))
(= (+ x (minf y k))
(minf (+ x y) (+ k (- (expo (+ x y)) (expo y))))))
:rule-classes ()
:hints (("goal" :in-theory (enable minf)
:use plus-trunc)))
;make alt form too?
; add to lib?
(defthm plus-rnd
(implies (and (rationalp x)
(>= x 0)
(rationalp y)
(>= y 0)
(integerp k)
(exactp x (+ -1 k (- (expo x) (expo y))))
(rounding-mode-p mode))
(= (+ x (rnd y mode k))
(rnd (+ x y)
mode
(+ k (- (expo (+ x y)) (expo y))))))
:rule-classes nil
:hints (("Goal" :in-theory (enable rnd ieee-mode-p ROUNDING-MODE-P)
:use (plus-near
plus-near+
plus-away
plus-trunc
plus-minf
plus-inf
(:instance exactp-<= (m (+ -1 k (- (expo x) (expo y))))
(n (+ k (- (expo x) (expo y)))))))))
(defthm rnd-rarely-zero
(implies (and (rationalp x)
(integerp k)
(case-split (< 0 k))
(case-split (rounding-mode-p mode)))
(equal (equal (rnd x mode k) 0)
(equal x 0)
))
:hints (("Goal" :in-theory (enable rnd near+ minf inf near ROUNDING-MODE-P ieee-mode-p))))
;add to lib?
(defthm flip-flip
(equal (flip (flip mode))
mode)
:hints (("Goal" :in-theory (enable flip))))
;add to lib?
(defthm inf-lower-bound
(implies (and (rationalp x)
(integerp n))
(>= (inf x n) x))
:hints (("Goal" :in-theory (enable inf)
:use trunc-upper-bound))
:rule-classes (:rewrite :linear))
;add to lib?
(defthm minf-upper-bound
(implies (and (rationalp x)
(integerp n))
(<= (minf x n) x))
:hints (("Goal" :in-theory (set-difference-theories
(enable minf)
'(abs-away))
:use away-lower-bound))
:rule-classes (:rewrite :linear))
;add to lib?
(defthm rnd-diff
(implies (and (rationalp x)
(integerp n)
(> n 0)
(rounding-mode-p mode))
(< (abs (- x (rnd x mode n))) (expt 2 (- (1+ (expo x)) n))))
:hints (("Goal" :in-theory (enable rnd near near+ inf minf ieee-mode-p rounding-mode-p)
:use (trunc-diff away-diff))))
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