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;; Cuong Chau <cuong.chau@arm.com>
;; January 2021
;; Establish the relationship between the partial remainder and
;; (rs57 + rc57). From this relationship, prove the remainder bound invariant.
;; rmd(j) = 2^-55 * (si(rs57(j), 57) + si(rc57(j), 57)),
;; -d <= rmd(j) < d,
;; where j >= 1.
;; We handle the first SRT iteration in this book.
(in-package "RTL")
(include-book "algorithm")
(local (arith-5-for-rtl))
;; ======================================================================
;; Remainder approximation
(defundd approx (i)
(+ (si (bits (rs57 i) 56 54) 3)
(si (bits (rc57 i) 56 54) 3)))
(defthm fmtrem-bounds
(and (<= 1 (fmtrem))
(<= (fmtrem) 3))
:hints (("Goal" :in-theory (enable fmtrem bits)))
:rule-classes :linear)
(bvecthm bvecp57-rs-0
(bvecp (rs-0) 57)
:hints (("Goal" :in-theory (enable rs-0))))
(bvecthm bvecp57-rc-0
(bvecp (rc-0) 57)
:hints (("Goal" :in-theory (enable rc-0))))
(defthmd rs-0-rewrite
(implies (not (specialp))
(equal (rs-0)
(case (fmtrem)
(3 (+ *2^2* (* *2^55* (x))))
(2 (+ *2^31* (* *2^55* (x))))
(1 (+ *2^44* (* *2^55* (x)))))))
:hints (("Goal"
:use (natp-2^prec-1*x
(:instance bits-shift-up-1
(x (* *2^52* (x)))
(i 56)
(j 3)
(k 3))
(:instance bits-shift-up-1
(x (* *2^23* (x)))
(i 56)
(j 32)
(k 32))
(:instance bits-shift-up-1
(x (* *2^10* (x)))
(i 56)
(j 45)
(k 45))
(:instance bits-shift-up-2
(x (* *2^52* (x)))
(i -2)
(k 3))
(:instance bits-shift-up-2
(x (* *2^23* (x)))
(i -2)
(k 32))
(:instance bits-shift-up-2
(x (* *2^10* (x)))
(i -2)
(k 45)))
:in-theory (enable rs-0 fmtrem x57-rewrite f cat))))
(local
(defthm-nl aux-1
(implies (and (integerp x)
(natp m)
(integerp n)
(< x (expt 2 m))
(< y (expt 2 n)))
(< (+ (* (expt 2 n) x) y)
(expt 2 (+ m n))))
:rule-classes :linear))
(defthm rs-0-lower-bounds
(and (implies (and (not (specialp))
(equal (fmtrem) 3))
(<= (+ *2^2* *2^55*) (rs-0)))
(implies (and (not (specialp))
(equal (fmtrem) 2))
(<= (+ *2^31* *2^55*) (rs-0)))
(implies (and (not (specialp))
(equal (fmtrem) 1))
(<= (+ *2^44* *2^55*) (rs-0))))
:hints (("Goal" :in-theory (enable rs-0-rewrite)))
:rule-classes :linear)
(defthm rs-0-upper-bound
(implies (not (specialp))
(< (rs-0) *2^56*))
:hints (("Goal"
:use (natp-2^prec-1*x
(:instance aux-1
(x (* (expt 2 (1- (prec (f))))
(x)))
(y (expt 2 (- 55 (prec (f)))))
(m (prec (f)))
(n (- 56 (prec (f))))))
:in-theory (enable rs-0-rewrite fmtrem f)))
:rule-classes :linear)
(defthmd rc-0-rewrite
(implies (not (specialp))
(equal (rc-0)
(case (fmtrem)
(3 (+ (- *2^2*)
*2^57*
(- (* *2^55* (q 1) (d)))))
(2 (+ (- *2^31*)
*2^57*
(- (* *2^55* (q 1) (d)))))
(1 (+ (- *2^44*)
*2^57*
(- (* *2^55* (q 1) (d))))))))
:hints (("Goal"
:use (natp-2^prec-1*d
(:instance bits-plus-mult-1
(x (1- *2^2*))
(y (+ -1 *2^55* (- (* *2^53* (d)))))
(k 2)
(n 56)
(m 2))
(:instance bits-plus-mult-1
(x (1- *2^31*))
(y (+ -1 *2^26* (- (* *2^24* (d)))))
(k 31)
(n 56)
(m 31))
(:instance bits-plus-mult-1
(x (1- *2^44*))
(y (+ -1 *2^13* (- (* *2^11* (d)))))
(k 44)
(n 56)
(m 44)))
:in-theory (enable rc-0
fmtrem
q
d57-rewrite
f
bits-lognot
cat
bvecp))))
(defthm rc-0-lower-bound
(implies (not (specialp))
(< (expt 2 56) (rc-0)))
:hints (("Goal"
:use (natp-2^prec-1*d
(:instance aux-1
(x (* (expt 2 (1- (prec (f))))
(d)))
(y (expt 2 (- 55 (prec (f)))))
(m (prec (f)))
(n (- 56 (prec (f))))))
:in-theory (enable rc-0-rewrite fmtrem f q)))
:rule-classes :linear)
(defthm rc-0-upper-bounds
(and (implies (and (not (specialp))
(equal (fmtrem) 3))
(<= (rc-0) (+ (- *2^2*) *2^57* (- *2^55*))))
(implies (and (not (specialp))
(equal (fmtrem) 2))
(<= (rc-0) (+ (- *2^31*) *2^57* (- *2^55*))))
(implies (and (not (specialp))
(equal (fmtrem) 1))
(<= (rc-0) (+ (- *2^44*) *2^57* (- *2^55*)))))
:hints (("Goal" :in-theory (enable rc-0-rewrite q)))
:rule-classes :linear)
(defthm natp-2^55*x
(implies (not (specialp))
(natp (* *2^55* (x))))
:hints (("Goal"
:use siga-rewrite
:in-theory (enable x)))
:rule-classes :type-prescription)
(defthm integerp-2^55*-d
(implies (not (specialp))
(integerp (* (- *2^55*) (d))))
:hints (("Goal"
:use sigb-rewrite
:in-theory (enable d)))
:rule-classes :type-prescription)
(defthmd rmd1-rewrite
(implies (not (specialp))
(equal (rmd 1)
(* (/ *2^55*)
(+ (si (rs57 1) 57)
(si (rc57 1) 57)))))
:hints (("Goal"
:use (rs-0-upper-bound
rc-0-lower-bound)
:in-theory (enable rmd-recurrence
rmd0-rewrite
rs57
rc57
rs-0-rewrite
rc-0-rewrite
q
si
bvecp))))
(defthm rmd1-bounds
(implies (not (specialp))
(and (< (- (d)) (rmd 1))
(< (rmd 1) (d))))
:hints (("Goal" :in-theory (enable rmd-recurrence rmd0-rewrite q)))
:rule-classes :linear)
(defthmd bits-siga-0
(implies (equal n (- 52 (prec (f))))
(equal (bits (siga) n 0)
0))
:hints (("Goal"
:use ((:instance bits-shift-up-1
(x (mana))
(i (- 52 (prec (f))))
(j 0)
(k (- 53 (prec (f)))))
(:instance bits-shift-up-1
(x (mana))
(i (- 52 (prec (f))))
(j 0)
(k (+ 53
(bits (clz53 (* (expt 2 (- 53 (prec (f))))
(mana)))
5 0)
(- (prec (f)))))))
:in-theory (enable f
siga
cat-for-gl
bits-logior
bvecp
normalize))))
(defthmd bits-sigb-0
(implies (equal n (- 52 (prec (f))))
(equal (bits (sigb) n 0)
0))
:hints (("Goal"
:use ((:instance bits-shift-up-1
(x (manb))
(i (- 52 (prec (f))))
(j 0)
(k (- 53 (prec (f)))))
(:instance bits-shift-up-1
(x (manb))
(i (- 52 (prec (f))))
(j 0)
(k (+ 53
(bits (clz53 (* (expt 2 (- 53 (prec (f))))
(manb)))
5 0)
(- (prec (f)))))))
:in-theory (enable f
sigb
cat-for-gl
bits-logior
bvecp
normalize))))
(defthmd bits-x57-0
(implies (equal n (- 55 (prec (f))))
(equal (bits (x57) n 0)
0))
:hints (("Goal" :in-theory (enable x57 f bits-siga-0))))
(defthmd bits-d57-0
(implies (equal n (- 55 (prec (f))))
(equal (bits (d57) n 0)
0))
:hints (("Goal" :in-theory (enable d57 f bits-sigb-0))))
(defthmd bits-rs-0-0
(implies (equal n (case (fmtrem)
(3 1)
(2 30)
(1 43)))
(equal (bits (rs-0) n 0)
0))
:hints (("Goal"
:use ((:instance bits-x57-0
(n (- 55 (prec (f)))))
(:instance bits-plus-bits
(x (x57))
(m 0)
(n (- 55 (prec (f))))
(p 2))
(:instance bitn-plus-bits
(x (x57))
(m 0)
(n (- 55 (prec (f))))))
:in-theory (enable rs-0 fmtrem f))))
(defthmd bits-rc-0-0
(implies (equal n (case (fmtrem)
(3 1)
(2 30)
(1 43)))
(equal (bits (rc-0) n 0)
0))
:hints (("Goal" :in-theory (enable rc-0))))
|
