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|
;; Cuong Chau <cuong.chau@arm.com>
;; May 2021
(in-package "RTL")
(include-book "final")
;; ======================================================================
;; We impose the following constraints on the inputs of fdivlane:
(defund input-constraints (opa opb fnum vec rin)
(and (bvecp opa 64)
(bvecp opb 64)
(member fnum '(1 2 3))
(bitp vec)
(natp rin)))
;; Our ultimate objective is the following theorem:
;; (defthm fdivlane-correct
;; (implies (input-constraints opa opb fnum vec rin)
;; (let* ((f (case fnum (1 (hp)) (2 (sp)) (3 (dp))))
;; (fmtw (+ 1 (expw f) (sigw f)))
;; (fzp (bitn rin 24))
;; (dnp (bitn rin 25))
;; (rmode (bits rin 23 22)))
;; (mv-let (data flags) (fdivlane opa opb fnum vec fzp dnp rmode)
;; (mv-let
;; (data-spec r-spec)
;; (arm-binary-spec 'div
;; (bits opa (1- fmtw) 0)
;; (bits opb (1- fmtw) 0)
;; rin
;; f)
;; (and (equal data data-spec)
;; (equal (logior rin flags) r-spec)))))))
;; In order to address the lack of modularity of the ACL2 code, we take the
;; following approach.
;; First, we introduce constrained constants representing the inputs:
(encapsulate
(((opa) => *)
((opb) => *)
((fnum) => *)
((vec) => *)
((rin) => *))
(local (defun opa () 0))
(local (defun opb () 0))
(local (defun fnum () 1))
(local (defun vec () 0))
(local (defun rin () 0))
(defthm input-constraints-lemma
(input-constraints (opa) (opb) (fnum) (vec) (rin))))
(defundd fzp () (bitn (rin) 24))
(defundd dnp () (bitn (rin) 25))
(defundd rmode () (bits (rin) 23 22))
(defundd f ()
(case (fnum)
(1 (hp))
(2 (sp))
(3 (dp))))
;; In terms of these constants, we shall define constants corresponding to the
;; local variables of fdivlane, culminating in the constant (result)
;; corresponding to the outputs. This can be done mechanically by applying the
;; CONST-FNS-GEN utility. See prelim.lisp for the details.
;; The constant definitions will be derived from that of fdivlane in such a way
;; that the proof of the following will be trivial:
(defthmd fdivlane-lemma
(equal (result)
(fdivlane (opa) (opb) (fnum) (vec) (fzp) (dnp) (rmode))))
(defundd data ()
(mv-nth 0 (mv-list 2 (result))))
(defundd flags ()
(mv-nth 1 (mv-list 2 (result))))
;; The real work will be the proof of the following theorem:
;; (defthm fdivlane-main
;; (let ((fmtw (+ 1 (expw (f)) (sigw (f)))))
;; (mv-let
;; (data-spec r-spec)
;; (arm-binary-spec 'div
;; (bits (opa) (1- fmtw) 0)
;; (bits (opb) (1- fmtw) 0)
;; (rin)
;; (f))
;; (and (equal (data) data-spec)
;; (equal (logior (rin) (flags)) r-spec)))))
;; The following is an immediate consequence of the four preceding events:
;; (defthm fdivlane-main-inst
;; (let* ((f (case fnum (1 (hp)) (2 (sp)) (3 (dp))))
;; (fmtw (+ 1 (expw f) (sigw f)))
;; (fzp (bitn (rin) 24))
;; (dnp (bitn (rin) 25))
;; (rmode (bits (rin) 23 22)))
;; (mv-let (data flags) (fdivlane (opa) (opb) (fnum) (vec) fzp dnp rmode)
;; (mv-let
;; (data-spec r-spec)
;; (arm-binary-spec 'div
;; (bits (opa) (1- fmtw) 0)
;; (bits (opb) (1- fmtw) 0)
;; (rin)
;; f)
;; (and (equal data data-spec)
;; (equal (logior (rin) flags) r-spec))))))
;; The desired theorem can then be derived by functional instantiation.
;; ======================================================================
;; Prove correctness for special inputs
(defundd specialp ()
(or (member (classa) '(0 1 2 3))
(member (classb) '(0 1 2 3))))
(defundd fmtw () (+ 1 (expw (f)) (sigw (f))))
(defundd opaw () (bits (opa) (1- (fmtw)) 0))
(defundd opbw () (bits (opb) (1- (fmtw)) 0))
(defundd opaz ()
(if (and (= (fzp) 1)
(denormp (opaw) (f)))
(zencode (sgnf (opaw) (f)) (f))
(opaw)))
(defundd opbz ()
(if (and (= (fzp) 1)
(denormp (opbw) (f)))
(zencode (sgnf (opbw) (f)) (f))
(opbw)))
(defthmd fdivlane-special-correct
(mv-let
(data-spec r-spec)
(arm-binary-spec 'div (opaw) (opbw) (rin) (f))
(implies (specialp)
(and (equal (data) data-spec)
(equal (logior (rin) (flags)) r-spec)))))
;; ======================================================================
;; Analyze function NORMALIZE
(defundd a () (decode (opaw) (f)))
(defundd b () (decode (opbw) (f)))
(defthmd quotient-formula
(implies (not (specialp))
(equal (abs (/ (a) (b)))
(* (expt 2 (- (si (expq-1) 13) (bias (f))))
(/ (siga) (sigb))))))
;; Define the normalized dividend x and normalized divisor d satisfying
;; 1 <= x < 2 and 1 <= d < 2.
(defundd x () (sig (a)))
(defundd d () (sig (b)))
(defthmd x57-rewrite
(implies (not (specialp))
(equal (x57)
(* *2^55* (x)))))
(defthmd d57-rewrite
(implies (not (specialp))
(equal (d57) (* *2^55* (d)))))
;; Partial quotients
(defundd quotient (i)
(if (zp i)
0
(+ (quotient (1- i))
(* (expt 2 (- 1 i)) (q i)))))
;; Partial remainders
(defundd rmd (i)
(* (expt 2 (1- i))
(- (x) (* (d) (quotient i)))))
(defthmd rmd-recurrence
(implies (posp j)
(equal (rmd j)
(- (* 2 (rmd (1- j)))
(* (q j) (d))))))
;; ======================================================================
;; 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.
;; Remainder approximation
(defundd approx (i)
(+ (si (bits (rs57 i) 56 54) 3)
(si (bits (rc57 i) 56 54) 3)))
;; The first SRT iteration
(defthmd rmd1-rewrite
(implies (not (specialp))
(equal (rmd 1)
(* (/ *2^55*)
(+ (si (rs57 1) 57)
(si (rc57 1) 57))))))
(defthm rmd1-bounds
(implies (not (specialp))
(and (< (- (d)) (rmd 1))
(< (rmd 1) (d))))
:rule-classes :linear)
(defthmd bits-rs-0-0
(implies (equal n (case (fmtrem)
(3 1)
(2 30)
(1 43)))
(equal (bits (rs-0) n 0)
0)))
(defthmd bits-rc-0-0
(implies (equal n (case (fmtrem)
(3 1)
(2 30)
(1 43)))
(equal (bits (rc-0) n 0)
0)))
;; The iterative phase
(defthmd bits-rs57&rc57-0
(implies (and (equal n (- 53 (prec (f))))
(not (equal (fmtrem) 3)))
(and (equal (bits (rs57 j) n 0) 0)
(equal (bits (rc57 j) n 0) 0))))
(defthmd rmd-rewrite
(implies (and (not (specialp))
(posp j))
(equal (rmd j)
(* (/ *2^55*)
(+ (si (rs57 j) 57) (si (rc57 j) 57))))))
(defthm rmd-bounds
(implies (and (not (specialp))
(posp j))
(and (<= (- (d)) (rmd j))
(< (rmd j) (d))))
:rule-classes :linear)
;; ======================================================================
;; Prove quot(j) = 2^54 * quotient(j),
;; and quotm1(j) = quot(j) - 2^(55 - j)
(defthmd quot"m1-rewrite
(implies (and (posp j)
(<= j (n)))
(and (equal (quot j)
(* *2^54* (quotient j)))
(equal (quotm1 j)
(- (quot j) (expt 2 (- 55 j)))))))
;; Connect qtrunc and stk with x/d and p
(defthmd quotf-to-x/d
(implies (not (specialp))
(equal (quotf)
(- (* *2^54*
(/ (x) (d)))
(* (expt 2 (- 55 (n)))
(/ (rmd (n)) (d)))))))
(defthmd quotm1f-to-x/d
(implies (not (specialp))
(equal (quotm1f)
(- (* *2^54*
(/ (x) (d)))
(* (expt 2 (- 55 (n)))
(1+ (/ (rmd (n)) (d))))))))
(defthmd qtrunc-rewrite-gen
(implies (not (specialp))
(equal (qtrunc)
(cond ((<= (si (expq-1) 13) 1)
(fl (* (expt 2 (+ (prec (f))
(si (expq-1) 13)))
(/ (x) (d)))))
((< (x) (d))
(if (equal (fnum) 2)
(fl (* (expt 2 (+ 2 (prec (f))))
(/ (x) (d))))
(* 2 (fl (* (expt 2 (1+ (prec (f))))
(/ (x) (d)))))))
(t (fl (* (expt 2 (1+ (prec (f))))
(/ (x) (d)))))))))
(defthmd stk-rewrite-gen-1
(implies (and (<= (si (expq-1) 13) 1)
(not (specialp)))
(equal (stk)
(if (and (integerp (* (expt 2 (1- (n)))
(/ (x) (d))))
(< -54 (si (expq-1) 13))
(equal (bits (* *2^54*
(/ (x) (d)))
(- (si (expq-1) 13))
0)
0)
(equal (bits (* (expt 2 (+ 53 (si (expq-1) 13)))
(/ (x) (d)))
(- 52 (prec (f)))
0)
0)
(equal (bitn (* (expt 2 (+ (prec (f))
(si (expq-1) 13)))
(/ (x) (d)))
0)
0))
0 1))))
(defthmd stk-rewrite-gen-2
(implies (and (< (x) (d))
(> (si (expq-1) 13) 1)
(not (specialp)))
(equal (stk)
(if (and (integerp (* (expt 2 (1- (n)))
(/ (x) (d))))
(equal (bits (* *2^55*
(/ (x) (d)))
(- 52 (prec (f)))
0)
0)
(equal (bitn (* (expt 2 (+ 2 (prec (f))))
(/ (x) (d)))
0)
0))
0 1))))
(defthmd stk-rewrite-gen-3
(implies (and (<= (d) (x))
(>= (si (expq-1) 13) 1)
(not (specialp)))
(equal (stk)
(if (and (integerp (* (expt 2 (1- (n)))
(/ (x) (d))))
(equal (bits (* *2^54*
(/ (x) (d)))
(- 52 (prec (f)))
0)
0)
(equal (bitn (* (expt 2 (1+ (prec (f))))
(/ (x) (d)))
0)
0))
0 1))))
;; Rounding
(defund rmode-prime (mode sign)
(declare (xargs :guard (and (symbolp mode)
(bitp sign))))
(cond ((and (equal mode 'RUP)
(equal sign 1))
'RDN)
((and (equal mode 'RDN)
(equal sign 1))
'RUP)
(t mode)))
(defund mode (rmode)
(declare (xargs :guard (natp rmode)))
(case rmode
(0 'rne)
(1 'rup)
(2 'rdn)
(3 'rtz)))
;; Normal rounding
(defthmd rnd-abs-a/b-to-qrnd
(implies (and (<= 1 (si (expq) 13))
(not (specialp)))
(equal (rnd (abs (/ (a) (b)))
(rmode-prime (mode (rmode)) (sign))
(prec (f)))
(* (expt 2 (+ 1
(si (expq) 13)
(- (bias (f)))
(- (prec (f)))))
(+ (expt 2 (1- (prec (f))))
(bits (qrnd) (- (prec (f)) 2) 0))))))
(defthmd inx-exact-a/b-rel
(implies (and (<= 1 (si (expq) 13))
(not (specialp)))
(equal (inx)
(if (equal (rnd (abs (/ (a) (b)))
(rmode-prime (mode (rmode)) (sign))
(prec (f)))
(abs (/ (a) (b))))
0
1))))
;; Subnormal rounding
(defthmd drnd-abs-a/b-to-qrnd
(implies (and (not (specialp))
(< (abs (/ (a) (b)))
(spn (f))))
(equal (drnd (abs (/ (a) (b)))
(rmode-prime (mode (rmode)) (sign))
(f))
(* (expt 2 (+ 2 (- (bias (f))) (- (prec (f)))))
(bits (qrnd) (1- (prec (f))) 0)))))
(defthmd inx-exact-a/b-denormal-rel
(implies (and (not (specialp))
(< (abs (/ (a) (b)))
(spn (f))))
(equal (inx)
(if (equal (drnd (abs (/ (a) (b)))
(rmode-prime (mode (rmode)) (sign))
(f))
(abs (/ (a) (b))))
0
1))))
;; The main lemma
(defthmd fdivlane-main
(mv-let
(data-spec r-spec)
(arm-binary-spec 'div (opaw) (opbw) (rin) (f))
(and (equal (data) data-spec)
(equal (logior (rin) (flags)) r-spec))))
;; The final theorem
(defthmd fdivlane-correct
(implies (input-constraints opa opb fnum vec rin)
(let* ((f (case fnum (1 (hp)) (2 (sp)) (3 (dp))))
(fmtw (+ 1 (expw f) (sigw f)))
(fzp (bitn rin 24))
(dnp (bitn rin 25))
(rmode (bits rin 23 22)))
(mv-let (data flags) (fdivlane opa opb fnum vec fzp dnp rmode)
(mv-let
(data-spec r-spec)
(arm-binary-spec 'div
(bits opa (1- fmtw) 0)
(bits opb (1- fmtw) 0)
rin
f)
(and (equal data data-spec)
(equal (logior rin flags) r-spec)))))))
|
