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|
;; Cuong Chau <cuong.chau@arm.com>
;; January 2021
;; This book analyzes function NORMALIZE.
(in-package "RTL")
(include-book "special")
(local (arith-5-for-rtl))
(local
(in-theory (disable acl2::reduce-multiplicative-constant-<
acl2::integerp-<-constant
acl2::|(< x (/ y)) with (< 0 y)|
neg-bitn-0
acl2::simplify-products-gather-exponents-<)))
;; ======================================================================
(defundd a () (decode (opaw) (f)))
(defundd b () (decode (opbw) (f)))
(defthm nonzero-a
(implies (not (specialp))
(not (equal (a) 0)))
:hints (("Goal" :in-theory (enable specialp
a
opaw
fmtw
decode
ddecode
ndecode
expf
manf
f
classa
analyze))))
(defthm nonzero-b
(implies (not (specialp))
(not (equal (b) 0)))
:hints (("Goal" :in-theory (enable specialp
b
opbw
fmtw
decode
ddecode
ndecode
expf
manf
f
classb
analyze))))
(bvecthm bvecp-expa
(bvecp (expa) (expw (f)))
:hints (("Goal" :in-theory (enable expa f analyze))))
(defthm expa-upper-bounds
(and (implies (equal (fnum) 1)
(< (expa) *2^5*))
(implies (equal (fnum) 2)
(< (expa) *2^8*))
(implies (equal (fnum) 3)
(< (expa) *2^11*)))
:hints (("Goal"
:use bvecp-expa
:in-theory (enable f)))
:rule-classes :linear)
(bvecthm bvecp-expb
(bvecp (expb) (expw (f)))
:hints (("Goal" :in-theory (enable expb f analyze))))
(defthm expb-upper-bounds
(and (implies (equal (fnum) 1)
(< (expb) *2^5*))
(implies (equal (fnum) 2)
(< (expb) *2^8*))
(implies (equal (fnum) 3)
(< (expb) *2^11*)))
:hints (("Goal"
:use bvecp-expb
:in-theory (enable f)))
:rule-classes :linear)
(bvecthm bvecp-mana
(bvecp (mana) (sigw (f)))
:hints (("Goal" :in-theory (enable mana f analyze))))
(defthm mana-upper-bounds
(and (implies (equal (fnum) 1)
(< (mana) *2^10*))
(implies (equal (fnum) 2)
(< (mana) *2^23*))
(implies (equal (fnum) 3)
(< (mana) *2^52*)))
:hints (("Goal"
:use bvecp-mana
:in-theory (enable f)))
:rule-classes :linear)
(bvecthm bvecp-manb
(bvecp (manb) (sigw (f)))
:hints (("Goal" :in-theory (enable manb f analyze))))
(defthm manb-upper-bounds
(and (implies (equal (fnum) 1)
(< (manb) *2^10*))
(implies (equal (fnum) 2)
(< (manb) *2^23*))
(implies (equal (fnum) 3)
(< (manb) *2^52*)))
:hints (("Goal"
:use bvecp-manb
:in-theory (enable f)))
:rule-classes :linear)
(bvecthm bvecp53-siga
(bvecp (siga) 53)
:hints (("Goal" :in-theory (enable siga normalize))))
(bvecthm bvecp53-sigb
(bvecp (sigb) 53)
:hints (("Goal" :in-theory (enable sigb normalize))))
(defthm natp-expashft
(natp (expashft))
:hints (("Goal" :in-theory (enable expashft)))
:rule-classes :type-prescription)
(defthm natp-expbshft
(natp (expbshft))
:hints (("Goal" :in-theory (enable expbshft)))
:rule-classes :type-prescription)
(bvecthm bvecp13-expq-1
(bvecp (expq-1) 13)
:hints (("Goal" :in-theory (enable expq-1 normalize))))
(defthm bs-rewrite
(equal (bs) (bias (f)))
:hints (("Goal" :in-theory (enable bs f))))
(defthm natp-bias
(natp (bias (f)))
:hints (("Goal" :in-theory (enable f)))
:rule-classes :type-prescription)
(local
(def-gl-rule clz53-expo
:hyp (and (bvecp x 53) (not (equal x 0)))
:concl (equal (clz53 x) (- 52 (expo x)))
:disabledp t
:g-bindings (gl::auto-bindings
(:nat x 53))))
(local
(defthm aux-1
(implies (and (rationalp x)
(not (equal x 0)))
(and (equal (expo (* *2^29* x))
(+ 29 (expo x)))
(equal (expo (* *2^42* x))
(+ 42 (expo x)))))
:hints (("Goal"
:use ((:instance expo-shift
(n 29))
(:instance expo-shift
(n 42)))
:in-theory (disable expo-shift)))))
(local
(defthm aux-2-hp
(implies (and (< (abs x) *2^10*)
(rationalp x))
(<= (expo x) 10))
:hints (("Goal" :use (:instance expo-monotone
(y *2^10*))))
:rule-classes :linear))
(local
(defthm aux-2-sp
(implies (and (< (abs x) *2^23*)
(rationalp x))
(<= (expo x) 23))
:hints (("Goal" :use (:instance expo-monotone
(y *2^23*))))
:rule-classes :linear))
(local
(defthm aux-2-dp
(implies (and (< (abs x) *2^52*)
(rationalp x))
(<= (expo x) 52))
:hints (("Goal" :use (:instance expo-monotone
(y *2^52*))))
:rule-classes :linear))
(local
(defthm-nl aux-3
(implies (bvecp x 52)
(equal (bits (* x (expt 2 (+ 52 (- (expo x)))))
52 0)
(* x (expt 2 (+ 52 (- (expo x)))))))
:hints (("Goal"
:use (aux-2-dp
expo-upper-bound)
:in-theory (enable bvecp)))))
(defthmd spec-fields
(and (equal (signa) (sgnf (opaw) (f)))
(equal (expa) (expf (opaw) (f)))
(equal (mana) (manf (opaw) (f)))
(equal (signb) (sgnf (opbw) (f)))
(equal (expb) (expf (opbw) (f)))
(equal (manb) (manf (opbw) (f))))
:hints (("Goal"
:in-theory (enable sgnf
expf
manf
opaw
opbw
fmtw
f
signa
signb
expa
expb
mana
manb
si
bitn-bits
analyze))))
;; a
(local
(defthmd abs-a-rewrite
(implies (not (specialp))
(equal (abs (a))
(* (expt 2 (+ (si (expashft) 13) (- (bias (f))) -52))
(siga))))
:hints (("Goal"
:in-theory (e/d (specialp
a
expashft
siga
siga-1
f
opaw
fmtw
expa
expf
mana
manf
classa
decode
ddecode
ndecode
clz53-expo
cat
si
bvecp
analyze
normalize)
(acl2::|(expt x (if a b c))|
acl2::default-expt-2
acl2::default-plus-1
acl2::default-plus-2
acl2::default-times-1
acl2::default-times-2))))))
(local
(defthmd si-expashft-aux-1
(implies (and (not (specialp))
(equal x (opaw))
(equal f (f))
(equal (expf x f) 0))
(denormp x f))
:hints (("Goal" :in-theory (enable specialp
denormp
opaw
fmtw
f
classa
expf
manf
encodingp
analyze)))))
(local
(defthmd si-expashft-aux-2
(implies (and (not (specialp))
(equal x (opaw))
(equal f (f))
(not (equal (expf x f) 0)))
(normp x f))
:hints (("Goal" :in-theory (enable specialp
normp
opaw
fmtw
f
classa
expf
manf
encodingp
analyze)))))
(local
(defthmd si-expashft
(implies (not (specialp))
(equal (si (expashft) 13)
(+ (expo (a)) (bias (f)))))
:hints (("Goal"
:in-theory (enable specialp
expashft
a
si
f
opaw
fmtw
expa
mana
classa
siga-1
decode
expf
manf
clz53-expo
si-expashft-aux-1
si-expashft-aux-2
cat
bvecp
analyze
normalize)))))
(defthmd siga-rewrite
(implies (not (specialp))
(equal (siga) (* *2^52* (sig (a)))))
:hints (("Goal"
:use abs-a-rewrite
:in-theory (e/d (si-expashft
fp-abs-rewrite)
(abs)))))
;; b
(local
(defthmd abs-b-rewrite
(implies (not (specialp))
(equal (abs (b))
(* (expt 2 (+ (si (expbshft) 13) (- (bias (f))) -52))
(sigb))))
:hints (("Goal"
:in-theory (e/d (specialp
b
expbshft
sigb
sigb-1
f
opbw
fmtw
expb
expf
manb
manf
classb
decode
ddecode
ndecode
clz53-expo
cat
si
bvecp
analyze
normalize)
(acl2::|(expt x (if a b c))|
acl2::default-expt-2
acl2::default-plus-1
acl2::default-plus-2
acl2::default-times-1
acl2::default-times-2))))))
(local
(defthmd si-expbshft-aux-1
(implies (and (not (specialp))
(equal x (opbw))
(equal f (f))
(equal (expf x f) 0))
(denormp x f))
:hints (("Goal" :in-theory (enable specialp
denormp
opbw
fmtw
f
classb
expf
manf
encodingp
analyze)))))
(local
(defthmd si-expbshft-aux-2
(implies (and (not (specialp))
(equal x (opbw))
(equal f (f))
(not (equal (expf x f) 0)))
(normp x f))
:hints (("Goal" :in-theory (enable specialp
normp
opbw
fmtw
f
classb
expf
manf
encodingp
analyze)))))
(local
(defthmd si-expbshft
(implies (not (specialp))
(equal (si (expbshft) 13)
(+ (expo (b)) (bias (f)))))
:hints (("Goal"
:in-theory (enable specialp
expbshft
b
si
f
opbw
fmtw
expb
manb
classb
sigb-1
decode
expf
manf
clz53-expo
si-expbshft-aux-1
si-expbshft-aux-2
cat
bvecp
analyze
normalize)))))
(defthmd sigb-rewrite
(implies (not (specialp))
(equal (sigb) (* *2^52* (sig (b)))))
:hints (("Goal"
:use abs-b-rewrite
:in-theory (e/d (si-expbshft
fp-abs-rewrite)
(abs)))))
;; |a/b|
(local
(defthmd si-expq-1-rewrite
(equal (si (expq-1) 13)
(+ (si (expashft) 13)
(- (si (expbshft) 13))
(bias (f))))
:hints (("Goal" :in-theory (enable expq-1
expashft
expbshft
f
si
siga-1
sigb-1
normalize
bvecp)))))
(local
(defthm si-expashft-bounds
(implies (not (specialp))
(and (<= -51 (si (expashft) 13))
(< (si (expashft) 13) *2^11*)))
:hints (("Goal" :in-theory (e/d (specialp
expashft
si
siga-1
expa
mana
classa
cat
bvecp
clz53-expo
analyze)
(acl2::default-less-than-1))))
:rule-classes :linear))
(local
(defthm si-expbshft-bounds
(implies (not (specialp))
(and (<= -51 (si (expbshft) 13))
(< (si (expbshft) 13) *2^11*)))
:hints (("Goal" :in-theory (e/d (specialp
expbshft
si
sigb-1
expb
manb
classb
cat
bvecp
clz53-expo
analyze)
(acl2::default-less-than-1))))
:rule-classes :linear))
(defthm si-expq-1-bounds
(implies (not (specialp))
(and (<= -2083 (si (expq-1) 13))
(<= (si (expq-1) 13) 3121)))
:hints (("Goal" :in-theory (enable si-expq-1-rewrite f)))
:rule-classes :linear)
(local
(defthm aux-4
(implies (and (rationalp x)
(rationalp y))
(equal (abs (/ x y))
(/ (abs x) (abs y))))))
(defthmd quotient-formula
(implies (not (specialp))
(equal (abs (/ (a) (b)))
(* (expt 2 (- (si (expq-1) 13) (bias (f))))
(/ (siga) (sigb)))))
:hints (("Goal"
:in-theory (e/d (abs-a-rewrite
abs-b-rewrite
si-expq-1-rewrite)
(abs)))))
|
