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;; Cuong Chau <cuong.chau@arm.com>
;; March 2021
(in-package "RTL")
(include-book "normalize")
(local (arith-5-for-rtl))
;; ======================================================================
;; 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)))
(defthm x-bounds
(and (implies (not (specialp))
(<= 1 (x)))
(< (x) 2))
:hints (("Goal" :in-theory (enable x)))
:rule-classes :linear)
(defthm d-bounds
(and (implies (not (specialp))
(<= 1 (d)))
(< (d) 2))
:hints (("Goal" :in-theory (enable d)))
:rule-classes :linear)
(defthm natp-2^prec-1*x
(implies (not (specialp))
(natp (* (expt 2 (1- (prec (f))))
(x))))
:hints (("Goal"
:use ((:instance exactp-sig
(x (bits (opa) 9 0))
(n 10))
(:instance exactp-sig
(x (bits (opa) 22 0))
(n 23))
(:instance exactp-sig
(x (bits (opa) 51 0))
(n 52)))
:in-theory (e/d (specialp
a-class
f
x
a
opaz
opaw
fmtw
snanp
qnanp
nanp
infp
zerp
decode
sig-ndecode
normp
denormp
encodingp
pseudop
unsupp
exactp2
manf)
(exactp-sig))))
:rule-classes :type-prescription)
(defthm natp-2^prec-1*d
(implies (not (specialp))
(natp (* (expt 2 (1- (prec (f))))
(d))))
:hints (("Goal"
:use ((:instance exactp-sig
(x (bits (opb) 9 0))
(n 10))
(:instance exactp-sig
(x (bits (opb) 22 0))
(n 23))
(:instance exactp-sig
(x (bits (opb) 51 0))
(n 52)))
:in-theory (e/d (specialp
b-class
f
d
b
opbz
opbw
fmtw
snanp
qnanp
nanp
infp
zerp
decode
sig-ndecode
normp
denormp
encodingp
pseudop
unsupp
exactp2
manf)
(exactp-sig))))
:rule-classes :type-prescription)
(defthmd x57-rewrite
(implies (not (specialp))
(equal (x57)
(* *2^55* (x))))
:hints (("Goal"
:use siga-rewrite
:in-theory (enable x57 x cat bvecp))))
(defthm x57-bounds
(and (implies (not (specialp))
(<= *2^55* (x57)))
(< (x57) *2^56*))
:hints (("Goal"
:use x57-rewrite
:in-theory (enable x57 cat)))
:rule-classes :linear)
(bvecthm bvecp56-x57
(bvecp (x57) 56)
:hints (("Goal" :in-theory (enable bvecp))))
(defthmd d57-rewrite
(implies (not (specialp))
(equal (d57) (* *2^55* (d))))
:hints (("Goal"
:use sigb-rewrite
:in-theory (enable d57 d cat bvecp))))
(defthm d57-bounds
(and (implies (not (specialp))
(<= *2^55* (d57)))
(< (d57) *2^56*))
:hints (("Goal"
:use d57-rewrite
:in-theory (enable d57 cat)))
:rule-classes :linear)
(bvecthm bvecp56-d57
(bvecp (d57) 56)
:hints (("Goal" :in-theory (enable bvecp))))
;; Partial quotients
(defundd quotient (i)
(if (zp i)
0
(+ (quotient (1- i))
(* (expt 2 (- 1 i)) (q i)))))
(local (in-theory (enable quotient)))
;; Partial remainders
(defundd rmd (i)
(* (expt 2 (1- i))
(- (x) (* (d) (quotient i)))))
(defthmd rmd0-rewrite
(equal (rmd 0) (/ (x) 2))
:hints (("Goal" :in-theory (enable rmd))))
(defthm rmd0-bound
(implies (not (specialp))
(< (abs (rmd 0)) (d)))
:hints (("Goal" :in-theory (enable rmd0-rewrite)))
:rule-classes :linear)
(defthmd rmd-recurrence
(implies (posp j)
(equal (rmd j)
(- (* 2 (rmd (1- j)))
(* (q j) (d)))))
:hints (("Goal" :in-theory (enable rmd))))
(defthm q-bounds
(and (<= -1 (q j))
(<= (q j) 1))
:hints (("Goal"
:expand (q j)
:in-theory (enable nextdigit)))
:rule-classes :linear)
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