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|
(in-package "RTL")
(include-book "quotient")
(local-defund lsb () (bitn (qtrunc) 1))
(local-defund grd () (bitn (qtrunc) 0))
(local-defund qrnd* ()
(if1 (logior1 (logior1 (logand1 (logand1 (log= (rmode) (rmodenear)) (grd))
(logior1 (lsb) (stk)))
(logand1 (logand1 (log= (rmode) (rmodeup))
(lognot1 0))
(logior1 (grd) (stk))))
(logand1 (logand1 (log= (rmode) (rmodedn)) 0)
(logior1 (grd) (stk))))
(bits (qinc) 53 1)
(bits (qtrunc) 53 1)))
(local-defund inx* () (logior1 (grd) (stk)))
(local (in-theory (disable (lsb) (grd) (qrnd*) (inx*))))
(local-defthmd rounder-lemma
(and (equal (qrnd) (qrnd*))
(equal (inx) (inx*)))
:hints (("Goal" :do-not '(preprocess) :expand :lambdas
:in-theory '(lsb grd qrnd* inx* qrnd inx rounder))))
(defund mode ()
(case (rmode)
(0 'rne)
(1 'rup)
(2 'rdn)
(3 'rtz)))
(in-theory (disable (mode)))
(local-defund z* () (* (expt 2 (1+ (p))) (qsqrt (x) (1+ (* 2 (n))))))
(local (in-theory (disable (z*))))
(defthmd expo-qsqrt-x
(implies (not (specialp))
(equal (expo (qsqrt (x) (+ 1 (* 2 (n)))))
-1))
:hints (("Goal" :in-theory (enable qsqrt-rto-sqrt n)
:use (fnum-vals x-bounds))))
(local-defthmd qrnd-1
(implies (not (specialp))
(equal (expo (z*)) (p)))
:hints (("Goal" :in-theory (enable z*)
:nonlinearp t
:use (p-vals expo-qsqrt-x
(:instance expo-shift (n (1+ (p))) (x (qsqrt (x) (1+ (* 2 (n))))))))))
(local-defthmd qrnd-2
(implies (not (specialp))
(and (rationalp (z*))
(> (z*) 0)))
:hints (("Goal" :in-theory (enable z* n)
:nonlinearp t
:use (p-vals x-bounds fnum-vals
(:instance qsqrt-pos (x (x)) (n (1+ (* 2 (n)))))))))
(local-defthmd qrnd-3
(implies (not (specialp))
(<= (expt 2 (p)) (z*)))
:hints (("Goal" :use (qrnd-1 qrnd-2 (:instance expo-lower-bound (x (z*)))))))
(defthmd qrnd-4
(implies (not (specialp))
(common-mode-p (mode)))
:hints (("Goal" :in-theory (enable rmode mode common-mode-p)
:use ((:instance bvecp-member (x (bits (rin) 23 22)) (n 2))))))
(local-defthmd qrnd-5
(implies (not (specialp))
(equal (bits (fl (z*)) (1- (p)) 0)
(bits (qtrunc) (1- (p)) 0)))
:hints (("Goal" :in-theory (enable bits z*)
:use (qtrunc-rewrite))))
(local-defthmd qrnd-6
(implies (not (specialp))
(equal (stk)
(if (integerp (z*)) 0 1)))
:hints (("Goal" :in-theory (enable z*)
:use (stk-rewrite))))
(local-defthmd qrnd-7
(implies (not (specialp))
(equal (expo (fl (z*))) (p)))
:hints (("Goal" :use (p-vals qrnd-1 qrnd-2
(:instance expo-lower-bound (x (z*)))
(:instance expo-upper-bound (x (z*)))
(:instance expo-unique (x (fl (z*))) (n (p)))))))
(local-defthmd qrnd-8
(implies (not (specialp))
(iff (exactp (z*) (p))
(and (= (stk) 0)
(= (bitn (fl (z*)) 0) 0))))
:hints (("Goal" :use (p-vals qrnd-2 qrnd-3 qrnd-4 qrnd-6 qrnd-7
(:instance roundup-pos-thm (mode (mode)) (z (z*)) (n (p)))))))
(local-defthmd qrnd-9
(implies (not (specialp))
(iff (exactp (qsqrt (x) (1+ (* 2 (n)))) (p))
(and (= (stk) 0)
(= (bitn (fl (z*)) 0) 0))))
:hints (("Goal" :use (p-vals qrnd-8
(:instance exactp-shift (k (1+ (p))) (n (p)) (x (qsqrt (x) (1+ (* 2 (n)))))))
:in-theory (enable z*))))
(defthmd inx-rewrite
(implies (not (specialp))
(equal (inx)
(if (exactp (qsqrt (x) (1+ (* 2 (n)))) (p))
0 1)))
:hints (("Goal" :use (p-vals qrnd-5 qrnd-9
(:instance bitn-bits (x (fl (z*))) (i (1- (p))) (j 0) (k 0))
(:instance bitn-bits (x (qtrunc)) (i (1- (p))) (j 0) (k 0)))
:in-theory (enable rounder-lemma inx* grd))))
(local-defthmd qrnd-10
(implies (not (specialp))
(equal (rnd (z*) (mode) (p))
(if (roundup-pos (fl (z*)) (p) (stk) (mode) (p))
(fp+ (rtz (fl (z*)) (p)) (p))
(rtz (fl (z*)) (p)))))
:hints (("Goal" :use (p-vals qrnd-2 qrnd-3 qrnd-4 qrnd-6 qrnd-7
(:instance roundup-pos-thm (mode (mode)) (z (z*)) (n (p)))))))
(in-theory (disable sgn-ddecode))
(local-defthmd qrnd-11
(implies (not (specialp))
(equal (qrnd)
(if (roundup-pos (fl (z*)) (p) (stk) (mode) (p))
(bits (qinc) 53 1)
(bits (qtrunc) 53 1))))
:hints (("Goal" :in-theory (enable stk-rewrite roundup-pos qrnd* rounder-lemma lsb grd mode
mode rmode rmodenear rmodedn rmodeup)
:use (qrnd-5 p-vals
(:instance bvecp-member (x (bits (rin) 23 22)) (n 2))
(:instance bitn-bits (x (fl (z*))) (i (1- (p))) (j 0) (k 0))
(:instance bitn-bits (x (qtrunc)) (i (1- (p))) (j 0) (k 0))
(:instance bitn-bits (x (fl (z*))) (i (1- (p))) (j 0) (k 1))
(:instance bitn-bits (x (qtrunc)) (i (1- (p))) (j 0) (k 1))))))
(local-defthmd qrnd-12
(implies (and (not (specialp)) (eql (p) k))
(equal (rtz (fl (z*)) k)
(* 2 (bits (fl (z*)) (p) 1))))
:hints (("Goal" :use (qrnd-2 qrnd-7 (:instance bits-rtz (x (fl (z*))) (n (1+ (p))) (k (p)))))))
(local-defthmd qrnd-13
(implies (not (specialp))
(equal (mod (rtz (fl (z*)) (p)) (expt 2 (p)))
(mod (* 2 (bits (fl (z*)) (1- (p)) 1)) (expt 2 (p)))))
:hints (("Goal" :in-theory (e/d (qrnd-12) (ACL2::|(equal (mod a n) (mod b n))|))
:use (p-vals
(:instance mod-mult (m (* 2 (bits (fl (z*)) (1- (p)) 1))) (a (bitn (fl (z*)) (p))) (n (expt 2 (p))))
(:instance bitn-plus-bits (x (fl (z*))) (n (p)) (m 1))))))
(local-defthmd qrnd-14
(implies (not (specialp))
(equal (mod (rtz (fl (z*)) (p)) (expt 2 (p)))
(mod (* 2 (bits (qtrunc) (1- (p)) 1)) (expt 2 (p)))))
:hints (("Goal" :use (qrnd-13 qrnd-5 p-vals
(:instance bits-bits (x (fl (z*))) (i (1- (p))) (j 0) (k (1- (p))) (l 1))
(:instance bits-bits (x (qtrunc)) (i (1- (p))) (j 0) (k (1- (p))) (l 1))))))
(local-defthmd qrnd-15
(implies (and (not (specialp))
(not (roundup-pos (fl (z*)) (p) (stk) (mode) (p))))
(equal (mod (rnd (z*) (mode) (p)) (expt 2 (p)))
(mod (* 2 (bits (qtrunc) (1- (p)) 1)) (expt 2 (p)))))
:hints (("Goal" :use (qrnd-14 qrnd-10))))
(local-defthmd qrnd-16
(implies (and (not (specialp))
(not (roundup-pos (fl (z*)) (p) (stk) (mode) (p))))
(equal (mod (rnd (z*) (mode) (p)) (expt 2 (p)))
(mod (* 2 (bits (qrnd) (- (p) 2) 0)) (expt 2 (p)))))
:hints (("Goal" :use (qrnd-15 qrnd-11 p-vals
(:instance bits-bits (x (qtrunc)) (i 53) (j 1) (k (- (p) 2)) (l 0))))))
(local-defthmd qrnd-17
(implies (not (specialp))
(equal (fp+ (rtz (fl (z*)) (p)) (p))
(+ (rtz (fl (z*)) (p)) 2)))
:hints (("Goal" :in-theory (enable fp+)
:use (p-vals qrnd-7))))
(local-defthmd qrnd-18
(implies (not (specialp))
(equal (fp+ (rtz (fl (z*)) (p)) (p))
(+ (* 2 (bits (fl (z*)) (p) 1)) 2)))
:hints (("Goal" :in-theory (enable qrnd-12) :use (qrnd-17 p-vals))))
(local-defthmd qrnd-19
(implies (not (specialp))
(equal (mod (+ (* 2 (bits (fl (z*)) (p) 1)) 2) (expt 2 (p)))
(mod (+ (- (bits (fl (z*)) (p) 0) (bitn (fl (z*)) 0)) 2) (expt 2 (p)))))
:hints (("Goal" :use (p-vals (:instance bits-plus-bitn (x (fl (z*))) (n (p)) (m 0))))))
(local-defthmd qrnd-20
(implies (not (specialp))
(equal (mod (+ (* 2 (bits (fl (z*)) (p) 1)) 2) (expt 2 (p)))
(mod (+ (- (bits (fl (z*)) (1- (p)) 0) (bitn (fl (z*)) 0)) 2) (expt 2 (p)))))
:hints (("Goal" :use (p-vals qrnd-19
(:instance bitn-plus-bits (x (fl (z*))) (n (p)) (m 0))
(:instance mod-mult (m (+ (- (bits (fl (z*)) (1- (p)) 0) (bitn (fl (z*)) 0)) 2))
(a (bitn (fl (z*)) (p)))
(n (expt 2 (p))))))))
(local-defthmd qrnd-21
(implies (not (specialp))
(equal (mod (+ (* 2 (bits (fl (z*)) (p) 1)) 2) (expt 2 (p)))
(mod (+ (- (mod (fl (z*)) (expt 2 (p))) (bitn (fl (z*)) 0)) 2) (expt 2 (p)))))
:hints (("Goal" :use (p-vals qrnd-20) :in-theory (enable bits))))
(local-defthmd qrnd-22
(implies (not (specialp))
(equal (mod (+ (* 2 (bits (fl (z*)) (p) 1)) 2) (expt 2 (p)))
(mod (+ (- (fl (z*)) (bitn (fl (z*)) 0)) 2) (expt 2 (p)))))
:hints (("Goal" :use (p-vals qrnd-21
(:instance mod-sum (a (- 2 (bitn (fl (z*)) 0))) (b (fl (z*))) (n (expt 2 (p))))))))
(local-defthmd qrnd-23
(implies (not (specialp))
(equal (mod (+ (* 2 (bits (fl (z*)) (p) 1)) 2) (expt 2 (p)))
(mod (- (mod (+ (fl (z*)) 2) (expt 2 (p))) (bitn (fl (z*)) 0)) (expt 2 (p)))))
:hints (("Goal" :use (p-vals qrnd-22
(:instance mod-sum (a (- (bitn (fl (z*)) 0))) (b (+ 2 (fl (z*)))) (n (expt 2 (p))))))))
(local-defthm qrnd-24
(implies (and (not (specialp))
(roundup-pos (fl (z*)) (p) (stk) (mode) (p)))
(or (= (stk) 1)
(= (bitn (qtrunc) 0) 1)))
:rule-classes ()
:hints (("Goal" :use (p-vals qrnd-5
(:instance bitn-bits (x (fl (z*))) (i (1- (p))) (j 0) (k 0))
(:instance bitn-bits (x (qtrunc)) (i (1- (p))) (j 0) (k 0)))
:in-theory (enable roundup-pos))))
(local-defthmd qrnd-26
(implies (and (not (specialp))
(roundup-pos (fl (z*)) (p) (stk) (mode) (p)))
(equal (mod (+ (fl (z*)) 2) (expt 2 (p)))
(mod (qinc) (expt 2 (p)))))
:hints (("Goal" :in-theory (enable z*) :use (p-vals qinc-rewrite))))
(local-defthmd qrnd-27
(implies (and (not (specialp))
(roundup-pos (fl (z*)) (p) (stk) (mode) (p)))
(equal (mod (+ (* 2 (bits (fl (z*)) (p) 1)) 2) (expt 2 (p)))
(mod (- (mod (qinc) (expt 2 (p))) (bitn (fl (z*)) 0)) (expt 2 (p)))))
:hints (("Goal" :in-theory (theory 'minimal-theory) :use (qrnd-23 qrnd-26))))
(local-defthm qrnd-28
(implies (and (not (specialp))
(roundup-pos (fl (z*)) (p) (stk) (mode) (p)))
(and (integerp (p))
(integerp 0)
(< 0 (p))))
:rule-classes ()
:hints (("Goal" :use (p-vals))))
(local-defthmd qrnd-29
(implies (and (not (specialp))
(roundup-pos (fl (z*)) (p) (stk) (mode) (p)))
(equal (bitn (qinc) 0)
(bitn (+ (fl (z*)) 2) 0)))
:hints (("Goal" :in-theory (theory 'minimal-theory)
:use (qrnd-26 qrnd-28 p-vals
(:instance bitn-mod (x (qinc)) (n (p)) (k 0))
(:instance bitn-mod (x (+ (fl (z*)) 2)) (n (p)) (k 0))))))
(local-defthmd qrnd-30
(implies (and (not (specialp))
(roundup-pos (fl (z*)) (p) (stk) (mode) (p)))
(equal (bitn (qinc) 0)
(bitn (fl (z*)) 0)))
:hints (("Goal" :use (qrnd-29
(:instance bitn-plus-mult (x (fl (z*))) (m 1) (k 1) (n 0))))))
(local-defthmd qrnd-31
(implies (and (not (specialp))
(roundup-pos (fl (z*)) (p) (stk) (mode) (p)))
(equal (mod (+ (* 2 (bits (fl (z*)) (p) 1)) 2) (expt 2 (p)))
(mod (- (mod (qinc) (expt 2 (p))) (bitn (qinc) 0)) (expt 2 (p)))))
:hints (("Goal" :use (qrnd-27 qrnd-30 p-vals))))
(local-defthmd qrnd-32
(implies (and (not (specialp))
(roundup-pos (fl (z*)) (p) (stk) (mode) (p)))
(equal (mod (+ (* 2 (bits (fl (z*)) (p) 1)) 2) (expt 2 (p)))
(mod (- (bits (qinc) (1- (p)) 0) (bitn (qinc) 0)) (expt 2 (p)))))
:hints (("Goal" :in-theory (enable bits) :use (qrnd-31 p-vals))))
(local-defthmd qrnd-33
(implies (and (not (specialp))
(roundup-pos (fl (z*)) (p) (stk) (mode) (p)))
(equal (mod (+ (* 2 (bits (fl (z*)) (p) 1)) 2) (expt 2 (p)))
(mod (* 2 (bits (qinc) (1- (p)) 1)) (expt 2 (p)))))
:hints (("Goal" :use (qrnd-32 p-vals (:instance bits-plus-bitn (x (qinc)) (n (1- (p))) (m 0))))))
(local-defthmd qrnd-34
(implies (and (not (specialp))
(roundup-pos (fl (z*)) (p) (stk) (mode) (p)))
(equal (mod (rnd (z*) (mode) (p)) (expt 2 (p)))
(mod (* 2 (bits (qinc) (1- (p)) 1)) (expt 2 (p)))))
:hints (("Goal" :use (qrnd-10 qrnd-18 qrnd-33))))
(local-defthmd qrnd-35
(implies (and (not (specialp))
(roundup-pos (fl (z*)) (p) (stk) (mode) (p)))
(equal (mod (rnd (z*) (mode) (p)) (expt 2 (p)))
(mod (* 2 (bits (qrnd) (- (p) 2) 0)) (expt 2 (p)))))
:hints (("Goal" :use (qrnd-34 qrnd-11 p-vals
(:instance bits-bits (x (qinc)) (i 53) (j 1) (k (- (p) 2)) (l 0))))))
(local-defthmd qrnd-36
(implies (not (specialp))
(equal (mod (rnd (* (expt 2 (1+ (p))) (qsqrt (x) (1+ (* 2 (n))))) (mode) (p)) (expt 2 (p)))
(mod (* 2 (bits (qrnd) (- (p) 2) 0)) (expt 2 (p)))))
:hints (("Goal" :use (qrnd-35 qrnd-16 p-vals)
:in-theory '(z*))))
(defthmd rnd-qsqrt-qrnd
(implies (not (specialp))
(equal (mod (rnd (* (expt 2 (1+ (p))) (qsqrt (x) (1+ (* 2 (n))))) (mode) (p)) (expt 2 (p)))
(* 2 (bits (qrnd) (- (p) 2) 0))))
:hints (("Goal" :use (qrnd-36 p-vals (:instance bits-bounds (x (qrnd)) (i (- (p) 2)) (j 0))))))
(local-defthmd rnd-1-1
(implies (and (not (specialp))
(not (= (mode) 'rup)))
(not (= (rnd (qsqrt (x) (1+ (* 2 (n)))) (mode) (p))
1)))
:hints (("Goal" :use (p-vals qsqrt-x-upper expo-qsqrt-x qrnd-4
(:instance rtz-upper-pos (x (qsqrt (x) (1+ (* 2 (n))))) (n (p)))
(:instance bvecp-member (x (bits (rin) 23 22)) (n 2))
(:instance rne-diff (x (qsqrt (x) (1+ (* 2 (n))))) (n (p))))
:in-theory (enable rmode mode rdn rnd))))
(local-defthmd rnd-1-2
(implies (and (not (specialp))
(= (quotf) (- 1 (expt 2 (- (1+ (p)))))))
(> (qsqrt (x) (1+ (* 2 (n))))
(- 1 (expt 2 (- (p))))))
:hints (("Goal" :use (fnum-vals quotf-error-b)
:in-theory (enable n p f dp sp hp prec))))
(local-defthmd rnd-1-3
(implies (and (not (specialp))
(= (quotf) (- 1 (expt 2 (- (1+ (p)))))))
(equal (rup (qsqrt (x) (1+ (* 2 (n)))) (p))
1))
:hints (("Goal" :use (p-vals rnd-1-2 expo-qsqrt-x x-bounds fnum-vals qsqrt-x-upper
(:instance qsqrt-pos (x (x)) (n (1+ (* 2 (n)))))
(:instance raz-squeeze (a (- 1 (expt 2 (- (p))))) (x (qsqrt (x) (1+ (* 2 (n))))) (n (p)))
(:instance raz-expo-upper (x (qsqrt (x) (1+ (* 2 (n))))) (n (p))))
:in-theory (enable rup n))))
(local-defthmd rnd-1-4
(implies (and (not (specialp))
(< (quotf) (- 1 (expt 2 (- (1+ (p)))))))
(< (* (expt 2 (* 2 (n))) (quotf))
(* (expt 2 (* 2 (n))) (- 1 (expt 2 (- (1+ (p))))))))
:hints (("Goal" :use (fnum-vals)
:nonlinearp t
:in-theory (enable n p f dp sp hp prec))))
(local-defthmd rnd-1-5
(implies (and (not (specialp))
(< (quotf) (- 1 (expt 2 (- (1+ (p)))))))
(<= (* (expt 2 (* 2 (n))) (quotf))
(1- (* (expt 2 (* 2 (n))) (- 1 (expt 2 (- (1+ (p)))))))))
:hints (("Goal" :use (fnum-vals rnd-1-4
(:instance int-quot (j (n))))
:in-theory (enable quotf n p f dp sp hp prec))))
(defthmd rnd-1-6
(implies (and (not (specialp))
(< (quotf) (- 1 (expt 2 (- (1+ (p)))))))
(<= (quotf)
(- (- 1 (expt 2 (- (1+ (p)))))
(expt 2 (- (* 2 (n)))))))
:hints (("Goal" :use (fnum-vals rnd-1-5)
:nonlinearp t
:in-theory (enable quotf n p f dp sp hp prec))))
(local-defthmd quotf-bounds
(implies (not (specialp))
(and (<= (quotf) 1)
(>= (quotf) 1/2)))
:hints (("Goal" :in-theory (enable quotf n inv quot-bnds-inv)
:use ((:instance inv-lemma (j (n)))))))
(local-defthmd rnd-1-7
(implies (and (not (specialp))
(< (quotf) (- 1 (expt 2 (- (1+ (p)))))))
(and (< 0 (+ (quotf) (* 2/3 (expt 4 (- (n))))))
(<= (+ (quotf) (* 2/3 (expt 4 (- (n)))))
(- 1 (+ (expt 2 (- (1+ (p)))) (* 1/3 (expt 2 (- (* 2 (n))))))))
(rationalp (+ (quotf) (* 2/3 (expt 4 (- (n))))))
(rationalp (- 1 (+ (expt 2 (- (1+ (p)))) (* 1/3 (expt 2 (- (* 2 (n))))))))
(rationalp (x))
(<= (x) (expt (+ (quotf) (* 2/3 (expt 4 (- (n))))) 2))))
:hints (("Goal" :use (fnum-vals rnd-1-6 quotf-error-a quotf-bounds)
:in-theory (e/d (expt n p f dp sp hp prec)))))
(local-defthmd rnd-1-8
(implies (and (rationalp x) (rationalp a) (rationalp b)
(< 0 a)
(<= a b)
(<= x (expt a 2)))
(<= x (expt b 2)))
:hints (("Goal" :in-theory (enable expt) :nonlinearp t)))
(local-defthmd rnd-1-9
(implies (and (not (specialp)) (>= (quotf) 1/2)
(< (quotf) (- 1 (expt 2 (- (1+ (p)))))))
(<= (x)
(expt (- 1 (+ (expt 2 (- (1+ (p)))) (* 1/3 (expt 2 (- (* 2 (n))))))) 2)))
:hints (("Goal" :use (rnd-1-7
(:instance rnd-1-8 (a (+ (quotf) (* 2/3 (expt 4 (- (n))))))
(b (- 1 (+ (expt 2 (- (1+ (p)))) (* 1/3 (expt 2 (- (* 2 (n))))))))
(x (x)))))))
(local-defthmd rnd-1-10
(implies (and (not (specialp))
(< (quotf) (- 1 (expt 2 (- (1+ (p)))))))
(< (x) (- 1 (expt 2 (- (p))))))
:hints (("Goal" :use (fnum-vals rnd-1-9)
:in-theory (enable n p f dp sp hp prec))))
(local-defthmd rnd-1-11
(implies (and (not (specialp))
(< (x) (- 1 (expt 2 (- (p))))))
(< (x) (* (- 1 (expt 2 (- (p)))) (- 1 (expt 2 (- (p)))))))
:hints (("Goal" :use (p-vals rnd-1-10 exactp-x
(:instance fp-2 (x (- 1 (expt 2 (- (p))))) (y (x)) (n (p)))))))
(local-defthmd rnd-1-12
(implies (and (not (specialp))
(< (x) (- 1 (expt 2 (- (p))))))
(< (qsqrt (x) (1+ (* 2 (n))))
(- 1 (expt 2 (- (p))))))
:hints (("Goal" :use (fnum-vals rnd-1-11
(:instance exactp-cmp-qsqrt (q (- 1 (expt 2 (- (p))))) (x (x)) (n (1+ (* 2 (n))))))
:in-theory (enable p prec n f sp dp hp))))
(local-defthmd rnd-1-13
(implies (and (not (specialp))
(< (x) (- 1 (expt 2 (- (p))))))
(< (rup (qsqrt (x) (1+ (* 2 (n)))) (p))
1))
:hints (("Goal" :use (fnum-vals rnd-1-12
(:instance qsqrt-pos (x (x)) (n (1+ (* 2 (n)))))
(:instance raz-monotone (x (qsqrt (x) (1+ (* 2 (n))))) (y (- 1 (expt 2 (- (p))))) (n (p))))
:in-theory (enable rup n p f dp sp hp prec))))
(defthmd rnd-1-rup-qmax
(implies (not (specialp))
(iff (equal (rnd (qsqrt (x) (1+ (* 2 (n)))) (mode) (p))
1)
(and (equal (mode) 'rup)
(equal (quotf) (- 1 (expt 2 (- (1+ (p)))))))))
:hints (("Goal" :use (rnd-1-1 rnd-1-3 rnd-1-10 rnd-1-13 quotf-upper)
:in-theory (enable rnd))))
(defthmd rnd-1-rup-xmax
(implies (not (specialp))
(iff (equal (rnd (qsqrt (x) (1+ (* 2 (n)))) (mode) (p))
1)
(and (equal (mode) 'rup)
(equal (x) (- 1 (expt 2 (- (p))))))))
:hints (("Goal" :use (fnum-vals rnd-1-1 rnd-1-10 rnd-1-13 quotf-upper rnd-1-rup-qmax x-bounds exactp-x
(:instance qsqrt-pos (x (x)) (n (1+ (* 2 (n)))))
(:instance fp-2 (x 1) (y (x)) (n (p))))
:in-theory (enable rnd rup n p f dp sp hp prec))))
(local-defthmd rup-expinc-1
(implies (not (specialp))
(equal (expinc-1)
(if (and (= (mode) 'rup)
(= (classa) 4)
(= (q 1) 0))
1 0)))
:hints (("Goal" :in-theory (enable expinc-0 expinc-1 q q-1 qn qn-1 firstiter rmode mode)
:use ((:instance bvecp-member (x (bits (rin) 23 22)) (n 2))))))
(local-defun allq0 (j)
(if (zp j)
t
(and (= (q j) 0)
(allq0 (1- j)))))
(local (in-theory (disable (allq0))))
(local-defthmd rup-expinc-2
(implies (and (not (specialp))
(not (zp j))
(< j (n)))
(equal (expinc j)
(if (and (= (mode) 'rup)
(= (classa) 4)
(allq0 j))
1 0)))
:hints (("Goal" :in-theory (enable expinc))
("Subgoal *1/1" :in-theory (enable expinc rup-expinc-1))
("Subgoal *1/2" :in-theory (enable expinc rup-expinc-1))))
(local-defthmd rup-expinc-3
(implies (not (specialp))
(equal (expinc (n))
(if (and (= (mode) 'rup)
(= (classa) 4)
(allq0 (1- (n)))
(= (q (n)) (if (= (fnum) 1) -2 -1)))
1 0)))
:hints (("Goal" :in-theory (enable n expinc)
:use (fnum-vals
(:instance expinc (j (n)))
(:instance rup-expinc-2 (j (1- (n))))))))
(local-defthmd rup-expinc-4
(implies (and (not (specialp))
(natp j)
(allq0 j))
(equal (quot j) 1))
:hints (("Subgoal *1/5" :expand ((quot j)))
("Subgoal *1/1" :expand ((quot 0)))))
(local-defthmd rup-expinc-5
(implies (and (not (specialp))
(= (expinc (n)) 1))
(and (equal (mode) 'rup)
(equal (quotf) (- 1 (expt 2 (- (1+ (p))))))))
:hints (("Goal" :in-theory (enable quotf n p prec f sp dp hp)
:use (fnum-vals rup-expinc-3
(:instance rup-expinc-4 (j (1- (n))))
(:instance quot (j (n)))))))
(local-defthmd rup-expinc-6
(implies (and (not (specialp))
(= (rnd (qsqrt (x) (1+ (* 2 (n)))) (mode) (p))
1))
(equal (x) (- 1 (expt 2 (- (p))))))
:hints (("Goal" :use (p-vals exactp-x x-bounds rnd-1-13 rnd-1-1
(:instance fp-2 (y (x)) (x 1) (n (p))))
:in-theory (enable rnd))))
(local-defthmd rup-expinc-7
(implies (and (not (specialp))
(natp j)
(allq0 j)
(not (zp k))
(<= k j))
(equal (q k) 0)))
(local-defthmd rup-expinc-8
(implies (and (not (specialp))
(natp j)
(allq0 j))
(equal (i j) 8))
:hints (("Goal" :in-theory (enable i i-1 firstiter q q-1)
:use ((:instance rup-expinc-7 (k 1))
(:instance rup-expinc-7 (k 2))))))
(local-defthmd rup-expinc-9
(implies (and (not (specialp))
(= (rnd (qsqrt (x) (1+ (* 2 (n)))) (mode) (p))
1)
(natp j)
(allq0 j))
(equal (r j) (- (expt 2 (- (* 2 j) (p))))))
:hints (("Goal" :in-theory (enable rup-expinc-4 rup-expinc-6 r)
:use (p-vals))))
(local-defthmd rup-expinc-10
(implies (and (not (specialp))
(= (rnd (qsqrt (x) (1+ (* 2 (n)))) (mode) (p))
1)
(natp j)
(< j (1- (n)))
(allq0 j))
(>= (* 4 (r j)) (m (i j) j 0)))
:hints (("Goal" :in-theory (enable m n p prec dp sp hp f rup-expinc-8 rup-expinc-9)
:nonlinearp t
:use (fnum-vals))))
(local-defthmd rup-expinc-11
(implies (and (not (specialp))
(= (rnd (qsqrt (x) (1+ (* 2 (n)))) (mode) (p))
1)
(natp j)
(< j (1- (n)))
(allq0 j))
(>= (q (1+ j)) 0))
:hints (("Goal" :in-theory (enable maxk m n p prec dp sp hp f inv approx-inv approx-bounds)
:use (fnum-vals rup-expinc-10 (:instance inv-lemma (j (1+ j)))))))
(local-defthmd rup-expinc-12
(implies (and (not (specialp))
(= (rnd (qsqrt (x) (1+ (* 2 (n)))) (mode) (p))
1)
(natp j)
(< j (1- (n)))
(allq0 j))
(allq0 (1+ j)))
:hints (("Goal" :in-theory (enable inv quot-bnds-inv quot rup-expinc-4 n p prec dp sp hp f)
:use (fnum-vals rup-expinc-11
(:instance inv-lemma (j (1+ j)))))))
(local-defthmd rup-expinc-13
(implies (and (not (specialp))
(= (rnd (qsqrt (x) (1+ (* 2 (n)))) (mode) (p))
1)
(natp j)
(< j (n)))
(allq0 j))
:hints (("Goal" :in-theory (enable quot) :induct (quot j))
("Subgoal *1/2" :in-theory (enable n)
:use (fnum-vals (:instance rup-expinc-12 (j (1- j)))))))
(local-defthmd rup-expinc-14
(implies (and (not (specialp))
(= (rnd (qsqrt (x) (1+ (* 2 (n)))) (mode) (p))
1))
(and (= (mode) 'rup)
(allq0 (1- (n)))
(= (q (n)) (if (= (fnum) 1) -2 -1))))
:hints (("Goal" :in-theory (enable quotf n quot f dp sp hp p prec)
:use (fnum-vals rnd-1-rup-qmax
(:instance rup-expinc-4 (j (1- (n))))
(:instance rup-expinc-13 (j (1- (n))))))))
(local-defthmd rup-expinc-15
(implies (and (not (specialp))
(not (= (classa) 4)))
(equal (expa) 0))
:hints (("Goal" :in-theory (enable normp denormp)
:use (a-class spec-class spec-fields))))
(local-defthmd rup-expinc-16
(implies (not (specialp))
(exactp (mana) (1- (p))))
:hints (("Goal" :in-theory (enable bvecp p prec sigw f dp sp hp)
:use (fnum-vals bvecp-mana
(:instance expo<= (x (mana)) (n (- (p) 2)))
(:instance exactp-<= (m (1+ (expo (mana)))) (n (1- (p))))
(:instance exact-bits-1 (x (mana)) (k 0) (n (1+ (expo (mana)))))))))
(local-defthmd rup-expinc-17
(implies (and (not (specialp))
(not (= (classa) 4)))
(exactp (siga) (1- (p))))
:hints (("Goal" :use (p-vals rup-expinc-15 rup-expinc-16 siga-mana
(:instance exactp-shift (x (mana)) (k (- 52 (expo (mana)))) (n (1- (p))))))))
(local-defthmd rup-expinc-18
(implies (and (not (specialp))
(not (= (classa) 4)))
(exactp (sig (a)) (1- (p))))
:hints (("Goal" :use (p-vals rup-expinc-17 siga-rewrite
(:instance exactp-shift (x (siga)) (k -52) (n (1- (p))))))))
(local-defthmd rup-expinc-19
(implies (and (not (specialp))
(not (= (classa) 4)))
(exactp (x) (1- (p))))
:hints (("Goal" :in-theory (enable x)
:use (p-vals rup-expinc-18
(:instance exactp-shift (x (sig (a))) (k -1) (n (1- (p))))
(:instance exactp-shift (x (sig (a))) (k -2) (n (1- (p))))))))
(local-defthmd rup-expinc-20
(implies (and (not (specialp))
(= (rnd (qsqrt (x) (1+ (* 2 (n)))) (mode) (p))
1))
(equal (classa) 4))
:hints (("Goal" :in-theory (enable rup-expinc-6)
:use (p-vals rup-expinc-19))))
(local-defthmd rup-expinc-21
(implies (and (not (specialp))
(= (rnd (qsqrt (x) (1+ (* 2 (n)))) (mode) (p))
1))
(equal (expinc (n)) 1))
:hints (("Goal" :in-theory (enable rup-expinc-3)
:use (rup-expinc-14 rup-expinc-20))))
(defthmd rup-expinc
(implies (not (specialp))
(equal (expinc (n))
(if (equal (rnd (qsqrt (x) (1+ (* 2 (n)))) (mode) (p))
1)
1 0)))
:hints (("Goal" :expand ((expinc (n)) (expinc-1) (expinc 1) (expinc 0))
:use (rnd-1-rup-qmax rup-expinc-5 rup-expinc-21))))
(defthmd rnd-inx-rewrite
(implies (not (specialp))
(equal (inx)
(if (= (rnd (qsqrt (a) (1+ (* 2 (n)))) (mode) (p))
(qsqrt (a) (1+ (* 2 (n)))))
0 1)))
:hints (("Goal" :use (qrnd-4 sqrt-a-x p-vals
(:instance exactp-shift (k (1+ (- (expq) (bias (f))))) (x (qsqrt (x) (1+ (* 2 (n))))) (n (p)))
(:instance rnd-exactp-b (x (qsqrt (a) (1+ (* 2 (n))))) (mode (mode)) (n (p))))
:in-theory (enable inx-rewrite))))
(local-defthmd rnd-qsqrt-a-1
(implies (and (not (specialp))
(not (= (rnd (qsqrt (x) (1+ (* 2 (n)))) (mode) (p)) 1)))
(equal (expo (rnd (qsqrt (x) (1+ (* 2 (n)))) (mode) (p)))
-1))
:hints (("Goal" :use (qrnd-4 expo-qsqrt-x p-vals
(:instance expo-rnd (x (qsqrt (x) (1+ (* 2 (n))))) (mode (mode)) (n (p)))))))
(local-defthmd rnd-qsqrt-a-2
(implies (and (not (specialp))
(not (= (rnd (qsqrt (x) (1+ (* 2 (n)))) (mode) (p)) 1)))
(equal (expo (rnd (* (expt 2 (1+ (p))) (qsqrt (x) (1+ (* 2 (n))))) (mode) (p)))
(p)))
:hints (("Goal" :use (qrnd-4 p-vals rnd-qsqrt-a-1
(:instance expo-shift (x (rnd (qsqrt (x) (1+ (* 2 (n)))) (mode) (p))) (n (1+ (p))))
(:instance rnd-shift (x (qsqrt (x) (1+ (* 2 (n))))) (k (1+ (p))) (n (p)) (mode (mode)))))))
(local-defthmd rnd-qsqrt-a-2
(implies (and (not (specialp))
(not (= (rnd (qsqrt (x) (1+ (* 2 (n)))) (mode) (p)) 1)))
(equal (expo (rnd (* (expt 2 (1+ (p))) (qsqrt (x) (1+ (* 2 (n))))) (mode) (p)))
(p)))
:hints (("Goal" :use (qrnd-4 p-vals rnd-qsqrt-a-1
(:instance expo-shift (x (rnd (qsqrt (x) (1+ (* 2 (n)))) (mode) (p))) (n (1+ (p))))
(:instance rnd-shift (x (qsqrt (x) (1+ (* 2 (n))))) (k (1+ (p))) (n (p)) (mode (mode)))))))
(local-defthmd rnd-qsqrt-a-3
(implies (and (not (specialp))
(not (= (rnd (qsqrt (x) (1+ (* 2 (n)))) (mode) (p)) 1)))
(> (rnd (* (expt 2 (1+ (p))) (qsqrt (x) (1+ (* 2 (n))))) (mode) (p)) 0))
:hints (("Goal" :use (qrnd-4 x-bounds p-vals
(:instance qsqrt-pos (x (x)) (n (1+ (* 2 (n)))))
(:instance rnd-positive (x (qsqrt (x) (1+ (* 2 (n))))) (mode (mode)) (n (p)))))))
(local-defthmd rnd-qsqrt-a-4
(implies (and (not (specialp))
(not (= (rnd (qsqrt (x) (1+ (* 2 (n)))) (mode) (p)) 1)))
(and (>= (rnd (* (expt 2 (1+ (p))) (qsqrt (x) (1+ (* 2 (n))))) (mode) (p))
(expt 2 (p)))
(< (rnd (* (expt 2 (1+ (p))) (qsqrt (x) (1+ (* 2 (n))))) (mode) (p))
(expt 2 (1+ (p))))))
:hints (("Goal" :use (rnd-qsqrt-a-2 rnd-qsqrt-a-3 p-vals
(:instance expo-upper-bound (x (rnd (* (expt 2 (1+ (p))) (qsqrt (x) (1+ (* 2 (n))))) (mode) (p))))
(:instance expo-lower-bound (x (rnd (* (expt 2 (1+ (p))) (qsqrt (x) (1+ (* 2 (n))))) (mode) (p))))))))
(local-defthmd rnd-qsqrt-a-5
(implies (and (not (specialp))
(not (= (rnd (qsqrt (x) (1+ (* 2 (n)))) (mode) (p)) 1)))
(equal (fl (/ (rnd (* (expt 2 (1+ (p))) (qsqrt (x) (1+ (* 2 (n))))) (mode) (p))
(expt 2 (p))))
1))
:hints (("Goal" :use (rnd-qsqrt-a-4 p-vals) :nonlinearp t)))
(local-defthmd rnd-qsqrt-a-6
(implies (and (not (specialp))
(not (= (rnd (qsqrt (x) (1+ (* 2 (n)))) (mode) (p)) 1)))
(equal (rnd (* (expt 2 (1+ (p))) (qsqrt (x) (1+ (* 2 (n))))) (mode) (p))
(+ (expt 2 (p)) (* 2 (bits (qrnd) (- (p) 2) 0)))))
:hints (("Goal" :use (rnd-qsqrt-a-5 rnd-qsqrt-qrnd p-vals
(:instance mod-def (x (rnd (* (expt 2 (1+ (p))) (qsqrt (x) (1+ (* 2 (n))))) (mode) (p)))
(y (expt 2 (p))))))))
(defthmd exprnd-rewrite
(implies (not (specialp))
(equal (exprnd)
(if (equal (rnd (qsqrt (x) (1+ (* 2 (n)))) (mode) (p))
1)
(1+ (expq))
(expq))))
:hints (("Goal" :use (expq-bounds fnum-vals)
:in-theory (enable dp sp hp f prec p expw bvecp exprnd rup-expinc))))
(local-defthmd rnd-qsqrt-a-8
(implies (and (not (specialp))
(not (= (rnd (qsqrt (x) (1+ (* 2 (n)))) (mode) (p)) 1)))
(equal (rnd (qsqrt (a) (1+ (* 2 (n)))) (mode) (p))
(* (expt 2 (- (exprnd) (+ (bias (f)) (1- (p)))))
(+ (expt 2 (1- (p))) (bits (qrnd) (- (p) 2) 0)))))
:hints (("Goal" :use (rnd-qsqrt-a-6 qrnd-4 fnum-vals sqrt-a-x
(:instance rnd-shift (x (* (expt 2 (1+ (p))) (qsqrt (x) (1+ (* 2 (n)))))) (mode (mode)) (n (p))
(k (- (exprnd) (+ (bias (f)) (p))))))
:in-theory (enable bias n dp sp hp f prec p expw exprnd-rewrite))))
(local-defthmd rnd-qsqrt-a-9
(implies (and (not (specialp))
(= (rnd (qsqrt (x) (1+ (* 2 (n)))) (mode) (p)) 1))
(equal (bits (qrnd) (- (p) 2) 0) 0))
:hints (("Goal" :use (rnd-qsqrt-qrnd qrnd-4 fnum-vals
(:instance rnd-shift (x (qsqrt (x) (1+ (* 2 (n))))) (mode (mode)) (n (p))
(k (1+ (p)))))
:in-theory (enable bias n dp sp hp f prec p expw exprnd-rewrite))))
(local-defthmd rnd-qsqrt-a-10
(implies (and (not (specialp))
(= (rnd (qsqrt (x) (1+ (* 2 (n)))) (mode) (p)) 1))
(equal (rnd (qsqrt (a) (1+ (* 2 (n)))) (mode) (p))
(* (expt 2 (- (exprnd) (+ (bias (f)) (1- (p)))))
(+ (expt 2 (1- (p))) (bits (qrnd) (- (p) 2) 0)))))
:hints (("Goal" :use (sqrt-a-x qrnd-4 fnum-vals rnd-qsqrt-a-9
(:instance rnd-shift (x (qsqrt (x) (1+ (* 2 (n))))) (mode (mode)) (n (p))
(k (- (exprnd) (bias (f))))))
:in-theory (enable bias n dp sp hp f prec p expw exprnd-rewrite))))
(defthmd rnd-qsqrt-a
(implies (not (specialp))
(equal (rnd (qsqrt (a) (1+ (* 2 (n)))) (mode) (p))
(* (expt 2 (- (exprnd) (+ (bias (f)) (1- (p)))))
(+ (expt 2 (1- (p))) (bits (qrnd) (- (p) 2) 0)))))
:hints (("Goal" :use (rnd-qsqrt-a-8 rnd-qsqrt-a-10))))
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