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|
(in-package "RTL")
(include-book "rounder")
(local-defthmd spec-1
(implies (not (specialp))
(equal (arm-sqrt-pre-comp-val (opaz) (rz) (f))
()))
:hints (("Goal" :in-theory (enable snanp qnanp specialp)
:use (a-fields a-class))))
(local-defthmd spec-2
(implies (not (specialp))
(equal (arm-sqrt-pre-comp-excp (opaz) (rz) (f))
(rz)))
:hints (("Goal" :in-theory (enable snanp qnanp specialp)
:use (a-fields a-class))))
(local-defthmd spec-3
(implies (not (specialp))
(equal (arm-sqrt-spec (opaw) (rin) (f))
(arm-sqrt-comp (opaz) (rz) (f))))
:hints (("Goal" :in-theory (enable arm-sqrt-spec-rewrite)
:use (spec-1 spec-2))))
(in-theory (disable arm-post-comp (arm-post-comp) arm-sqrt-spec (arm-sqrt-spec)))
(local-defthmd spec-4
(implies (not (specialp))
(equal (decode (opaz) (f)) (a)))
:hints (("Goal" :in-theory (enable a specialp))))
(local-defthmd spec-5
(implies (not (specialp))
(not (equal (a) 0))))
(local-defthmd spec-6
(implies (not (specialp))
(equal (arm-sqrt-spec (opaw) (rin) (f))
(arm-post-comp (qsqrt (a) (+ (p) 2)) (rz) (f))))
:hints (("Goal" :in-theory (enable p specialp spec-3)
:use (spec-4 spec-5 a-class))))
(local-defthm spec-7
(implies (not (specialp))
(equal (prec (f)) (p)))
:hints (("Goal" :in-theory (enable p prec f sp dp hp)
:use (fnum-vals))))
(local-defthm spec-8
(implies (not (specialp))
(equal (fpscr-rc (rz)) (mode)))
:hints (("Goal" :in-theory (enable fpscr-rc mode rmode bitn-logior a-flags)
:use ((:instance bitn-plus-bits (x (rin)) (n 23) (m 22))
(:instance bitn-plus-bits (x (rz)) (n 23) (m 22))))))
(local-defthm spec-9
(implies (not (specialp))
(common-mode-p (mode)))
:hints (("Goal" :in-theory (enable mode rmode)
:use ((:instance bvecp-member (x (bits (rin) 23 22)) (n 2))))))
(local-defthmd spec-10
(implies (not (specialp))
(equal (rnd (qsqrt (a) (1+ (* 2 (n)))) (mode) (p))
(rnd (qsqrt (a) (+ (p) 2)) (mode) (p))))
:hints (("Goal" :in-theory (enable n f dp sp hp p prec)
:use (n fnum-vals
(:instance rnd-qsqrt-equal (x (a)) (m (1+ (* 2 (n)))) (n (+ (p) 2)) (k (p)) (mode (mode)))))))
(local-defthmd spec-11
(implies (not (specialp))
(equal (expo (qsqrt (x) (+ (p) 2)))
-1))
:hints (("Goal" :in-theory (enable qsqrt-rto-sqrt)
:use (p-vals x-bounds))))
(local-defthmd spec-12
(implies (not (specialp))
(equal (qsqrt (a) (+ (p) 2))
(* (expt 2 (1+ (- (expq) (bias (f)))))
(qsqrt (x) (+ (p) 2)))))
:hints (("Goal" :in-theory (enable n a-x dp sp hp p prec bias f)
:use (fnum-vals x-bounds
(:instance qsqrt-shift (x (x)) (n (+ (p) 2)) (k (1+ (- (expq) (bias (f))))))))))
(local-defthmd spec-13
(implies (not (specialp))
(equal (expo (qsqrt (a) (+ (p) 2)))
(- (expq) (bias (f)))))
:hints (("Goal" :in-theory (enable n a-x dp sp hp p prec bias f)
:use (fnum-vals spec-11 spec-12
(:instance expo-shift (x (qsqrt (x) (+ (p) 2))) (n (1+ (- (expq) (bias (f))))))))))
(local-defthmd spec-14
(implies (not (specialp))
(>= (qsqrt (a) (+ (p) 2))
(spn (f))))
:hints (("Goal" :in-theory (enable expw spn n a-x dp sp hp p prec bias f)
:nonlinearp t
:use (fnum-vals spec-13 expq-bounds x-bounds
(:instance qsqrt-pos (x (a)) (n (+ (p) 2)))
(:instance expo-lower-bound (x (qsqrt (a) (+ (p) 2))))))))
(local-defthmd spec-15
(implies (not (specialp))
(<= (abs (rnd (qsqrt (a) (+ (p) 2)) (mode) (p)))
(lpn (f))))
:hints (("Goal" :in-theory (enable expw lpn n dp sp hp p prec bias f exprnd-rewrite)
:nonlinearp t
:use (fnum-vals spec-10 expq-bounds rnd-qsqrt-a
(:instance bits-bounds (x (qrnd)) (i (- (p) 2)) (j 0))))))
(local-defthmd spec-16
(implies (not (specialp))
(iff (= (rnd (qsqrt (a) (+ (p) 2)) (mode) (p))
(qsqrt (a) (+ (p) 2)))
(= (rnd (qsqrt (a) (1+ (* 2 (n)))) (mode) (p))
(qsqrt (a) (1+ (* 2 (n)))))))
:hints (("Goal" :in-theory (enable n dp sp hp p prec f)
:use (fnum-vals
(:instance qsqrt-exact-equal (x (a)) (k (p)) (m (1+ (* 2 (n)))) (n (+ (p) 2)))
(:instance qsqrt-exact-equal (x (a)) (k (p)) (n (1+ (* 2 (n)))) (m (+ (p) 2)))
(:instance rnd-exactp-b (x (qsqrt (a) (1+ (* 2 (n))))) (mode (mode)) (n (p)))
(:instance rnd-exactp-b (x (qsqrt (a) (+ (p) 2))) (mode (mode)) (n (p)))))))
(local-defthmd spec-17
(implies (not (specialp))
(equal (arm-sqrt-spec (opaw) (rin) (f))
(mv (nencode (rnd (qsqrt (a) (1+ (* 2 (n)))) (mode) (p)) (f))
(if (= (rnd (qsqrt (a) (1+ (* 2 (n)))) (mode) (p))
(qsqrt (a) (1+ (* 2 (n)))))
(rz)
(set-flag (ixc) (rz))))))
:hints (("Goal" :in-theory (enable spec-6 arm-post-comp)
:use (spec-16 spec-14 spec-15 spec-10))))
(local-defthmd spec-18
(implies (not (specialp))
(equal (arm-sqrt-spec (opaw) (rin) (f))
(mv (nencode (rnd (qsqrt (a) (1+ (* 2 (n)))) (mode) (p)) (f))
(if (= (inx) 0)
(rz)
(set-flag (ixc) (rz))))))
:hints (("Goal" :in-theory (enable rnd-inx-rewrite)
:use (spec-17))))
(local-defthmd sqrtpow2-1
(implies (and (not (specialp))
(= (classa) 4)
(= (mana) 0))
(not (equal (expa) 0)))
:hints (("Goal" :in-theory (enable expa normp)
:use (a-class a-fields))))
(local-defthmd sqrtpow2-2
(implies (and (not (specialp))
(= (classa) 4)
(= (mana) 0))
(equal (siga) (expt 2 52)))
:hints (("Goal" :in-theory (enable si normalize siga)
:use (sqrtpow2-1 fnum-vals))))
(local-defthmd sqrtpow2-3
(implies (and (not (specialp))
(= (classa) 4)
(= (mana) 0))
(equal (sig (a)) 1))
:hints (("Goal" :use (sqrtpow2-2) :in-theory (enable siga-rewrite))))
(local-defthmd sqrtpow2-4
(implies (not (specialp))
(equal (expodd)
(if (oddp (expshft)) 1 0)))
:hints (("Goal" :in-theory (enable bitn-rec-0 normalize expodd si))))
(local-defthmd sqrtpow2-5
(implies (and (not (specialp))
(= (classa) 4)
(= (mana) 0))
(equal (a)
(if (= (expodd) 1)
(expt 2 (* 2 (- (expq) (bias (f)))))
(* 2 (expt 2 (* 2 (- (expq) (bias (f)))))))))
:hints (("Goal" :use (a-x) :in-theory (enable si f sp dp hp bias x sqrtpow2-3 sqrtpow2-4))))
(local-defthmd sqrtpow2-6
(implies (and (not (specialp))
(= (classa) 4)
(= (mana) 0))
(equal (qsqrt (a) (1+ (* 2 (n))))
(if (= (expodd) 1)
(expt 2 (- (expq) (bias (f))))
(* (expt 2 (- (expq) (bias (f))))
(qsqrt 2 (1+ (* 2 (n))))))))
:hints (("Goal" :use (sqrtpow2-5 fnum-vals
(:instance qsqrt-shift (x 1) (k (- (expq) (bias (f)))) (n (1+ (* 2 (n)))))
(:instance qsqrt-shift (x 2) (k (- (expq) (bias (f)))) (n (1+ (* 2 (n))))))
:in-theory (enable n f sp dp hp bias x))))
(local-defthmd sqrtpow2-7
(implies (and (not (specialp))
(= (classa) 4)
(= (mana) 0))
(equal (rnd (qsqrt (a) (1+ (* 2 (n)))) (mode) (p))
(if (= (expodd) 1)
(expt 2 (- (expq) (bias (f))))
(* (expt 2 (- (expq) (bias (f))))
(rnd (qsqrt 2 (1+ (* 2 (n)))) (mode) (p))))))
:hints (("Goal" :use (sqrtpow2-6 fnum-vals
(:instance bvecp-member (x (bits (rin) 23 22)) (n 2))
(:instance rnd-shift (x 1) (mode (mode)) (n (p)) (k (- (expq) (bias (f)))))
(:instance rnd-shift (x (qsqrt 2 (1+ (* 2 (n))))) (mode (mode)) (n (p)) (k (- (expq) (bias (f))))))
:in-theory (enable mode rmode n f sp dp hp bias x p prec))))
(local-defthmd sqrtpow2-8
(implies (and (not (specialp))
(= (classa) 4)
(= (mana) 0))
(equal (ndecode (d-sqrtpow2) (f))
(rnd (qsqrt (a) (1+ (* 2 (n)))) (mode) (p))))
:hints (("Goal" :use (sqrtpow2-7 fnum-vals expq-bounds (:instance bvecp-member (x (bits (rin) 23 22)) (n 2)))
:in-theory (enable mode rmode bvecp sgnf manf expf ndecode d-sqrtpow2 sqrtpow2 sqrtpow2-4 n
f sp dp hp bias x p prec))))
(local-defthmd sqrtpow2-9
(implies (and (not (specialp))
(= (classa) 4)
(= (mana) 0))
(equal (expf (d-sqrtpow2) (f))
(expq)))
:hints (("Goal" :use (fnum-vals expq-bounds)
:in-theory (enable mode rmode bvecp sgnf manf expf ndecode d-sqrtpow2 sqrtpow2 n
normp expw encodingp f sp dp hp bias x p prec))))
(local-defthmd sqrtpow2-10
(implies (and (not (specialp))
(= (classa) 4)
(= (mana) 0))
(encodingp (d-sqrtpow2) (f)))
:hints (("Goal" :use (fnum-vals expq-bounds (:instance bvecp-member (x (bits (rin) 23 22)) (n 2)))
:in-theory (enable mode rmode bvecp sgnf manf expf ndecode d-sqrtpow2 sqrtpow2 n
cat encodingp f sp dp hp bias x p prec))))
(local-defthmd sqrtpow2-11
(implies (and (not (specialp))
(= (classa) 4)
(= (mana) 0))
(normp (d-sqrtpow2) (f)))
:hints (("Goal" :use (fnum-vals sqrtpow2-9 sqrtpow2-10 expq-bounds)
:in-theory (enable normp expw f sp dp hp bias x p prec))))
(local-defthmd sqrtpow2-12
(implies (and (not (specialp))
(= (classa) 4)
(= (mana) 0))
(equal (d-sqrtpow2)
(nencode (rnd (qsqrt (a) (1+ (* 2 (n)))) (mode) (p)) (f))))
:hints (("Goal" :use (fnum-vals sqrtpow2-11 sqrtpow2-8
(:instance nencode-ndecode (x (d-sqrtpow2)) (f (f))))
:in-theory (enable f sp dp hp p prec))))
(local-defthmd sqrtpow2-13
(implies (and (not (specialp))
(= (classa) 4)
(= (mana) 0))
(iff (= (rnd (qsqrt (a) (1+ (* 2 (n)))) (mode) (p))
(qsqrt (a) (1+ (* 2 (n)))))
(= (expodd) 1)))
:hints (("Goal" :use (fnum-vals sqrtpow2-6 sqrtpow2-7
(:instance bvecp-member (x (bits (rin) 23 22)) (n 2)))
:in-theory (enable mode rmode n bias f sp dp hp p prec))))
(local-defthmd sqrtpow2-14
(implies (and (not (specialp))
(= (classa) 4)
(= (mana) 0))
(equal (flags-sqrtpow2)
(if (= (expodd) 1) 0 16)))
:hints (("Goal" :use (fnum-vals)
:in-theory (enable flags-sqrtpow2 sqrtpow2 expodd))))
(local-defthmd sqrtpow2-15
(implies (and (not (specialp))
(= (classa) 4)
(= (mana) 0))
(equal (rz) (rin)))
:hints (("Goal" :use (a-class)
:in-theory (enable rz))))
(local-defthmd sqrtpow2-16
(implies (and (not (specialp))
(= (classa) 4)
(= (mana) 0))
(equal (arm-sqrt-spec (opaw) (rin) (f))
(mv (d-sqrtpow2)
(logior (rin) (flags-sqrtpow2)))))
:hints (("Goal" :in-theory (enable set-flag sqrtpow2-12 sqrtpow2-14 sqrtpow2-15 spec-17)
:use (sqrtpow2-13))))
(local-defthmd sqrtpow2-lemma
(implies (and (not (specialp))
(= (classa) 4)
(= (mana) 0))
(equal (arm-sqrt-spec (opaw) (rin) (f))
(mv (data) (logior (rin) (flags)))))
:hints (("Goal" :use (signa-0-1) :in-theory (enable sqrtpow2-16 data flags specialp))))
(local-defund selmaxnorm ()
(logior1 (logior1 (logand1 (log= (rmode) (rmodedn))
(lognot1 0))
(logand1 (log= (rmode) (rmodeup)) 0))
(log= (rmode) (rmodezero))))
(local-defund ovf-dp () (log>= (si (exprnd) 13) 2047))
(local-defund ovf-d-dp ()
(if1 (selmaxnorm)
(setbits (setbits (setbitn (bits 0 63 0) 64 63 0) 64 62 52 2046) 64 51 0 4503599627370495)
(setbits (setbits (setbitn (bits 0 63 0) 64 63 0) 64 62 52 2047) 64 51 0 0)))
(local-defund unf-dp () (log<= (si (exprnd) 13) 0))
(local-defund unf-d-dp ()
(if1 (fzp)
(setbitn (bits 0 63 0) 64 63 0)
(let* ((exp (bitn (qrndden) 52))
(d (setbits (setbitn (bits 0 63 0) 64 63 0) 64 62 52 exp)))
(setbits d 64 51 0 (bits (qrndden) 51 0)))))
(local-defund norm-d-dp ()
(setbits (setbits (setbitn (bits 0 63 0) 64 63 0) 64 62 52 (si (exprnd) 13)) 64 51 0 (bits (qrnd) 51 0)))
(local-defund ovf-sp () (log>= (si (exprnd) 13) 255))
(local-defund ovf-d-sp ()
(if1 (selmaxnorm)
(setbits (setbits (setbitn (bits 0 63 0) 64 31 0) 64 30 23 254) 64 22 0 8388607)
(setbits (setbits (setbitn (bits 0 63 0) 64 31 0) 64 30 23 255) 64 22 0 0)))
(local-defund unf-sp () (log<= (si (exprnd) 13) 0))
(local-defund unf-d-sp ()
(if1 (fzp)
(setbitn (bits 0 63 0) 64 31 0)
(let* ((exp (bitn (qrndden) 23))
(d (setbits (setbitn (bits 0 63 0) 64 31 0) 64 30 23 exp)))
(setbits d 64 22 0 (bits (qrndden) 22 0)))))
(local-defund norm-d-sp ()
(setbits (setbits (setbitn (bits 0 63 0) 64 31 0) 64 30 23 (si (exprnd) 13)) 64 22 0 (bits (qrnd) 22 0)))
(local-defund ovf-hp () (log>= (si (exprnd) 13) 31))
(local-defund ovf-d-hp ()
(if1 (selmaxnorm)
(setbits (setbits (setbitn (bits 0 63 0) 64 15 0) 64 14 10 30) 64 9 0 1023)
(setbits (setbits (setbitn (bits 0 63 0) 64 15 0) 64 14 10 31) 64 9 0 0)))
(local-defund unf-hp () (log<= (si (exprnd) 13) 0))
(local-defund unf-d-hp ()
(if1 (fzp)
(setbitn (bits 0 63 0) 64 15 0)
(let* ((exp (bitn (qrndden) 10))
(d (setbits (setbitn (bits 0 63 0) 64 15 0) 64 14 10 exp)))
(setbits d 64 9 0 (bits (qrndden) 9 0)))))
(local-defund norm-d-hp ()
(setbits (setbits (setbitn (bits 0 63 0) 64 15 0) 64 14 10 (si (exprnd) 13)) 64 9 0 (bits (qrnd) 9 0)))
(local-defund ovf-flags ()
(setbitn (setbitn (flags-a) 8 (ofc) 1) 8 (ixc) 1))
(local-defund unf-flags ()
(if1 (fzp)
(setbitn (flags-a) 8 (ufc) 1)
(let ((flags (setbitn (flags-a) 8 (ixc) (logior1 (bitn (flags-a) (ixc)) (inxden)))))
(setbitn flags 8 (ufc) (logior1 (bitn flags (ufc)) (inxden))))))
(local-defund norm-flags ()
(setbitn (flags-a) 8 (ixc) (logior1 (bitn (flags-a) (ixc)) (inx))))
(local-defund data* ()
(case (fnum)
(2 (if1 (ovf-dp)
(ovf-d-dp)
(if1 (unf-dp)
(unf-d-dp)
(norm-d-dp))))
(1 (if1 (ovf-sp)
(ovf-d-sp)
(if1 (unf-sp)
(unf-d-sp)
(norm-d-sp))))
(0 (if1 (ovf-hp)
(ovf-d-hp)
(if1 (unf-hp)
(unf-d-hp)
(norm-d-hp))))
(t (bits 0 63 0))))
(local-defund flags* ()
(case (fnum)
(2 (if1 (ovf-dp)
(ovf-flags)
(if1 (unf-dp)
(unf-flags)
(norm-flags))))
(1 (if1 (ovf-sp)
(ovf-flags)
(if1 (unf-sp)
(unf-flags)
(norm-flags))))
(0 (if1 (ovf-hp)
(ovf-flags)
(if1 (unf-hp)
(unf-flags)
(norm-flags))))
(t (flags-a))))
(local (in-theory (disable (ovf-dp) (ovf-d-dp) (unf-dp) (unf-d-dp) (norm-d-dp) (ovf-sp) (ovf-d-sp) (unf-sp) (unf-d-sp) (norm-d-sp)
(selmaxnorm) (ovf-hp) (ovf-d-hp) (unf-hp) (unf-d-hp) (norm-d-hp) (ovf-flags) (unf-flags) (norm-flags)
(data*) (flags*))))
(local-defthmd final-lemma
(and (equal (data-final) (data*))
(equal (flags-final) (flags*)))
:hints (("Goal" :do-not '(preprocess) :expand :lambdas
:in-theory '(ovf-dp ovf-d-dp unf-dp unf-d-dp norm-d-dp ovf-sp ovf-d-sp unf-sp unf-d-sp norm-d-sp ovf-hp
ovf-d-hp unf-hp unf-d-hp norm-d-hp ovf-flags unf-flags norm-flags data* flags* data-final
selmaxnorm flags-final final))))
(local-defthm no-ovf-unf
(implies (not (specialp))
(and (equal (data-final)
(case (fnum)
(2 (norm-d-dp))
(1 (norm-d-sp))
(0 (norm-d-hp))))
(equal (flags-final)
(norm-flags))))
:hints (("Goal" :in-theory (enable exprnd-rewrite final-lemma expw f dp sp hp ovf-dp unf-dp ovf-sp unf-sp
bvecp si ovf-hp unf-hp flags* data*)
:use (fnum-vals expq-bounds))))
(local-defthmd exprnd-bounds
(implies (not (specialp))
(and (> (exprnd) 0)
(< (exprnd) (1- (expt 2 (expw (f)))))))
:hints (("Goal" :in-theory (enable f dp sp hp p prec expw exprnd-rewrite)
:use (fnum-vals expq-bounds))))
(local-defthmd final-1
(implies (not (specialp))
(equal (sgnf (data-final) (f)) 0))
:hints (("Goal" :in-theory (enable f dp sp hp p prec bias no-ovf-unf norm-d-dp norm-d-sp norm-d-hp
ndecode manf sgnf expf expw sigw bvecp si)
:use (fnum-vals exprnd-bounds))))
(local-defthmd final-2
(implies (not (specialp))
(equal (expf (data-final) (f))
(exprnd)))
:hints (("Goal" :in-theory (enable f dp sp hp p prec bias no-ovf-unf norm-d-dp norm-d-sp norm-d-hp
ndecode manf sgnf expf expw sigw bvecp si)
:use (fnum-vals exprnd-bounds))))
(local-defthmd final-3
(implies (not (specialp))
(equal (manf (data-final) (f))
(bits (qrnd) (- (p) 2) 0)))
:hints (("Goal" :in-theory (enable f dp sp hp p prec bias no-ovf-unf norm-d-dp norm-d-sp norm-d-hp
ndecode manf sgnf expf expw sigw bvecp si)
:use (fnum-vals exprnd-bounds))))
(local (in-theory (disable no-ovf-unf)))
(local-defthmd final-4
(implies (not (specialp))
(equal (ndecode (data-final) (f))
(rnd (qsqrt (a) (1+ (* 2 (n)))) (mode) (p))))
:hints (("Goal" :in-theory (enable f dp sp hp p prec bias ndecode expw sigw)
:use (rnd-qsqrt-a final-1 final-2 final-3 fnum-vals))))
(local-defthm final-5
(implies (and (natp n) (natp m) (natp x) (natp y))
(<= (cat x n y m) (+ (* (expt 2 m) x) y)))
:rule-classes ()
:hints (("Goal" :in-theory (enable bits-mod cat))))
(local-defthmd final-6
(implies (not (specialp))
(encodingp (data-final) (f)))
:hints (("Goal" :in-theory (enable encodingp norm-d-dp norm-d-sp norm-d-hp no-ovf-unf f dp sp hp p
cat-0 bvecp prec bias ndecode expw sigw)
:use (fnum-vals
(:instance final-5 (x (bits (si (exprnd) 13) (1- (expw (f))) 0))
(y (bits (qrnd) (- (p) 2) 0))
(m (sigw (f)))
(n 54))
(:instance final-5 (x (bits (si (exprnd) 13) (1- (expw (f))) 0))
(y (bits (qrnd) (- (p) 2) 0))
(m (sigw (f)))
(n 41))
(:instance bits-bounds (x (si (exprnd) 13)) (i (1- (expw (f)))) (j 0))
(:instance bits-bounds (x (qrnd)) (i (- (p) 2)) (j 0))))))
(local-defthmd final-7
(implies (not (specialp))
(normp (data-final) (f)))
:hints (("Goal" :in-theory (enable normp f dp sp hp expw p prec)
:use (fnum-vals exprnd-bounds final-2 final-6))))
(local-defthmd final-8
(implies (not (specialp))
(equal (data-final)
(nencode (rnd (qsqrt (a) (1+ (* 2 (n)))) (mode) (p)) (f))))
:hints (("Goal" :in-theory (enable normp f dp sp hp expw p prec)
:use (fnum-vals final-4 final-7
(:instance nencode-ndecode (x (data-final)) (f (f)))))))
(local-defthmd final-9
(implies (not (specialp))
(equal (data-final)
(nencode (rnd (qsqrt (a) (1+ (* 2 (n)))) (mode) (p)) (f))))
:hints (("Goal" :in-theory (enable normp f dp sp hp expw p prec)
:use (fnum-vals final-4 final-7
(:instance nencode-ndecode (x (data-final)) (f (f)))))))
(local-defthmd final-9-a
(implies (not (specialp))
(equal (data-final)
(mv-nth 0 (mv-list 2 (arm-sqrt-spec (opaw) (rin) (f))))))
:hints (("Goal" :in-theory (enable final-9 spec-18))))
(local-defthmd final-10
(implies (and (not (specialp))
(natp b)
(< b 8))
(equal (bitn (flags-final) b)
(if (= b (ixc))
(logior (bitn (flags-a) (ixc)) (inx))
(bitn (flags-a) b))))
:hints (("Goal" :in-theory (enable bitn-cat bitn-bits rnd-inx-rewrite no-ovf-unf norm-flags))))
(local-defthmd bvecp-flags-final
(implies (not (specialp))
(bvecp (flags-final) 8))
:hints (("Goal" :in-theory (enable no-ovf-unf norm-flags a-flags))))
(local-defthmd final-11
(implies (and (not (specialp))
(natp b)
(< b 8))
(equal (bitn (logior (flags-final) (rin)) b)
(if (= b (ixc))
(logior (bitn (rz) (ixc)) (inx))
(bitn (rz) b))))
:hints (("Goal" :use (bvecp-flags-final)
:in-theory (enable a-flags bitn-cat bitn-bits final-10 rz-flags-a bitn-logior))))
(local-defthmd final-12
(implies (and (not (specialp))
(natp b))
(equal (bitn (logior (flags-final) (rin)) b)
(if (= b (ixc))
(logior (bitn (rz) (ixc)) (inx))
(bitn (rz) b))))
:hints (("Goal" :use (bvecp-flags-final (:instance bvecp-bitn-0 (x (flags-final)) (n b)))
:cases ((< b 8))
:in-theory (enable bvecp a-flags bitn-cat bitn-bits final-10 rz-flags-a bitn-logior))))
(local-defthmd inx-0-1
(implies (not (specialp))
(member (inx) '(0 1)))
:hints (("Goal" :in-theory (enable rnd-inx-rewrite))))
(local-defthmd final-13
(implies (and (not (specialp))
(natp b))
(equal (bitn (logior (flags-final) (rin)) b)
(bitn (mv-nth 1 (mv-list 2 (arm-sqrt-spec (opaw) (rin) (f)))) b)))
:hints (("Goal" :use (inx-0-1 bvecp-flags-final)
:in-theory (enable bitn-logior a-flags final-12 spec-18))))
(local-defthmd final-14
(and (integerp (logior (flags-final) (rin)))
(integerp (mv-nth 1 (mv-list 2 (arm-sqrt-spec (opaw) (rin) (f))))))
:hints (("Goal" :in-theory (enable arm-post-comp arm-sqrt-spec))))
(local-defthmd final-15
(implies (not (specialp))
(equal (logior (rin) (flags-final))
(mv-nth 1 (mv-list 2 (arm-sqrt-spec (opaw) (rin) (f))))))
:hints (("Goal" :use (final-14
(:instance final-13 (b (bit-diff (logior (flags-final) (rin))
(mv-nth 1 (mv-list 2 (arm-sqrt-spec (opaw) (rin) (f)))))))
(:instance bit-diff-diff (x (logior (flags-final) (rin)))
(y (mv-nth 1 (mv-list 2 (arm-sqrt-spec (opaw) (rin) (f))))))))))
(local-defthmd final-16
(implies (not (specialp))
(equal (arm-sqrt-spec (opaw) (rin) (f))
(mv (data-final) (logior (rin) (flags-final)))))
:hints (("Goal" :use (spec-18 final-15 final-9-a))))
(local-defthmd final-17
(implies (not (specialp))
(equal (arm-sqrt-spec (opaw) (rin) (f))
(mv (data) (logior (rin) (flags)))))
:hints (("Goal" :use (signa-0-1 final-16 sqrtpow2-lemma) :in-theory (enable specialp data flags))))
(local-defthmd final-18
(implies (not (specialp))
(mv-let (d r) (arm-sqrt-spec (opaw) (rin) (f))
(and (= d (data))
(= r (logior (rin) (flags))))))
:hints (("Goal" :use (final-17))))
(local-defthm not-specialp-main
(let ((fmtw (+ 1 (expw (f)) (sigw (f)))))
(mv-let (data-spec r-spec) (arm-sqrt-spec (bits (opa) (1- fmtw) 0) (rin) (f))
(implies (not (specialp))
(and (equal (data) data-spec)
(equal (logior (rin) (flags)) r-spec)))))
:hints (("Goal" :use (final-18)
:in-theory (enable opaw fmtw))))
(defthm fsqrt64-main
(let ((fmtw (+ 1 (expw (f)) (sigw (f)))))
(mv-let (data-spec r-spec) (arm-sqrt-spec (bits (opa) (1- fmtw) 0) (rin) (f))
(and (equal (data) data-spec)
(equal (logior (rin) (flags)) r-spec))))
:hints (("Goal" :use (not-specialp-main specialp-main))))
(defthmd lemma-to-be-functionally-instantiated
(let* ((f (case (fnum) (0 (hp)) (1 (sp)) (2 (dp))))
(fmtw (+ 1 (expw f) (sigw f)))
(dnp (bitn (rin) 25))
(fzp (bitn (rin) 24))
(rmode (bits (rin) 23 22)))
(mv-let (data flags) (fsqrt64 (opa) (fnum) fzp dnp rmode)
(mv-let (data-spec r-spec) (arm-sqrt-spec (bits (opa) (1- fmtw) 0) (rin) f)
(and (equal data data-spec)
(equal (logior (rin) flags) r-spec)))))
:hints (("Goal" :use (fsqrt64-main fsqrt64-lemma)
:in-theory (enable f fzp dnp rmode))))
(defmacro ic ()
'(input-constraints opa fnum rin))
(defthm fsqrt64-correct
(implies (input-constraints opa fnum rin)
(let* ((f (case fnum (0 (hp)) (1 (sp)) (2 (dp))))
(fmtw (+ 1 (expw f) (sigw f)))
(dnp (bitn rin 25))
(fzp (bitn rin 24))
(rmode (bits rin 23 22)))
(mv-let (data flags) (fsqrt64 opa fnum fzp dnp rmode)
(let ((r (logior rin flags)))
(mv-let (data-spec r-spec)
(arm-sqrt-spec (bits opa (1- fmtw) 0) rin f)
(and (equal data data-spec)
(equal r r-spec)))))))
:rule-classes ()
:hints (("Goal" :use ((:functional-instance lemma-to-be-functionally-instantiated
(opa (lambda () (if (ic) opa (opa))))
(fnum (lambda () (if (ic) fnum (fnum))))
(rin (lambda () (if (ic) rin (rin))))))
:in-theory (disable arm-sqrt-spec))
("Subgoal 1" :use (input-constraints-lemma))))
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