ii
Cyril
A lb
erg
a
Ro
y
A dler
Ric
har
d
A nder
so
n
Geor
ge
A ndr
ews
Henr
y
Ba
k
er
Stephen
Ba
lzac
Y
ur
ij
Bar
ansk
y
Da
vid
R.
Ba
rto
n
Ger
ald
Baumga
rtner
Gil b
er
R.
Roy
RoyR.
yro0(b)-20c(er)]Tj32.56 0 Tk(a)Tj76.72 0 (Mumga)Tj5.04 0 Tn(k)Tj29.68 0 TuelGerGersoykyGer
Da
vidR.Day
C
o n
t en
t s
1
Ax i
om
F
ea tures
1
1.1
In
tro d uctio
n
t o
Axio
m
iv
CO
NTENTS
3.2
. 4
Sym
b
ols,
V
a
ria
bles,
Assig
nm eb9.31000(4)]TD18.24 [cla 0 Td9q 0.12 0 0 0.12 0 0 cm/R7 gs0 G0 gq8.33333 0 0 8.33333 0 0 cm BT/R81 0.12 Tf1 0 0 -1 160.68 736.08 Tm(iv)Tj/R257 0.12 Tf287.16 0 Td(CO)Tj14q 0.12 0 0 0.12 0 0 cm/R7 gs0 G0 gq8.33333 0 0 8.33333 0 0 cm BT/R81 0.12 t04[000(4)]TD18.24 [cla 0 Td9q 0.12 0 0 0v
vi
CO
NTENTS
7
I nput
Fi
les
an d
Output
St
yles
22
1
7.1
Inp ut
Files
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
2
21
7.2
The
.ax
iom.inp ut
File
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
2
22
7.3
Common
F
ea
t ur
es
of
Using
Outp ut
F
o
rmats
.
.
.
.
.
.
.
.
.
.
.
.
2
23
7.4
Monos
p a
ce
Tw
o-
D imensio
nal
Ma
t hema
t ica
l
F
or
mat
.
.
.
.
.
.
.
2
24
7.5
T
eX
F
oTd(of)Tj11.4 0 T0 Td(23)Tj-318.84 12 Td(7.4)Tj22.92 0 Td(Monos)Tj28.56 0 Td[(p)-1000(a)]TJ10.56 0 Td(ce)Tj12.12 0 Td(Tw)Tj14.16 0 Td(o-)Tj8.28 0 Td[(D)-1000(imensio)]TJ40.32 0 Td(nal)Tj16.68 0 Td(Ma)Tj14.04 0 Td[(t)-1000(hema)]TJ27.12 0 Td[(t)-1000(ic0t(0j14.04 0 Td(ce)Tj12.12 0 Td(Tw)Tj14.16 0 Td(o-)Tj8.28 0 Td[(D)-1000(imensio)]TJ40.32 0 Td(nal)Tj16.68 0 Td(Ma)Tj14.04 0 Td[(ce)Tj12.12 0 Td(Tw)Tj14.16 0 Td(o-)Tj8.28 0 Td[(D)-1000(imensio)]TJ40.32 0 Td(nal)Tj16.68 0 Td(Ma)Tj14.04 0 Td[(ce)Tj12.12 0 Td(Tw)Tj14.16 0 Td(o-)Tj8.28 0 Td[(D)-1000(imensio)]TJ40.32 0 Td(nal)Tj16.68 0 Td(Ma)Tj14.04 0 Td[(ce)Tj12.l)Tj16.683 d(nal)Tj16.68 0 Td(Ma)Tj14.04 0 Td[(t)-1000(hema)]TJ27.12 0 Td[(t)-1000(ic0t(0j14.04 0SM0(ic0327.12 0 TSc24 0 (.)8 0 Td(.i18.84 1pj7.68al)7.12 0 T8.64 0 Td(2)Tj4.92 0 Td(23)Tj-(mat)T(23)Tjula7.8 0 T57.12 0 T8.64 0 Td(2)Tj424 0 Td(F)Tj5[(m5.64 0 Td(or).)Tj7.68 0t)Tj1692 Td(Ma)Tj14.04 0 Td[(ce)Tj12.12 0 Td(Tw)Tj14.16 0 Td(o-)Tj8.28 0 Td[(D)-1000(imensio)]TJ40.32 0 Td(nal)Tj16.68 0 Td(Ma)Tj14.04 0 Td[(ce)Tj12.l)Tj16.683 d(nal)Tj16.68 0 Td(Ma)T Td(nal)Tj16.68 0 Td(Ma)Tj14.04 0 Td[(ce)Tj12.l)Tj16.683 d(nal)Tj16.68 0 Td(Ma)Tj14.04 0 Td[(t)-1000(hema)6TJ27.12 0 Td[(t)-1070(ic0t(0j14.04 Fsio)]TTj16.68 OR8 0 Td1000(ic0tTRAN8 0 32Ma)Tj14.00 Td(Ma)Tj14.04 0 Td[(t)-1000(hema)]TJ27.12 0 Td[(t7T57.12 0 T[(ce)Tj12.l)Tj16.683 d(nal)Tj16.68 0 Td(Ma)T Td(nal)Tj16.68 0 Td(Ma)Tj14.04 0 Td[(ce)Tj12.l)Tj16.683 d(nal)Tj16.68 0 Td(Ma)T Td(nal)Tj16.68 0 Td(Ma)Tj14.04 0 Td[(ce)Tj12.l)Tj16.683 d(nal)Tj16.68 0 Td(Ma)T Td(nal)Tj16.68 0 Td(Ma)Tj14.04 0 Td[(ce)Tj12.lnal
Ma
CO
NTENTS
vii
New
F
o rew
o rd
O
n
Octob
er
1
,
20
01
Axio
m
w
as
withdra
wn
f r
om
th e
ma
rk
et
a
nd
ended
life
a
s
a
commer
cial
pro
duct.
C
hapte r
1
Axi
om
F
eat ures
1.1
In
tro
du
t ion
to
Axiom
W
elco
m e
to
t he
w
or
ld
o
f
A x
i o
m.
W
e
ca
ll
Axiom
a
2
CHAPTE
R
1.
A XIO
M
FEA
TUR E
S
whic
h
w
ould
g
1
.1.
IN TR
ODU CT
ION
TO
AX IO
M
3
liter
ally
d o
zens
of
kinds
of
n
u m
b
er
s
to
co
mp ute
wit h.
Th e
se
r
ang
e
fro
m
v
ar
-
io
us
kind s
o
f
in
00inr
4
CHAPTE
R
1.
A XIO
M
FEA
TUR E
S
i
nve
rs
e(%
)
1
x
+i
0
1
2
x
+2
i
1
2
Ty
pe
:
Un
io
n(M
at
rix
F
ra
ti
on
Po
ly
nom
ia
l
Com
pl
ex
In
te
ger
,.
..)
1.1.4
Hyp
er
Do
Figure
1.1
:
Hyp
er
do
c
o
p
e
n ing
men
u
Hyp
erDo
c
pres
en
ts
y
o
u
wind o
ws
o
1
.1.
IN TR
ODU CT
ION
TO
AX IO
M
5
k
1
.1.
IN TR
ODU CT
ION
TO
AX IO
M
7
d
ra
w(5
*b
es
sel
J(
0,s
qr
t(
x**
2+
y**
2)
),
x=
-2
0..
20
,
y
=-
20
..2
0)
F ig
ure
1.2
:
J
0
(
p
8
CHAPTE
R
1.
A XIO
M
FEA
TUR E
S
1
;
3
x ;
1
5
1
.1.
IN TR
ODU CT
ION
TO
AX IO
M
9
Ty
pe:
Ex
pre
ss
ion
I
nt
ege
r
No
t e
the
use
of
\%"
h er
e.
This
m e
ans
the
v
a
lue
of
t he
la
st
ex
p r
ess
ion
w
e
c
omput ed.
In
t his
case
it
is
the
lo
ng
expres
sion
a
b
o
v
e.
1.1.8
P
a tt
ern
Mat
hi8yTj106P8yTja4.32 0 T(9)TjTj10.08 0 Td(the)Tje
1
0
CHAPTE
R
1.
A XIO
M
FEA
TUR E
S
Using
i
nput
les
and
the
) read
co
m ma
nd ,
y
ou
ca
n
c
rea
t e
y
our
o
I2y
ou)ourourr84 0 TTj20.4 0 ud(the)Tj18.72 0 Td(o)Tjs12 07.2 0 Td(r84 0 TTj20.4 (elev14.886)]TJ12.72an000(XIO]TJ12.72t1C2 96 Td(our)To18.72 01Td(o)Tj4.68 0 Td(I7Tj4rea)Tja4.68 0 Td(Ipplications,4.68530.12 Tf2 Td(our)Then Td(y2.58 0 Td(selectiv14.8310 Td(ou)Te000(e)]TJ12.7[(l(our)Ty18.71)T0 Td[(TUa4.68 0 Td(I[ppd(the)Ty Td(y2.]TJ12.72t00(rea-31 04 0r84 0 TTj20.4 0 ud(the)Tj18.72 0 Td(o)Tjs12 07.3Td(o)Tj4.68 0 Td(ITd(y)T Td(Ineed.(read)34517 0.12 Tf4 0 Td320.1226.28 1.94.68 10.12 Tf27Pj42.84 0 lo)Tjol000(XIO412 Tf27ymor84 0 0M)Tj12.48 pd(the)hi14.88 0 TJ21.6 0 000(XIO412 Tf27AlTd(y)T Td(Igori12 0 2. Td(ou)Tthm12 0 384 0 lo)Tjsg)Tj/R217 0.12 Tf180M)Tj 2. Tdf27All14.886)56 Tcompd2ur)Tonen Td(43. Td(ou) Td(our)Ts18.71)T12 Tf27o1C1)Tj4rea)Tjt00(rea7T12 Tf27Axiom12 03204rea)Tja4.68 0 Td(Ilgey)Tj4.92 a4.688Tj4rea)TjCHAPTE)Tj42.84 0 lo)Tj12 T8CHAPTE)Tj42.84 0 lo)Tjej42.8.12 Tf27wr1C1oury401Td(o)Tjin1C1rea7T12 Tf27Axiom12 03204reaououruag0 Td(M)Tj12.4jej429(c)Tj4.32 0 Td(rea)Tjlled 0 Td(nput)Tj/R8108.12 Tf27Spades
1
2
C
hapte r
2
T
en
F
undamen
tal
Ideas
Axioeas
1
4
CHA P
TER
2.
TE
NNAMENTj1037.0 Td(N)ALj10.7.04 Td(N)IDEASj/R25345.12 Tf13-312.72 27.96d(2.)T12 Tj4.97.70 Td(TE)j4.9
15an/R7 gs0Myth e
1
8
CHA P
TER
2.
TE
N
FUN D
AMENT
AL
IDEAS
Ty
pe
:
Ma
tri
x
Pol
yn
om
ial
F
ra
ti
on
In
te
ger
the
in
t er
preter
1
9
Axio
m's
use
o
f
a
bstra
ct
da
tat
yp
es
clea
rly
separ
ates
the
exp
or
ts
of
a
domain
(wha
t
o
p
er
ations
a
re
dened)
fro
j10.44 a-1TJ1t.88 0 15(domain)Timpleme1.52 4d(6at)Tj12.430.4 0 Td(the)Tj1(h88 0 Td(ations)Tw8 0 Td(j10.44 8.24 0 Td[(yp)-2000ob)-5TJ1ject.88 0 35d(Tj4.92 0 Td(re)Tj13.08 0 27.5.28(wha)Tj8 0 T3(a)Tj4.9epr.72 0 T(ct)Tj12e1.52 9(the)Tj18ed2 0 Td68Tj4.92 0 Td(re)Tj13nd8 0 Td(re)Tj132 0 Td[(p)-3000(er)]TJ1488 0 10(ct)Tj12.2 0 T8(j10.44[64 -1TJ1.4 8 0 11(o)Tj4.9n24 0 Td(of)Tj12a.08 0 T6(ct)Tj12)Tj39.6..52 4d(j10.44 User4 0 Td68Tj4.924 0 6(the)Tj1.48 0 T0.8Tj4.92 0 T7.8Tj4.9j-311.52 3d(re)Tj13c6 0 Td(domain)Tn 0 T8tat)Tj12h 0 Td1re
2
0
CHA P
TER
2.
TE
N
FUN D
AMENT
AL
IDEAS
C
hapte r
3
Starting
Axi
om
W
elco
m e
t o
th e
A x
i o
m
en
vir
onmen
t
for
in
t e
rac
t ivvir
2
2
CHAPTE
R
3.
ST
A R
TING
AXIOM
If
y
ou
a
re
running
Axiom
under
th e
X
W indo
w
System,
there
ma
y
b
e
t
w
o
windo
ws:
t he
c
onso
le
wind o
w
(as
j ust
descr
ib
ed)
a
n d
the
Hyp
e
rDo
c
main
men
u.
Hyp
erDo
c
is
a
m
u ltiple-windo
w
h
yp
ertext
sys
t em
that
lets
y
o
u
view
A x
iom
do
cumen
tatio
n
a
nd
examples
o
n-line,
execute
A x
iom
expre
ssions
,
a
nd
g
ener
ate
g
ra
p hics
.
If
y
o
u
ar
e
in
a
g
ra
2
4
CHAPTE
R
3.
ST
A R
TING
AXIOM
1
+
2
-
3
/
4
*
3
**
2
-
1
1
9
4
T
yp
e:
F
ra
ti
on
In
te
ger
The
a
b
o
v
e
ex
p r
essio
n
is
equiv
alen
t
to
th is
.
(
(1
+
2)
-
((3
/
4
)
*
(
3
*
*
2)
))
-
1
1
9
4
T
yp
e:
F
ra
ti
on
In
te
ger
If
a
n
expr
essio
n
co
n
t a
ins
s
ub
expr
essio
ns
enclo
sed
i n
par
en
theses,
t he
3
.2.
TH E
AXIOM
L
A NGUA
G E
2
5
99
99
99
99
99
Ty
pe
:
Po
si
tiv
eI
nt
ege
r
This
is
the
la
st
r
esult .
%
%(
-1)
99
99
99
99
99
Ty
pe
:
Po
si
tiv
eI
nt
ege
r
This
is
the
r
esult
fro
m
step
n
um
b
er
1.
%
%(
1)
10
00
00
00
000
Ty
pe
:
Po
si
tiv
eI
nt
ege
r
3.2.3
Some
T
yp
es
E
v
e
rything
i n
Axiom
ha
s
a
t
y
p
e.
The
t
yp
e
determines
what
op
era
t io
ns
y
ou
can
p
er
f o
rm
o
n
a
n
ob
ject
a
nd
ho
w
the
ob
j e[
2
6
CHAPTE
R
3.
ST
A R
TING
AXIOM
x
**8
x
8
T
yp
e:
P
ol
yno
mi
al
In
te
ger
Here
a
nega
t iv
e
in
teg
er
exp
o
nen
t
pro d uce
s
a
fra
ct io
3
.2.
TH E
AXIOM
L
A NGUA
G E
2
7
This
giv
es
the
v
a
lue
z
+
3
=5
(a
2
8
CHAPTE
RA yp
ti als anrl=i(b)-2[(yp)-300038 Tm(2)Tjg736.08 Tm(2)Tjiv736.7..22 7e.14 0 3.5n.18.tog14 0 3..22 7ether 0 de5..22 7with 0 de3 A.71.1a736.08 Tm(2)Tj79.04 0 n.9 0as92 0((CHAPTE)Tj[(92 0[(tg)-3007..22 7nme.14 0e3 5n.9 0t.92 0n.9 0The14 0e0.76]TJ19.4.92 0 dela.22 7- 0 d-340.32 8)TT2 739.04 0HAPTE)Tjation 0 de5.5n.18.736.083n.9 0a.14 0 3. Tm(2)Tja736.08 Tm(2)Tjssis79.04210HAPTE
3
.2.
TH E
AXIOM
L
A NGUA
G E
2
9
T
ype
:
F
loa
t
Use
3
0
CHAPTE
R
3.
ST
A R
TING
AXIOM
3.2.6
Ca l
l
i
ng
F
un
ti
ons
As
w
e
sa
w
ear
lier,
when
y
ou
w
an
t
to
add
or
subtra
ct
t
w
o
v
alues,
y
ou
pla
ce
the
ar
it hmetic
op
era
t o
r
\
+"tt
3
.2.
TH E
AXIOM
L
A NGUA
G E
3
1
An
o
p
e
ra
t io
ns
that
r
eturns
a
B
oo
lea
n
v
a
lue
(that
is,
t
rue
o
r
fal
se
)
3
.3.
U SING
AXIOM
AS
A
P
OCK
ET
CA L
CU LA
TO
R
3
3
r
4(3)Tj4.9[(b)-2J27elong 0 Td8.4s7PAS
3
4
CHAPTE
R
3.
ST
A R
TING
AXIOM
3.3.2
T
yp
e
Con
v
3
6
CHAPTE
R
3.
ST
A R
TING
AXIOM
3.3.3
Us eful
F
unti
ons
T
o
obtain
the
a
bsolute
v
alue
o
f
a
n
3
.3.
U SING
AXIOM
AS
A
P
OCK
ET
CA L
CU LA
TO
R
3
7
Ty
pe
:
Po
si
tiv
eI
nt
ege
r
T
es
t s
on
v
alues
ca
n
b
e
done
using
v
a
rious
functions
whic
h
ar
e
gener
ally
mo
re
e
Æ cien
t
than
using
r
elationa
l
o
p
era
t o
rs
s
uc
h
as
=
pa
rticular
ly
if
the
v
a
lue
is
a
ma
t r
ix.
E
xamples
of
s12 Td(ma)Tj13mue
of
thses
3
8
CHAPTE
R
3.
ST
A R
TING
AXIOM
Ty
pe
:
Bo
ol
ean
e
3
.4.
U SING
AXIOM
AS
A
SYMBO
LIC
CALCULA
TO
R
4
1
3.4.2
Com
p l
ex
Num
b
ers
F
o
r
man
y
scien
tic
calc
u la
ti o
ns
r
eal
n
um
b
ers
ar
en' t
suÆcien
t
a
n d
s
u pp
or
t
f o
r
c
omplex
n
um
b
er
s
is
a
lso
re
quired.
Complex
n
um
b
ers
ar
e
ha
ndled
in
a
n
in
tuitiv
e
ma
nn e
r.
A xio
m
us
es
t he
%
i
macr
o
to
r
eprese
n
t
the
squar
%
4
2
CHAPTE
R
3.
ST
A R
TING
AXIOM
Ty
pe:
Com
pl
ex
In
te
ger
f
at
or
(%)
i
3
.4.
U SING
AXIOM
AS
A
SYMBO
LIC
CALCULA
TO
R
4
4
CHAPTE
R
3.
ST
A R
TING
AXIOM
r
adi
x(
3/2
1,
5)
0:0
32
41
2
T
yp
e:
R
41
2
T
yp41T41
3
.4.
U SING
AXIOM
AS
A
SYMBO
LIC
CALCULA
TO
R
4
5
om
pa
tF
ra
ti
on
(%)
6
3
4
6
CHAPTE
R
3.
ST
A R
TING
AXIOM
3
3
.5.
GENERAL
P
OINTS
ABO
U T
A XIO
M
4
7
The
rst
e
xample
s
h o
uld
b
e
rea
d
a
s:
Le
t
x
be
of
ty
pe
P
rim
eF
iel
d(
7)
an
d
ass
ig
n
t
o
it
th
e
val
ue
5
No
t e
that
it
is
o
nl y
p
os
sible
t o
in
v
er
t
no
n -
zer
o
v
alues
if
the
ar
it hmetic
is
p
er
-
fo
rmed
mo
dulo
a
prime
n
u m
b
er
.
Th
u s
ar
it hmetic
mo
dulo
a
no
n -
prime
in
teg
er
is
p
os
sible
bu t
the
r
ecipro cal
op
er
ation
is
unde ned
a
sn
3
.5.
GENERAL
P
OINTS
ABO
U T
A XIO
M
4
9
3.5.4
Com
men
ts
3
.5.
GENERAL
P
OINTS
ABO
U T
A XIO
M
5
1
le.
T
o
g
et
A x
iom
t o
rea
d
t his
le,
y
ou
use
the
system
command
)
rea
d
m
y.
inp
ut
.
If
y
o
u
need
to
m a
k
e
c
h a
nges
to
y
our
appr
oa
c
h
or
deniti o
ns,
g
o
in
t o
y
our
f a
v
or
it e
edit o
r,
c
hang
e
m
y
.i
nput ,
t hen
)re
ad
m
y.i
np
ut
ag
ain.
O
th e
r
sys
t em
co
m ma
nd s
in clude:
)
hi
sto
ry
,
t o
d is
pl a
y
pr
evious
input
a
nd/o
r
o
utp ut
lines;
)d
isp
la
y,
to
displa
y
pro
p
e
rties
and
v
alues
of
w
or
kspa
ce
v
ar
iables;
a
nd
)w
ha
t.
Is
sue
)w
ha
t
to
ge
t
a
li s
t
o
f
Axio
m
o
b
jects
thar5N8.0(e)]TJ10.2 0 Td(0 Td[(li)-[(pl)-1000(a)]TJalues)Tj24.48 0ye12.6 0 T5000(jecwbstriTd(e)42nd)Tj18.9
5
2
CHAPTE
5
6
CHAPTE
R
3.
ST
A R
TING
AXIOM
AXIOMTp11.88 0yTCHAitTCHAteT
3
.6.
D
A
T
A
STR
UCTURES
IN
A XIOM
5
7
r
ev
ers
e(
[7
,2,
-1
,2℄
)
[2;
1;
2;
7
]
T
yp
e:
L
ist
I
nt
ege
r
s
or
t([
7,
2,
-1,
2℄
)
[
1
;
2;
2;
7
]
T
yp
e:
L
ist
I
nt
ege
r
r
em
ove
Du
pl
ia
te
s([
1,
5,
3,5
,1
,1,
2℄
)
[1;
5
;
3;
2
]
Ty
pe
:
Li
st
Po
si
tiv
eI
nt
ege
r
#
[7
,2,
-1
,2
℄
4
Ty
pe
:
Po
si
tiv
eI
nt
ege
r
L
i s
t s
in
A xio
m
ar
e
m
ut a
ble
and
so
th e
i r
co
n
t e
n
t s
( the
elemen
5
8
CHAPTE
R
3.
ST
A R
TING
AXIOM
[9;
2
;
4;
7
;
1
;
5;
4
2]
Typ
e:
Lis
t
Pos
it
iv
eIn
te
ger
e
ndO
fu
:=
r
est
(u
,4
)
[1;
5
;
4
2]
Typ
e:
Lis
t
Pos
it
iv
eIn
te
ger
p
art
Of
u
:
=
res
t(
u,
2)
[4
;
7;
1
;
5
;
42
]
Typ
e:
Lis
t
Pos
it
iv
eIn
te
ger
s
etr
es
t!(
en
dOf
u,
pa
rtO
fu
);
u
9;
2
;
6
0
CHAPTE
R
3.
ST
6
2
CHAPTE
R
3.
ST
A R
TING
AXIOM
[9;
9
9;
20
;
7]
Typ
e:
Lis
t
Pos
it
iv
eIn
te
ger
In
th e
pr
evious
ex
ample
a
new
e6j30.96 0 ad(9)T10.68 0 y(ex)T210.44 0 of(In)21.12 0 constd(pr266 23.08 0 Tu[(th)-1c00(e) Tf9.96 0 Td[(th)-1ing00(e))Tj210.44 0 ld(Lis)Tj9.36 0 Td[(th)-1s00(e)]Tj11.88 0 Td(e6j42.84 0 a(Pos)2j30.96 0 gd(9)Tj4.92 0 Td(iv Tf7.8 0 end(3.))T24.04 0 T(iv T13.08 0 T[(th)-1is00(e)]6Tf4.32 0 d(Lis)Tj10.68 0 ad(9-338.76 1 0. 0 Tp)-2h)-1o00(e)]0.561.88 0 Td(e6j42.84 0 erful(pr236 23.08 0 Tmetho)-2h)-1d00(e)36j42.84 0 whic(3.))T561.88 0 h(Lis)T15.48 0 gTd(iv)2j15.72 0 T(Pos)2j249.36 0 Td[(th)-1h000(e)]TJ7.n9
3
.6.
D
A
T
A
STR
UCTURES
IN
A XIOM
6
3
3.6.2
Segmen
ted
Li
sts
A
se
gmen
ted
list
is
one
in
whic
h
so
me
of
the
elemen
ts
ar
e
ra
nges
of
v
alues.
The
e
x pand
fun c
t io
n
con
v
er
t s
lists
o
f
th is
t
yp
e
in
to
or
dinary
lists:
[
1.
.10
℄
[1::10
]
Ty
pe:
Lis
t
3
.6.
D
A
T
A
STR
UCTURES
IN
A XIOM
6
5
T
o
crea
te
t he
s
eries
the
windo
w
is
p la
ced
at
the
star
0
3
.6.
D
A
T
A
STR
UCTURES
IN
A XIOM
6
7
y
:
=
xy36 0 0 Td(:)Tj5.16 0pe Td(26)Tj10.56 On Td(:)Tj5.16 0eDi Td(5)Tj6.36 0me Td(:)Tj5.16 0nsi Td(5)Tj6.36 0on Td(:)Tj5.16 0alA Td(5)Tj6.36 0rr Td(:)Tj5.16 0ay Td(5)Tj6.36 0Po Td(:)Tj5.16 0si Td(:)Tj5.16 0tiv Td(5)Tj6.36 0eI Td(:)Tj5.16 0nt Td(:)Tj5.16 0ege Td(5)Tj6.36 0r Td96 Td4 32.44 0 Td=
5
6
8
CHAPTE
R
3.
ST
A R
TING
AXIOM
s
wap
!(
b,2
,3
);
b
[2;
4
;
3;
5
;
6]
T
yp
e:
O
ne
Dim
en
sio
na
lA
rra
y
Pos
it
iv
eIn
te
ger
opy
In
to!
(a
,b,
3)
[4;
4
;
2
lArr73r3;
4
2
;
4
;
2
5
;
6]
T
yp
e:
O
ne
Dim
en
sio
na
lA
rra
y
3.30 Td(a 0.12 Tf4.44 0 T124.25.72 0 Td(In)Tj10.44 0 Td(to!)Tj15.72 0 Td((a)Tj10.44 0 Td(,b,)Tj15.72 0 Td(3))Tj/R81 0.12 Tf51 28.56 Td([4)Tj/R24 0.12 Tf7.8 0 Td(;)Tj/R81 0.12 Tf4.44 0 Td(4)Tj/R24 0.12 Tf4.92 0 Td(;)Tj/R81 0.12 Tf4.44 0 Td(2)Tj/R24 0.12 Tf)Tj10.44 0 Td(lA)Tj10.44 0 Td(rr73r3;)Tj/R84 0.12 Tf4.92 0 Td(;)Tj/R81 0.12 Tf4.44 0 Td(4)Tj/R26 0.12 Tf4.92 0 Td(;)Tj/R81 0.12 Tf4.44 0 Td(2)Tj/R24 0.12 Tf4.92 0 Td(;)Tj/R81 0.12 Tf4.44 0 Td(4)Tj/R24 0.12 Tf4.92 0 Td(;)Tj/R81 0.12 Tf4.44 0 Td(2)Tj/R24 0.12 Tf5.04 0 Td87.6/R8880.12 Tf4.44 0 Td(5)Tj/R24 0.12 Tf4.92 0 Td(;)Tj/R81 0.12 Tf4.44 0 Td(6℄)Tj/R83 0.12 Tf-64.08 23.76 Td(T)Tj5.16 0 Td(yp)Tj10.44 0 Td(e:)Tj21 0 Td(O)Tj5.16 0 Td(ne)Tj10.44 0 Td(Dim)Tj15.72 0 Td(en)Tj10.44 0 Td(sio)Tj15.72 0 Td(na)Tj10.44 0 Td(lA)Tj10.44 0 Td(rra)Tj15.72 0 Td(y)Tj10.44 0 Td(Po4 0 (yv81 0.12 Tf4.44etj10.44 0 Td(rrorj/R83 0.12 Tf-3([1j10.44 0 Td(rr/24 0 Td(ger)Tj-31/.96 36.24 Td()T,4 0 Td(ger)Tj-1/.96 3d(ger)Tj-14 0.12 Td(te)Tj105.16 03625.04 0 Td3881 019(geTj1 0.12 Tf4.44 0 T68 737r3;)Tj-1 0.1ETQq 4-4 2630.9 3452.9 BI/IM true/W 1/H 1/BPC 1ID EI Q3 Td(t40481 0.m/R24 0.12 Tf7.8 0 TPos)T-60 m/R81 0.12 Tf4.44 0 T5.6)T-607;)Tj-1 0.1ETQq 4-4 27APT9 3452.9 BI/IM true/W 1/H 1/BPC 1ID EI Q327.640481 0.m/R4 0.12 Tf)Tj10.4Po4 0-60 m/R81 0.12 Tf4.44 0 T812 T-607;)Tj-1 0.1ETQq m B-4 28 0 T 3452.9 BI/IM true/W 1/H 1/BPC 1ID EI Q339.30 40481 0.m/R1 0.1d(;)Tj/R845.16 03625.04 0 TdPos)T-208880.12 0.12 Tf5.04 0 Td3 48 73.12 Tf4.44 0 Td(5)Tj/R24 0.12 Tf4.92 0 Td(;)Tj/V81 0.12 Tf4.44e44 0 Td(eIn)Tj15r 0 Td(;)Tj/F81 0.12 Tf4.44raj10.44 0 Td(rrai44 0 Td(eIn)TjoTj10.44 0 Td(rr0.44 0 Td(eIn)Tj1)Tj15.72 0 Td(y)Tj10.44 0 Td(3.30 Td("81 0.12 Tf4.44Helj10.44 0 Td(rrlo4 0 Td(ger)Tj-34 0 Td5 Tf4.44W72 0 Td(it)Tj1rj/R83 0.12 Tf-ld"81 07781 0.72 0 Td("H/R24 0.12 Tf4.9lj/R83 0.12 Tf-lo4 0 Td(ger)Tj-34 0 Td5 Tf4.44W72 0 Td(it)Tj1rj/R83 0.12 Tf-ld"81 083(;)/R8880.12 Tf4.44 0 Td(5)Tj/R24 0.12 Tf4.92 0 Td(;)Tj/Sf4.44 0 Td(5)trj/R83 0.12 Tf-ing10.44 0 Td(Pos)Tj15bf4.44 0 Td(5)itTj10.44 0 Td(si 0 mTd(ger)Tj-3trj/R844 0 Td(siu)Tj15.72 0 Td(y05.1688.25.72 0 Td("1.96 3d(ger)Tj-11.96 3d(ger)Tj-111j10.44 0 Td(rr11.96 3d(ger)Tj-"81 0938 73/R81 0.12 Tf4.44 0 Td(5)Tj/R24 0.12 Tf4.92 0 Td(;)Tj/Bf4.44 0 Td(5)itTj1044 0 Td2681 0.12 )Tj5.4.92 0 Tdyv81 0d(;)Tj/R[4etjTj/R2o7 0.12781 0f4.44r5.1683.)TTd(5)isTj5.4 0 Td(5)similar5.16(3.8(eIn)Tj15Tj5.3r3;)Tj/R8j10.9.30 it)Tj181 0d(;)Tj/R[4njTj/R2e-7 0.123r3;)Tj/R[(d.12Tj/R2e7 0.1d(;)Tj[4njTj/R2s7 0.198 736.08 Tionalj/R250.12 Tf4.ar5.1683.92 0 TdyTj10.8.6)Td(Dim)Tj1964.08 23.76xj10. Tf179.04 eptj10.8.30 it)Tjthaj10. T.)TTd(5)t10.8..12 Tf-if/R24 02 Td(5)itTj1015 Td(5)o10.9.30 it)Tj[(mp)-3/R2o7 0.129)Tjf4.44 0Tj10.442(eIn)Tj1s10.44353.92.4.92)Tj[(b)-20/R2elong 0.131!eIn.8(eIn)Tjing10.16 Tf1 0 0 th0Tj1022 Tf1 0 0 a81 0d(;)Tj/R[4rithmetjTj/R2i 0.143.;
3
.6.
D
A
T
A
STR
UCTURES
IN
A XIOM
6
9
3.6.5
Flexi
bl
e
Ar
ra
ys
Flexible
ar
ra
ys
ar
e
desig
ned
to
pr
o
v
id e
th e
eÆciency
o
f
one-
d imensio
nal
a
rr
a
ys
while
r
ri-T39(pr)Tj9.3the 0 Td[1r
7
0
CHAPTE
R
3.
ST
A R
TING
AXIOM
T
ype
:
F
lex
ib
leA
rr
ay
In
te
ger
d
ele
te
!(f
,5
)
[4
;
3;
4
2;
8;
2
;
28
]
T
ype
:
F
lex
ib
leA
rr
ay
In
te
ger
g
:=f
(3
..5
)
[42
;
8;
2]
T
ype
:
F
lex
ib
leA
rr
ay
In
te
ger
g
.2:
=7
;
f
[4
;
3;
4
2;
8;
2
;
28
]
T
ype
:
F
lex
ib
leA
rr
ay
In
te
ger
i
7
2
CHAPTE
R
3.
ST
A R
TING
AXIOMTING
3
.7.
FU NCTIO
N S,
CHO
ICES,
AND
LOO
PS
7
3
:=a
+
b
)
2:8
28
42
71
24
7
4
61
90
09
76
T
ype
:
F
loa
t
No
t e
tha
t
inden
tatio
n
i s
ext rem
el
y
imp
or
t a
n
t.
If
the
exa
mple
ab
o
v
7nTu.12 Tf9. .
7
4
CHAPTE
R
3.
ST
A R
TING
AXIOM
Er
ro
r
A:
Mi
ss
in
g
m
at
e.
Li
ne
2:
a:
=3.
0
Li
ne
3:
b:
=1.
0
Li
ne
4:
:
=a
+
b
Li
ne
5:
Li
ne
6:
)
...
..
..
..A
Er
ro
r
A:
(f
ro
m
A
u
p
to
A)
I
gno
re
d.
Er
ro
r
A:
Im
pr
op
er
sy
nta
x.
Er
ro
r
A:
sy
nt
ax
er
ro
r
a
t
to
p
l
ev
el
Er
ro
r
A:
Po
ss
ib
ly
mi
ssi
ng
a
)
5
e
rro
r(
s)
pa
rs
ing
a
similar
er
ro
r
will
b
e
r
aise
d .
F ina
lly
,
the
\
) "
m
3
.7.
FU NCTIO
N S,
CHO
ICES,
AND
LOO
PS
7
5
3
:0
T
ype
:
F
loa
t
b
:=
1.0
1
:0
T
ype
:
F
loa
t
:=
a
+
b
4
:0
T
ype
:
F
loa
t
s
qr
t(4
.0
+
)
2:8
28
42
71
24
7
4
61
90
09
76
T
ype
:
F
loa
t
whic
h
a
c
hiev
es
the
same
r
esult
and
is
eas
ier
to
under
st a
7
6
CHAPTE
RA
3
.7.
FU NCTIO
N S,
CHO
ICES,
AND
LOO
PS
7
7
with
s
ome
in
v
o
ca
t io
ns
o
f
t hes
e
fun c
t io
ns:
f
()
Com
pi
li
ng
fu
nt
io
n
f
w
it
h
t
yp
e
(
)
->
Li
st
In
te
ge
r
[
]
T
yp
e:
L
ist
I
nt
ege
r
g
(4
)
Com
pi
li
ng
fu
nt
io
n
g
w
it
h
t
yp
e
I
nt
eg
er
->
Li
st
I
nte
ge
r
[4
]
T
yp
e:
L
ist
I
nt
ege
r
h
(2
,9)
Com
pi
li
ng
fu
nt
io
n
h
w
it
h
t
yp
e
(
7
8
CHAPTE
R
3.
ST
A R
TING
AXIOM
p
:
I
nte
ge
r
-
>
In
teg
er
T
yp
e:
V
oid
p
x
==
(a
8
0
CHAPTE
R
3.
ST
A R
TING
AXIOM
i
:=1
r
epe
at
if
i
>
4
the
n
br
eak
ou
tp
ut(
i)
i:
=i
+1
the
)read
yields:
i
:=1
1
Ty
pe:
Pos
it
iv
eIn
te
ger
r
epe
at
if
i
>
4
the
n
br
eak
ou
tp
ut(
i)
i:
=i
+1
1
2
3
4
T
yp
e:
V
oid
It
w
as
m e
n
t io
3
.7.
FU NCTIO
N S,
CHO
ICES,
AND
LOO
PS
8
1
0
Ty
pe:
No
nNe
ga
tiv
eI
nt
ege
r
r
ep
eat
i
:=
i
+
1
i
f
i
>
6
th
en
br
ea
k
8
2
CHAPTE
R
3.
8
4
CHAPTE
R
3.
ST
A R
TING
AXIOM
T
ypd(T)Tj.4 0 Tde:G
3
.7.
FU NCTIO
N S,
CHO
ICES,
AND
LOO
PS
8
5
4
Ty
pe
:
Po
si
tiv
eI
nt
ege
r
r
:
=
1
1
Ty
pe
:
Po
si
tiv
eI
nt
ege
r
w
hi
le
r
<=
la
st
row
r
ep
eat
:=
1
-
-
I
nd
ex
of
f
irs
t
ol
um
n
w
hil
e
<=
la
st
ol
r
epe
at
if
e
lt
(m,
r,
)
<
0
the
n
ou
tp
ut
[r
,,
el
t(
m,r
,
)℄
r
:=
la
st
row
br
ea
k
-
-
3
.7.
FU NCTIO
N S,
CHO
ICES,
AND
LOO
PS
8
7
Typ
e:
3
.8.
N UMBE
RS
8
9
the
)read
yields:
f
or
a
in
1
..4
f
or
b
in
8.
.5
by
-
1
r
ep
ea
t
o
utp
ut
[
a,b
℄
[1,
8℄
[2,
7℄
[3,
6℄
[4,
5℄
Typ
e:
Voi
d
No
t e
t ha
t
wit ho
ut
the
\
b
y
-
1"
t he
segmen
t
8
..5Typ
9
0
CHAPTE
R
3.
ST
A R
TING
AXIOM
T
yp
e:
F
at
or
ed
In
te
ger
In
teg
ers
ca
n
a
lso
b
e
di s
pla
y
ed
t o
ba
ses
o
t her
t ha
n
1
0.
This
is
an
in
teger
in
bas
e
1
1.
r
adi
x(
259
37
424
60
1,
11)
1
00
00
00
00
00
Ty
pe
:
Ra
dix
Ex
pa
nsi
on
11
Roma
n
n
umera
ls
a
rTj27.12 ma
n
n
umera
3
.8.
N UMBE
RS
9
1
Ty
pe
:
Si
ngl
eI
nt
ege
r
Ma
c
hine
double-pr
ecision
oa
tin g
-p
oin
t
n
um
b
ers
a
re
a
lso
a
v
aila
b le
for
n
umeric
a
nd
gr
aphica
l
applica
t io
ns.
1
23
.21
D
ou
ble
Fl
oat
12
3:2
10
00
000
00
00
01
Ty
pe:
Dou
bl
eF
loa
t
The
nor
mal
o
ating-
p
o
in
t
t
yp
e
in
A x
iom,
Fl
oa
t,
is
a
so
f t
w
ar
e
implemen
tation
o
f
o
ating-
p
o
in
t
n
u m
b
er
s
in
whic
h
the
exp
onen
t
and
the
m a
n
tissa
m a
y
ha
v
e
a
n
y
9
2
CHAPTE
R
3.
ST
A R
TING
AXIOM
d
igi
ts
(40
);
ex
p(
%p
i
*
s
qrt
1
63
.0)
2
62
53
74
1
2
64
07
68
74
3:9
99
99
99
99
9
992
50
07
25
9
76
Ty
pe
:
Fl
oat
Here
ar
e
co
mpl e
x
n
um
b
ers
with
ra
tional
n
um
b
er
s
3
.8.
N UMBE
RS
9
3
u
+
v
i
Typ
e:
C
om
ple
x
Po
lyn
om
ial
I
nt
ege
r
O
f
cour
se,
y
ou
can
do
co
m plex
ar
it hmetic
with
these
also
.
%
*
*
2
v
2
+
u
2
+
2
u
v
i
Typ
e:
C
om
ple
x
Po
lyn
om
ial
I
nt
ege
r
E
v
e
ry
r
ationa
l
n
u m
b
er
haor6 Tf105 26.16 T8ehaor6 Tf105
3
.8.
N UMBE
RS
9
5
Since
7
is
prime,
y
ou
can
in
v
er
t
nonzer
o
v
a
lues.
1
/x
3
T
yp
e:
P
rim
eF
ie
ld
7
Y
o
u
can
also
co
m pute
mo
dulo
an
i n
t eg
er
th a
t
is
n o
t
a
prime.
y
:
I
nt
eg
erM
od
6
:=
5
5
T
yp
e:
I
nte
ge
rM
od
6
All
o
f
th e
usual
ar
i thmetic
o
p
er
ations
ar
e
a
v
aila
bl e
.
y
**
3
5
T
yp
e:
I
nte
ge
rM
od
6
In
v
er
sion
is
noTd(eI(e:)Tj21 0 (e:)Tj4b:)Tj21Td(5)Tj/R81 0.12yp
Qd(5)Tj/RnfTodnoT1
9
6
CHAPTE
R
3.
ST
A R
TING
AXIOM
This
denes
a
to
b
e
a
n
alg
ebra
i c
n
um
b
e
r,
that
is,
a
3
.9.
D
A
T
A
STR
UCTURES
9
7
2
/%
+1
0
a
4
a
3
+
2
a
2
a
+
1
b
3
+
a
4
a
3
+
2
a
2
a
+
1
b
2
+
a
4
a
3
+
2
a
2
a
+
1
b
+
a
4
a
3
+
2
a
2
a
+
3
1
A
0
a
4
a
3
+
2
a
2
a
+
1
b
3
+
a
4
a
3
+
2
a
2
a
+
1
b
2
+
a
4
a
3
+
2
a
2
a
+
1
b
+
a
4
a
3
+
2
a
2
a
+
1
1
A
Ty
pe:
Ex
pre
ss
ion
I
nt
ege
r
B
u t
w
e
need
t o
r
ationa
9
8
CHAPTE
R
3.
ST
A R
TING
AXIOM
u
:=
[
1,-
7,
11℄
3
.9.
D
A
T
A
STR
UCTURES
9
9
1;
7;
1
1;
9
T
yp
e:
L
ist
I
nt
ege
r
A
st
r
e
a m
is
a
structure
that
(p
o
t e
n
t ia
lly)
h a
s
a
n
innite
n
um
b
er
of
disti nct
e
l e
m en
t s
.
Think
o
f
a
strea
m
as
an
\
in nite
list"
where
elemen
ts
a
re
co
mpu ted
s
uccessiv
ely
.
Cr
ea
t e
an
in nite
strea
m
of
f a
ct o
re
d
in
teger
s.
Only
a
certa
i n
n
um
b
1
00
CHAPTE
R
3.
ST
A R
TING
AXIOM
O
n e
-dim e
n s
ional
ar
ra
ys
a
re
also
m
u ta
ble:
y
ou
c
an
c
hang
e
their
cons
t ituen
t
elemen
ts
\in
pla
ce."
a
.3
:=
11
;
a
1
;
7
;
11
;
3
2
Ty
pe
:
On
eD
ime
ns
ion
al
Ar
ray
F
ra
ti
on
In
te
ger
Ho
w
ev
er,
one-
dim e
n s
ional
ar
ra
ys
a
re
not
exible
str
u c
t ur
es.
Y
o
u
canno
t
d e
-
str
u c
t iv
ely
ona t!
them
tog
et her
.
on
at
!(a
,o
neD
im
en
sio
na
lAr
ra
y
[1,
-2
℄)
T
he
re
ar
e
5
e
xp
ose
d
and
0
u
nex
po
sed
l
ibr
ar
y
ope
ra
tio
ns
n
ame
d
on
a
t!
hav
in
g0.12 Tf-308.28ysedOn
3
.9.
D
A
T
A
STR
UCTURES
10
1
A
exi blea rr
3
.9.
D
A
T
A
STR
UCTURES
10
3
Typ
e:
Mul
ti
set
I
nt
ege
r
A
t
a ble
i s
conceptually
a
set
of
\
k
ey{
v
alue"
p a
irs
and
is
a
gener
aliza
t io
n
of
a
m
u ltiset.
F
or
e
xamples
of
ta
b le
s,
s
1
04
CHAPTE
R
3.
ST
A R
TING
AXIOM
d
ani
el
:
R
eo
rd
(a
ge
:
In
te
ge
r,
sa
lar
y
:
3
.10
.
E
X P
AN DING
TO
H IGHER
D IME
N SIO
N S
10
5
3.10
Expa n
ding
to
Higher
Di mensions
T
o
g
et
higher
dim e
n s
ional
ag
gr
ega
t e
s,
y
3
.11
.
WRITING
Y O
UR
O
W N
FUNCTIONS
10
7
n
u m
b
er
s
a
s
co
eÆcien
ts.
Mor
eo
v
er,
t he
libr
ar
y
pro
vides
a
w
1
08
CHAPTE
R
3.
ST
A R
TING
AXIOM
This
f unction
is
less
3
.11
.
WRITING
Y O
UR
O
W N
FUNCTIONS
10
9
Ty
pe
:
Po
si
tiv
eI
nt
ege
r
The
libr
ar
y
v
er
sio
n
use
s
an
a
lgo
rithm
that
is
dieren
1
10
CHAPTE
R
3.
ST
A R
TING
AXIOM
Cr
eate
a
n
exa
mpl e
matrix
to
p
er
m
ut e.
m
:=
m
atr
ix
[
[4
*i
+
j
for
j
i
n
1
..
4℄
fo
r
i
i
n
0..
3℄
2
6
6
4
1
2
3
4
5
6
7
8
9
10
1
1
1
2
13
14
1
5
1
6
3
7
7
5
T
ype
:
Ma
tr
ix
In
te
ger
In
ter
c
hang
e
the
se
cond
a
nd
3
.11
.
WRITING
Y O
UR
O
W N
FUNCTIONS
11
1
1
:0
T
ype
:
F
loa
t
Her
e
w
e
dene
o
u r
o
wn
(user
-dened)
f unction.
os
inv
(y
)
==
o
s(1
/y
)
Typ
e:
Voi
d
P
a
ss
t his
f unction
a
s
an
a
rg
um en
t
t o
t.
t
(
osi
nv
,
5.2
05
8)
1:
3
3
.12
.
P
OL
YNOMIALS
11
3
m
:
MP
OL
Y(
[x,
y℄
,IN
T)
:
=
(
x*
*2-
x*
y**
3+
3*
y)*
*2
x
4
2
y
3
x
3
+
y
6
+
6
y
x
2
6
y
4
x
+
9
y
2
Ty
pe
:
Mu
lt
iva
ri
ate
Po
lyn
om
ia
l([
1
14
CHAPTE
R
3.
ST
A R
TING
3
.14
.
SE
R IE
S
11
5
l
im
it(
sq
rt
(y*
*2
)/y
,y
=
0)
[l ef
tH
andLi
mit
=
1
;
r
ig
htH
andL
imi
t
=
1]
Ty
pe:
Uni
on
(R
eo
rd
(le
ft
Ha
ndL
im
it:
Uni
on
(O
rde
re
dCo
mp
le
tio
n
Ex
pr
ess
io
n
Int
eg
er,
"f
ail
ed
")
,ri
gh
tHa
nd
Li
mit
:
U
nio
n(
Ord
er
ed
Com
pl
eti
on
E
xpr
es
sio
n
Int
eg
er
,"f
ai
led
")
),
...
)
AsTy6Td())Tj/R81 0.915.72 0pprTd(s)Tj
1
16
CHAPTE
R
3.
ST
A R
TING
AXIOM
T
yp
e:
U
ni
var
ia
te
Pui
se
uxS
er
ies
Tnr
te
3
.14
.
SE
R IE
S
11
7
f
*
*
2
1
+
2
1
18
CHAPTE
R
3.
ST
A R
TING
AXIOM
E
v
a
luate
the
serie
s
a
t
the
v
3
.15
.
DERIV
A
TIVES
11
9
Y
o
u
ca
n
als
o
comput e
par
tial
d er
iv
ativ
es
b
y
sp
ecifying
t he
or
der
of
diere
n
t ia
-
tio
n .
g
:
=
s
in
(x
**2
+ Td(n)Tj5.28 -1000(0/R81 0.12 Tf262.2 0 Td(11)Tj9.96 0 Td(9)Tj-338.64 2 0 27(x)Tj2
1
20
CHAPTE
R
3.
ST
A R
TING
AXIOM
Y
ou
c
an
use
F,
x ,
and
y
in
expr
essio
ns.
a
:=
F
(x
z,
y
z,
z
**2
)
+
x
y
(z
+1)
x
(y
( z
+
1))
+
F
x
(z
0 Td(+)Tj10.567[(; 0 4Td[yA)-10 0 cm BT/R81 12 Tf Tf6.84 0 Td(()Tj/R24 0.12 Tf3.96 0 Td(z)Tj/R815sio
0 Td(+)Tj10.567[(; 0 4Td[zA)-10 0366
F
3
.16
.
INTEGRA
TION
12
1
0
B
B
B
B
2
z
2
+
2
z
1
22
CHAPTE
R
3.
ST
A R
TING
AXIOM
omp
le
xIn
te
gra
te
(1
/(x
**
2
+
a
),
x)
lo
g
x
p
a+a
p
a
log
x
p
a
a
p
a
2
p
a
T
yp
e:
E
xp
res
si
on
In
te
ger
The
f o
llo
wing
t
w
o
examples
illu s
t r
ate
t he
limit a
t io
ns
o
f
table-ba
sed
appro
ac
hes.
The
t
w
o
in
tegr
ands
ar
e
v
ery
similar
,
but
the
a
nsw
er
to
one
of
them
r
equires
the
a
d dition
of
t
w
o
new
alg
eb r
aic
n
um
b
ers.
This
o
ne
is
t he
e
asy
o
ne.
The
next
one
lo
o
ks
v
ery
s
i mila
r
but
the
a
nsw
er
is
m
uc
h
more
co
m plica
t ed.
i
nte
gr
ate
(x
**3
/
(
a+b
*x
)**
(1
/3
),x
)
12
0
b
3
x
3
13
5
a
b
2
x
2
+
1
62
a
2
b
x(3)Tj/R395 0.12 T 0 i72 Tf6.72 3.72 T4(b)Tj/R366 0.12 1
62
3
.16
.
INTEGRA
TION
12
3
c
onclusiv
e
ly
pr
o
v
es
that
an
in
tegr
al
canno
t
b
e
expres
sed
i n
terms
o
f
elemen
tar
y
functions.
When
A x
iom
returns
a
n
i n
t eg
ra
l
sign,
it
has
pr
o
v
ed
th a
t
no
answ
er
exists
a
s
an
e
l e
m en
t a
ry
f unction.
i
nt
egr
at
e(
log
(1
+
sq
rt
(a*
x
+
b
))
/
x,
x)
Z
x
log
p
b
+
%Q
a
+
1
%Q
d%Q
T
yp
e:
U
nio
n(
Exp
re
ss
ion
I
nte
ge
r,
...
)
Axio
m
can
ha
ndl e
complica
t ed
mixed
f unctions
m
u c
h
b
ey
o
n d
wh a
t
y
o
u
can
nd
in
1
24
CHAPTE
R
3.
ST
A R
TING
AXIOM
1.
If
x
=
tan
t
and
g
=
t a
n(t=3
)
then
the
follo
wing
a
lgebr
aic
r
elation
is
t r
3
.17
.
DIFF E
RENTIAL
E
QUA
TIONS
12
5
y
:
=
o
pe
ra
tor
'
y
y
Ty
pe
:
Ba
si
Op
er
ato
r
Her
e
w
e
so
lv
e
a
third
or
d e
r
eq
u a
tion
with
p
olyno
mial
co eÆ c
ien
ts.
d
eq
:=
x
**
3
*
D
(y
x,
x
,
3
)
+
x
**
2
*
D
(y
x,
x
,
2
)
-
2
*
x
*
D(
y
x
,
x
)
+
2
*
y
x
=
2
*
x
**
4
x
3
y
;;;
( x)
+
x
2
y
;;
( x)
2
x
y
;
( x)
+
2
y
(x)
=
2
x
4
T
ype
:
Eq
ua
tio
n
Ex
pre
ss
ion
I
nt
ege
r
s
ol
ve(
de
q,
y,
x
)
h
par
tiu
l
ar
=
x
5
10
x
3
+20
x
2
+4
15
x
;
b
asis
=
2
x
3
3
x
2
+
1
3
.18
.
SOL
U TIO
N
OF
E
QUA
1
28
CHAPTE
R
3.
ST
A R
TING
AXIOM
T
yp
e:
V
oid
Find
the
re
al
r
o
ots
of
S
(1
9)
with
r
ationa
l
ar
it hmetic,
co
rr
ect
to
wit hin
3
.18
.
SOL
U TIO
N
OF
E
QUA
TIONS
12
9
e
qn
s
:
=
[x
**2
-
y
+
z,
x**
2*
z
+
x
**4
-
b
*y,
y
**2
*
z
-
a
-
b*
x℄
*
C
hapte r
4
G
raphis
Figure
4.1
:
1
32
CH AP
TER
4
.
GRAPHICS
13
3
Pl
otting
2D
gra phs
of
1
v
ariable
The
gener
al
fo
rmat
fo
r
dr
a
w
i ng
a
fu nctio
n
dened
b
y
a
for
m
ula
f
(x )
is
:
d
raw
(f
(x
),
x
=
a
..
b,
o ptions)
wher
e
a
: :b
denes
th e
r
ange
o
f
x ,
and
wher
e
1
34
CH AP
TER
4
.
GRAPHICS
Plo
t ti
ng
2D
para m
13
5
Pl
otting
2D
a l
gebrai
urv
es
The
g
enera
l
f o
rma
t
fo
r
dra
wing
a
non-
singular
so
lu tio
n
c
u r
v
e
giv
e
n
b
y
a
p
olyno
mial
o
f
th e
f o
rm
p( x;
y
)
=
0
is:
dr
aw
(p
(x,
y)
=
0,
x,
y
,
ran
ge
==
[
a.
.b,
..d
℄,
op t
i ons)
wher
e
th e
s
econd
a
n d
third
ar
gumen
ts
na
m e
the
rs
t
a
nd
s
econd
indep
e
n den
t
v
a
riables
of
p.
A
r
an
ge
option
is
a
lw
a
ys
g
iv
en
to
designa
te
a
b
ounding
r
ectang
u la
r
r
egio
n
of
the
plane
a
x
b
;
y
d.
Z
ero
or
mor
e
additi o
nal
o
pt io
ns
a
s
des
crib
ed
in
4
.0.1
on
pa
ge
13
6
m a
y
b
e
giv
en.
A
third
k
i nd
o
f
t
w
o-
dim e
n s
ional
g
ra
1
36
CH AP
TER
4
.
GRAPHICS
co
me
to
a
p
o
in
t
(cusp).
Algebr
aica
lly
13
7
adaptiv
e
T
h e
ad
apt
iv
e
o
pti o
n
turns
ada
p tiv
e
plotting
o
n
o
r
o
.
Ada
p tiv
e
plotting
us
es
a
n
1
38
CH AP
TER
4
.
GRAPHICS
13
9
Fig
u r
e
4
.6:
Tw
o-dimensio
nal
co
n
tro
l-panel.
14
1
Pi
k :
k :
1
42
CH AP
TER
4
.
GRAPHICS
ax e
sCol
orD e
f ault
([
o l
or (dar k
bl ue()) ])
sets
o
r
indica
t es
th e
def a
ult
colo
r
o
f
t he
axes
in
a
t
w
o-
dim e
n s
ional
gr
aph
viewp
o
rt.
li
pP
oi
n
ts
darda(ul)
(
[o
1
44
CH AP
TER
4
.
GRAPHICS
regi
on
(
viewp
o rt,
int e
ge r(1) ,
str i
ng ("of f"))
d e
clar
es
whether
g
ra
ph
i nt
e
ger
is
or
is
no
t
to
b
e
displa
y
ed
wit h
a
b
o
undi ng
rectang
le.
rese
t
(vi
e wp
or
1
46
CH AP
TER
4
.
GRAPHICS
p
8
:
=
poi
nt
[.
5,
1℄
$(P
oi
nt
DF
LO
AT)
1
48
CH AP
TER
4
.
GRAPHICS
p
3
:=
pa
st
el
ye
ll
ow(
)
[ Hue:
1
1W
eigh
t:
1
:0]
fr
om8i48 20 TdasTj4.6 te
1
5068 762571 0.12 Tf216.482 0 Td[(CH)-1000(AP)]TJ284.92 0 TdTER068 24.792 0 Td41
15
3
f
or
p
in
lp
r
epe
at
o
mp
on
ent
(g
,p,
po
in
tCo
lo
rDe
fa
ult
()
,l
ine
Co
lor
De
fa
ult
()
,
po
in
tSi
ze
De3f0sCdfafa
inin
lorp
lorplorin
ofa
m(in)Tj15.72 0ak())Tj10.44 0 TdViinine
on
inlorpoinDe3f0sCdDe
,3ze)
De
1
54
CH AP
TER
4
.
GRAPHICS
15
5
Pl
otting
3D
fun tio
ns
of
2
v
a riabl
es
The
gener
al
f o
rma
t
f o
r
dra
wing
a
surfac
e
dened
a
1
56CH AP1t tipara metr ispac ec urv
15
7
Pl
otting
3D
pa ram
etri
surfa es
1
58
15
9
4.0.9
Three- Di
m
en si
ona l
Con
trol
-P
ane l
O
nce
y
oC7ron
1
60
CH AP
TER
4
.
GRAPHICS
ob
je
t:
The
ob
jet
button
indicates
that
the
ro
tation
is
to
o
ccur
with
res
p
ec
t
to
t he
cen
ter
o
f
v
o
l ume
o
f
t he
o
b
ject,
indep
enden
t
of
the
ax
es'
or
igin
p
os
it io
n.
Sa l
e:
A
scaling
t r
ans
f o
rmatio
n
o ccurs
b
y
clic
king
t he
mo
use
1
62
CH AP
TER
4
.
GRAPHICS
BW
co
n
v
er
ts
a
co
lor
viewp
or
t
to
b la
c
k
and
whi te,
or
vic
e-v
e
rsa
.
When
this
b utton
is
selected
th e
con
tr
ol-pa
nel
and
viewp
or
t
switc
h
to
a
n
imm
ut a
ble
colo
rmap
co
mp
o
sed
o
f
a
ra
nge
o
f
gr
ey
sca
le
patterns
or
t iles
t ha
t
ar
e
used
wh e
rev
eTj8.88 s is
neces
sar
y
.
Ligh
t
ta
k
esj8.88 s
16
3
Vi
ew
V
ol
um
e
The
V i
ew
V
o
lum
e
but to
n
c
hang
es
the
co
n
1
16
5
s
etAda pti
v
e3D
(b
o
ol
e 0 Td e(o)Tj4.39[Tjt6 0 Tdrue)D16
16
7
vi
ewS ale
D e
f aul
t
([o
a t
1
68
CH AP
TER68TER
C
1
70
CHAPTE
R
5.
U SING
TYPE
S
AND
M O
D E
S
-
3
3
Ty
pe
:
In
te
ger
Here
w
e
crea
te
a
r
ationa
l
n
um
b
er
bu t
i t
lo
oks
lik
e
the
las
t
res
ult .
5
.1.
TH E
BASIC
I
D E
A
17
1
An
y
doma
in
can
b
e
re
ned
t o
a
su
b
d omai n
b
y
a
mem
b
er
ship
p
re
di
at
e.
A
p
re
di
at
e
is
a
f unction
t ha
t ,
when
a
pp244 0 an
1
72
CHAPTE
R
5
.1.
TH E
BASIC
I
D E
A
17
3
1
74
CHAPTE
R
5.
U SING
TYPE
S
AND
M O
D E
S
P
oly
n o
mial
Squa
reMa
trix(7,Co
mp le
x
In
teg
er)
T
yp
e:
D
om
ain
Another
common
c
atego
ry
is
F
iel
d,
the
cla
ss
o
f
al24 0 4oelds68 0 29 0.12 Tf5A6 0 Td(o)]TJ10eld0 Td(S36y)Tj12.24 0 3 0.12 Tf9a4 0 Td(ss)Tj11Td(An 0.12 Tf9ing 0 Td((om)Tj10with 0 T1 04S36yTd88 Tf9a4 0 40(O)]TJ16.d 0 Td[ditiona-1000(6y)Tj12l28 0 T7,Co)Tj39o6 02Td[era-100004S1O)]TJ16.t 0 Td[i[(n)-101Td(x)Tj8ns68 0 d((o)]TJ10.84 0 Td(x)Tj8or2 0 Tdd(,)Tj5example.28 041Td(x)Tj8a4 0 7Td(o)Tj4eld0 Td(S36y)Tj12ha6 0 Td(om)Tj10s92 0 T9er)Tj38.74 0 3 3,Co
M
5
.1.
TH E
BASIC
I
D E
A
17
5
1.
a
n a
me
(f o
r
e
xample,
R
in
g),
use
d
t o
1
76
CHAPTE
R
5.
U SING
5
.2.
W RITING
TYP
ES
AND
M O
DES
17
7
When
migh
t
y
5
.2.
W RITING
TYP
ES
AND
M O
DES
17
9
If
the
t
y
p
e
it se
lf
ha
s
pa
ren
t hese
s
a
ro
u nd
it
and
w
e
ar
e
no
t
in
t he
cas
e
o
f
the
r
st
exa
mpl e
ab
o
v
e,
t hen
t he
p a
ren
t hes
es
ca
n
usua
ll y
b
e
omitted.
(
2/
3)
Fr
a
tio
n(
Pol
yn
om
ial
I
nte
ge
r)
2
3
T
ype
:
Fr
a
tio
n
Po
lyn
om
ial
I
nt
ege
r
If
t he
t
yp
e
i s
use
d
in
a
d e
clar
ation
and
t he
ar
gumen
t
is
a
single-
w
or
d
t
y
p
e,
in
t eg
er
or
sym
b
o
l,
th e
n
t he
par
en
theses
can
usua
lly
b
e
o
mi tted.
(
d,
f,g
)
:
Co
mp
lex
P
ol
yno
mi
al
In
teg
er
Typ
e:
Voi
e:
1
80
CHAPTE
R
5.
U SING
TYPE
S
AND
M O
D E
S
?
(In
te
ger
),
Ma
tr
ix(
?
(P
ol
yno
mi
al
)),
Sq
ua
reM
at
ri
x(?
,
Int
eg
er
)
( it
re
-
quir
es
a
n
um e
ric
ar
gumen
t)
a
n d
Sq
ua
reM
at
rix
(?
,
?)
a
re
all
in
v
alid.Fhe 0 Td26.5j-340.32 14 0 Td9re
1
82
CHAPTE
R
5.
U SING
TYPE
S
AND
M O
D E
S
Y
ou
can
alw
a
ys
co
m
bine
a
decla
ra
ti o
n
with
an
a
ssig
nmen
t.
When
y
ou
do,
it
is
equiv
ale
n
t
t o
r
st
giv
i ng
a
declar
atio
n
statemen
t,
then
g
i v
in g
a
n
Scan(n76TJ16.68 0 Td4ar)Tj26:0 Td(sine)Tj21I16 05(st)Tj1(nte0 Td(sine)Tj21s(n)Tj96TJ16.68yV[T8f1e)Tj21I16 05(yp)Tj21s(n)Tj9e:Tj8.0n
5
.3.
D E
CLARA
TIO
N S
18
3
(
p,
q,r
)
:
Ma
tr
ix
Po
ly
nom
ia
l
?
Typ
e:
Voi
d
h(ia)Tj10.44 0 Tdhis7.3.72 0 Tdnis8.0.44 0 Tdeg(?)Tj..72 0 Td(tr)Tj138.62 0 Tdrea(?)Tj.3.72 0 Td(l)T6 0.12 0 Tdan(d)T1910.44 0 Tdmaoir(ly)Tj218.48 0 Tdp(N)-100a(E)]TJ10.56 0 Tdrts.(3)Tj/R83 0.12 Tf2-3388 24 0.12 Td)
:l:
d5d3
1
84
CHAPTE
R
5.
U SING
TYPE
S
AND
M O
D E
S
i
x
+
1
7
y
+
4
i
T
yp
e:
M
atr
ix
Po
ly
no
mia
l
Com
pl
ex
In
te
ger
Note
the
dierence
b
et
w
een
this
a
nd
t he
n e
xt
example.
This
is
a
complex
ob
ject
with
p
olyno
m ia
l
r
eal
a
nd
imag
inary
par
ts.
f
:
C
OMP
LE
X
P
OL
Y
?
:
=
(x
+
7
5
.4.
RECO
R DS
18
5
Ty
pe
:
Re
or
d(
a:
I
nt
ege
r,
b:
S
tr
ing
)
T
o
acce
ss
a
co
mp
o
nen
a
5
.4.
RECO
R DS
18
7
Rec
ords
ma
y
b
e
nested
and
the
selec
t o
r
na
mes
ca
n
b
e
shar
ed
at
dieren
t
lev
els.
r
:
R
e
or
d(a
:
R
e
or
d(b
:
In
te
ger
,
:
I
nt
ege
r)
,
b:
I
nte
ge
r)
Typ
e:
Voi
d
The
reco
rd
d
5
.5.
U NIO
N S
18
9
It
is
p
os
sible
to
cre
ate
unio
n s
lik
e
Un
io
n(
Int
eg
er,
P
osi
ti
ve
Int
eg
er)
but
they
a
re
diÆ c
u lt
to
w
o
rk
with
b
e
cause
of
the
o
v
erla
p
in
the
bra
nc
h
t
1
90
CHAPTE
R
5.
U SING
TYPE
S
AND
M O
D E
S
1.
A xio
m
nor
mally
co
n
v
erts
a
r
esult
t o
the
tar
get
v
a
lu e
b
efo
re
pa
ssing
it
t o
t he
funct io
n.
If
w
e
left
the
decla
ra
t io
n
in fo
rmation
o
ut
o
f
t his
functi o
n
d e
nition
th e
n
the
s
a
yBr an
h
5
.5.
U NIO
N S
19
1
3
Ty
pe:
Un
ion
(I
1
92
CHAPTE
R
5.
U SING
TYPE
S
AND
M O
D E
S
5.5.2
Un i
ons
Wi
t
h
Sel
e
tors
Lik
e
r
ecor
ds,
y
ou
ca
n
wr
it e
1
94
CHAPTE
R
5.
U SING
TYPE
S
AND
M O
D E
S
1
;
7:2
;
3
2
;
x
2
;
"w
al
ly"
Typ
e:
Lis
t
Any
When
w
e
ask
for
th e
elemen
ts,
Axio
m
displa
ys
these
t
yp
e
s.
u
.1
1
Ty
pe:
Pos
it
iv
eIn
te
ger
Actually
,
th e
se
ob
jects
b
elong
to
A
ny
but
A x
iom
automa
t ica
lly
5
.7.
CONVERSION
19
5
B
y
default,
3
has
t he
t
yp
e
Po
si
tiv
eI
nt
ege
r.
3
1
96
CHAPTE
R
5.
U SING
TYPE
S
AND
M O
D E
S
x
3
i
4
y
2
z
+
1
2
3
i
7
y
4
x
60
9
i
5
Ty
pe:
Sq
uar
eM
atr
ix
(2
,Po
ly
nom
ia
l
F
ra
t
ion
C
omp
le
x
Int
eg
er)
In
ter
c
hang
e
the
P
oly
no
mia
l
and
the
F
ra
ti
on
lev
e
l s
.
m
2
:
=
m1
::
S
qu
ar
eMa
tr
ix(
2,
FR
AC
PO
LY
CO
MPL
EX
I
NT)
"
4
x
3
i
4
2
y
2
z
+1
2
3
i
y
4
7
x
7
60
9
i
5
#
Ty
pe:
Sq
uar
eM
atr
ix
(2
,Fr
a
tio
n
Pol
yn
om
ial
C
omp
le
x
Int
eg
er)
In
ter
c
hang
e
the
P
oly
no
mia
l
and
the
C
omp
le
x
le
v
els.
m
3
:
=
m2
::
S
qu
ar
eMa
tr
ix(
2,
FR
AC
CO
MPL
EX
PO
LY
I
NT)
"
4
x
3
i
4
2
y
2
z
+1
2
5
.7.
CONVERSION
19
7
c
atego
ry
19
9
F
ur
t her
m o
re
,
unl e
ss
y
o
u
a
re
a
ssig
n ing
an
in
t e
Tm-e.84 0 Tdtore
aaTm-).84 Td(GAIN)Tj/100012 06ug1(Tf22.84 Td3GAIN)Tj/59cm B24.
2
00
CHAPTE
R
5.
U SING
TYPE
S
AND
M O
D E
S
T
yp
e:
F
ra
ti
on
In
te
ger
It
mak
es
sense
t hen
tha
t
t his
is
a
list
o
f
2
02
CHAPTE
R
5.
U SING
TYPE
S
AND
M O
D E
S
Ty
pe
:
Fl
oat
P
erha
ps
w
e
a
5
.9.
P
A
CK
A
GE
CALLING
AND
T
A R
G E
T
TYP
ES
20
3
So
met imes
it
m a
k
es
sense,
a
s
in
this
expr
essio
n,
to
s
a
y
\c
ho
os
e
th e
op
era
tions
in
t his
expres
sion
so
t ha
t
t he
nal
r
esult
is
Flo
at.
(
2/
3)
Fl
oa
t
0:6
66
66
66
66
666
66
66
66
67
T
ype
:
F
loa
t
Her
e
w
e
used
\
2
04
CHAPTE
R
5.
U SING
TYPE
S
AND
M O
D E
S
This
sa
ys
th a
t
the
o
p
e
ra
t io
ns
s
hould
b
e
c
hos
en
so
that
t he
re
sult
is
a
Po
lyn
om
ial
o
b
j e
ct .
(
(x
+
y
*
%
i)*
*2
)
(Po
ly
nom
ia
l
Com
pl
ex
In
teg
i(eU.%m0(e)(re)Tj8.2i 0 Td(s)Tj3.8x0 Td(is)Tjy Td(ial)Tj/R81 7eU.pfyTd(+ Td(i(eU.%m0(e)9 Tf0)10.4x 0 Tlud66eU.%m0(e)NG)℄T-d()10.42 Td(ia)℄TJ/R83 03d(s)TYP3327.96T 0 Td(%)Tj5.1ype Td(teg)Tj15.7: Td(teg)Tj15.72 0 Td(Po)Tj10.44 0 Td(ly)Tj10.44 0 Td(ly)Tj10.4m2 0 Td(Po)Tj15.72 0 Td(l)Tj10.44 0 Td(Com)Tj15.72 0 Td(pl)Tj10.44 0 Td(ex)Tj15.72 0 Td(In)Tj10.44 0 Td(In)Tj10.4g2 0 Td(ial)Tj/R81 0.12 Tf%(Com)TW 0 Td(t7 Tf0)10.4d 0 Td3(2)Tj4.9y 0 Td0)Tj10.4 12 Td(o)Tj4.(u0 Td(ra)Tj8.88 0 Td[(hink)-100oul)Tj10.4m2ghTd(Thi2)℄TJ14.4 0 T0.12 Tf7.3h 0 Td(ia)Tj10.92 0 Td[(p0 Td[(bn)-100o4om)Tj15.72f 0 T9 2)℄TJ14.w 0 Td(s)Tj3.8e 0 T0.a
5
.10
.
RE
S O
L
VING
TYPE
S
20
5
1
8
1
6
1
4
1
9
Ty
pe:
Mat
2
08
CHAPTE
R
5.
U SING
TYPE
S
AND
M O
D E
S
ate
go
rie
s
A
be
lia
nG
ro
up
A
BE
LGR
P
A
be
lia
nM
on
oid
A
BE
LMO
N
A
be
lia
nM
on
oid
Ri
ng
A
MR
A
be
lia
nS
em
iGr
ou
p
A
BE
LSG
A
gg
reg
at
e
A
GG
A
lg
ebr
a
A
LG
EBR
A
A
lg
ebr
ai
a
lly
Cl
ose
dF
ie
ld
A
CF
A
lg
ebr
ai
a
lly
Cl
ose
dF
un
ti
on
Spa
e
A
CF
S
A
r
Hyp
er
bo
li
Fu
nt
io
nC
ate
go
ry
A
HY
P
.
..
F
or
eac
h
co
nstructor
in
a
gr
oup,
the
f ull
name
a
n d
the
abbre
viation
is
g
i v
en.
Ther
e
a
re
other
gr
oups
in
e
x p
o
sed.
ls
p
but
init ia
lly
o
nly
the
constr
u c
t o
rs
in
exp
os
ure
g
ro
ups
\
basic"
\
categ
or
ies"
\na
glink"
and
\
anna
"
a
re
exp
osed.
As
an
in
t er
activ
e
user
o
f
Axiom,
y
o
u
do
5
.12
.
CO
MM AND S
F
OR
SNOO
PING
20
9
This
is
a
p
olyno
mi a
l.
x
+
x
2
x
Ty
pe:
Po
lyn
om
ial
I
nt
ege
r
E
xp
ose
Ou
tpu
tF
or
m.
)
se
t
e
xp
os
e
a
dd
o
ns
tr
ut
or
Ou
tp
utF
or
m
Out
pu
tF
orm
i
s
n
ow
e
xpl
i
itl
y
exp
os
ed
in
f
ram
e
G8
232
2
This
is
wha
t
w
e
get
when
O
ut
pu
tFo
rm
is
a
utomatica
l ly
a
v
ailable.
x
+
x
x
+
x
T
ype
:
Ou
tp
ut
For
m
Hide
O
ut
put
Fo
rm
so
w
e
don't
run
in
t o
pro
blems
with
a
n
y
la
ter
e
xamples!
)
se
t
e
xp
os
e
d
ro
p
on
st
ru
to
r
O
ut
put
Fo
rm
Out
pu
tF
orm
i
s
n
ow
e
xpl
i
itl
y
hid
de
n
in
fr
ame
G
82
322
Fina
l ly
,
exp
os
u r
e
is
d o
ne
on
a
f r
ame-
b
y-fra
me
ba
sis.
A
f r
a me
is
one
of
p
o
2
10
CHAPTE
R
5.
U SING
TYPE
S
AND
M O
D E
S
o
p
er
ations
.
T
h e
mos
t
p
o
5
.12
.
CO
MM AND S
F
OR
SNOO
PING
21
1
RM
ATC
AT
-
Re
ta
ngu
la
rM
atr
ix
Cat
eg
ory
&
RM
ATR
IX
Re
ta
ngu
la
rM
atr
ix
SM
ATC
AT
-
Squ
ar
eMa
tr
ix
Cat
eg
ory
&
SQ
MAT
RI
X
Squ
ar
eMa
tr
ix
Similar
ly
,
if
y
ou
wis
h
to
see
a
ll
p a
c
k
a
ges
w
h o
se
na
m e
s
con
t a
in
\g
aus
s"
,
en
ter
this.
)
wh
at
pa
k
age
g
aus
s
-
--
---
--
--
---
--
---
--
--
Pa
k
age
s
---
--
--
---
--
---
--
--
---
-
P
a
kag
es
w
ith
n
ame
s
ma
th
in
g
p
at
ter
ns
:
g
au
ss
GA
USS
FA
C
Gau
ss
ian
Fa
t
ori
za
tio
nP
ak
ag
e
This
co
m ma
2
12
CHAPTE
R
5.
U SING
TYPE
S
AND
M O
D E
S
)
dis
pl
ay
op
era
ti
on
o
mp
lex
T
her
e
is
on
e
e
xp
os
ede
C
hapte r
6
U
si
ng
Hy
p
erDo
Fig
ure
6.1:
The
Hyp
er
D o c
r
o
ot
windo
w
pa
ge
Hyp
erDo
c
is
the
ga
t ew
a
y
to
Axio
m.
It's
b
oth
an
on-line
tut o
ria
l
a
nd
a
n
on-line
r
eference
ma
n
ual.
It
also
enables
y
ou
to
use
Axio
m
s
im ply
b
y
using
the
m o
use
a
nd
llin g
in
6
.3.
SCR
OLL
BARS
21
5
Do
wn
Arr o
w
Scro
ll
do
wn
one
line.
P
a g
e
Up
Scr
oll
up
o
ne
pag
e.
P
a g
e
ro
ll
Sr
oll
do
wn
o
ne
pag
2
16
CHAPTE
R
6
.
USIN G
HYPE
RD O
C
The
inpu t
ar
ea
g
ro
ws
to
a
ccommo
date
as
m a
n
y
c
ha
ra
cters
a
s
y
o
u
t
yp
e.
Use
the
Ba
k s
pa e
k
ey
to
era
se
c
ha
ra
ct er
s
to
t he
left .
T
o
mo
dif y
wha
t
y
o
u
t
y
p
e,
u s
e
the
r
igh
t -
ar
ro
w
!
6
.7.
EXAMPLE
P
A
GE
S
21
7
The
glos
sar
y
ha
s
an
input
ar
ea
a
t
it s
b
o
tt o
m.
W
e
review
t he
v
a
rio
u s
kinds
o
f
s
ear
c
h
s
t r
ings
y
o
u
can
en 0 Td
2
18
CHAPTE
R
6
.
USIN G
HYPE
RD O
C
its
t ex
t !
Wh en
y
o
u
d o
,
the
example
lin e
is
co
p ie
d
in
to
a
new
in
tera
cti v
e
A x
iom
bu e
r
for
this
Hyp
erDo
c
page
.
Sometimes
one
exa
mple
line
ca
n no
t
b
e
run
b
efor
e
y
o
u
r
un
a
n
ea
rlier
one.
Don't
w
o
r0 Td[(U2F[(d)-1000(o)]T447.4 0 Tmaticalld(y)4349.44 0 Td(sne.)Tjj12 0 Tdllits)Tj4.92 0 Td(the)T7j8.16 0 Tdc(un)Tj20.28 0 Tesstera
C
hapte r
7
I
nput
Fil
es
and
Output
St
y
les
In
this
c
ha
p ter
7
.3.
COMMON
FE
A
TURE
S
OF
USING
OUTP
U T
F
ORMA
TS
22
3
2
24
CHA P
TER
7.
7
.5.
TEX
F
O
RMA
T
22
5
T
ur
n
T
E
X
output
on
a
ga
in.
)
se
t
o
ut
pu
t
t
ex
on
The
c
har
acters
u s
ed
f o
r
the
m a
trix
bra
c
k
ets
ab
o
v
e
a
re
ra
ther
ugly
.
Y
ou
g
et
this
c
har
acter
set
when
y
ou
iss
u e
)s
et
ou
tp
ut
h
ara
t
ers
p
la
in.
This
c
ha
ra
cter
s
et
should
b
e
used
wh e
n
y
o
u
ar
e
r
u nning
on
a
mac
hine
th a
t
do
es
n o
t
supp
o
rt
the
IBM
ex
t ended
ASCI
I
c
har
ac
t er
set.
If
y
o
u
ar
e
running
o
n
an
IBM
w
or
k-
s
t a
tion,
for
example,
i s
sue
)se
t
out
pu
t
ha
ra
te
rs
d
efa
ul
t
t o
get
b
et
2
26
CHA P
TER
7.
IN P
UT
FILES
A ND
OUTP
U T
STY LE
S
\
def
\
sh
{\
mat
ho
p{
\rm
sh
}\
no
lim
it
s}
\
def
\e
rf{
\m
ath
op
{\
rm
er
f}\
no
li
mit
s}
\
def
\z
ag#
1#
2{
{{
\h
fil
l
\le
ft
.
{#1
}
\ri
gh
t|
}
\
ov
er
{
\l
eft
|
{#2
}
\r
igh
t.
\h
fi
ll
}
}
}
7. 6
IBM
Sript
F
o rm
ula
F
o rmat
Axiom
can
pr
o
duce
IBM
Script
F
or
m
ula
F
o
rmat
o
7
.7.
F
OR
TRAN
F
ORMA
T
22
7
Since
so
me
v
er
sio
n s
of
F
OR
TRAN
ha
v
e
restr
ict io
ns
on
the
n
um
b
e
r
of
lines
p
er
sta
t emen
t ,
Axiom
brea
ks
long
expr
essio
ns
in
t o
s
egmen
ts
wit h
a
maxim
um
o
f
13
20
c
har
acter
s
(20
li nes
of
66
c
ha
ra
ct er
s)
p
e
r
seg
men
t.
If
y
ou
w
an
t
c
longj4.92 0 Td8n s6cwayTj10.44 , Td(w)Tj6.84 0 Td[(t)-1000(o)]ysegchara66haued(seg)th
2
28
7
.7.
F
OR
TRAN
F
ORMA
T
22
9
Ty
pe:
Po
lyn
om
ial
I
nt
ege
r
This
c
2
30
CHA P
TER
7.
IN P
UT
FILES
A ND
OUTP
U T
STY LE
S
R8=
SI
N(E
XP
C
hapte r
8
Axi
om
Sy
stem
Commands
This
c
ha
p ter
descr
ib
es
sys
t e
m
commands,
the
co
mmand-line
facilities
u s
ed
to
c
on
8
.2.
) ABB
R E
VIA
TIO
N
23
5
8.2
)a b
brevi ati on
Use
r
Lev
e
l
R
equired:
compiler
2
36CH AP
8
.5.) CL
238
8
.7.
) CO
MPILE
23
9
)
l
ear
v
al
ue
al
l
)
l
ear
v
a
ll
This
re
t a
ins
wha
t e
v
er
decla
ra
t io
ns
the
ob
jects
h a
d.
T
o
r
emo
v
e
d enitions
and
v
a
lues
for
the
sp
ecic
o
b
jects
x
,
y
a
nd
f,
i s
sue
)
l
ear
v
al
ue
x
y
f
)
l
ear
v
x
y
f
T
o
re
m o
v
e
th e
declar
2
40
CH AP
TER
8
.
AXIOM
SYSTEM
COMMAN DS
)o
mp
ile
l eName.a
l
)o
mp
ile
di r
e
Desriptio(mp)51.20.44 0 Tn:ile
8
.38) CO
2
42
CH AP
TER
8
.
AXIOM
SYSTEM
COMMAN DS
-
O
-
Fa
sy
-F
ao
-F
ls
p
-
la
xio
m
-M
no-
AX
L_W
_W
ill
Ob
so
let
e
-DA
xi
om
These
options
mean:
-O:
p
er
f o
rm
all
o
pt imiza
t io
ns,
-Fa
sy
:
gener
ate
a
.
asy
le,
-Fa
o:
g
enera
t e
a
.a
o
le,
-Fl
sp
:
gener
ate
a
.
8
.7.
) CO
MPILEMPILE
2
44
CH AP
TER
8
.
AXIOM
SYSTEM
COMMAN DS
)
om
pi
le
ma
tri
x.
sp
ad
)
edi
t
)
om
pi
le
will
call
the
c
ompiler,
edit,
and
t hen
ca
l l
the
compiler
a
ga
in
o
n
the
le
m
a-
trix.s
pa d.
If
y
o
u
do
no
t
s
p
ecify
a
d ir
e
t
8
.8.
) DISPL
A
Y
24
5
8.8
)displa
y
Use
r
Lev
e
l
R
equired:
i n
t er
preter
Co
m
m
a nd
Syn
tax:
)d
is
pla
y
all
)d
is
pla
y
pro
pe
rti
es
)d
is
pla
y
pro
pe
rti
es
a
ll
)d
is
pla
y
pro
pe
rti
es
[tax:es
[
2
46
CH AP
TER
8
.
AXIOM
SYSTEM
COMMAN DS
8
.10
.
)FIN
24
7
)
sy
ste
m
em
as
/
et
/r
.
tp
ip
c
alls
e
ma
s
2
48
CH AP
TER
8
.
AXIOM
SYSTEM
COMMAN DS
Some
fra
m es
a
re
crea
ted
b
y
the
Hyp
erDo
c
pro
gr
am
2
50
CH AP
TER
8
.
AXIOM
SYSTEM
COMMAN DS
)
hel
p
le
ar
will
displa
y50)
8
.14
.
)HISTOR
Y
25
1
ha
s
b
een
is
sued.
Issuing
eit her
)
se
t
h
is
to
ry
of
f
)
hi
sto
ry
)
off
will
d is
con
tin
ue
t he
reco
rding
of
infor
m a
tion.
Whether
the
facilit
y
is
disa
b led
or
not,y
2
52
CH AP
TER
8
.
AXIOM
SYSTEM
COMMAN DS
)
res
et
will
us
h
the
in
t er
nal
list
o
f
the
mo
st
r
ecen
t
w
o
rkspa
ce
calcula
t io
ns
so
t ha
t
t he
d a
ta
s
t r
uct ur
es
ma
y
b
e
g
ar
bag
e
co
llect e
d
b
y
the
underly
i ng
Common
Lisp
sy
stem.
Lik
e
)h
ist
or
y
)
h
ang
e,
t his
option
only
has
rea
l
e ect
w
h en
hi s
t o
ry
data
is
b
eing
sa
v
ed
in
a
le.
)
res
to
re
[sav e
d HistoryName
]
co
mp letely
clear
s
the
en
v
iro
n men
t
a
nd
re
store
s
i t
t o
a
s
a
v
ed
s
essio
n ,
if
p
o
ssible.
The
)s
av
e
option
8
.16
.
)LISP
25
3
)l
ib
rar
y
)no
ex
pos
e
Co
m
m
a nd
Des
ripti
on:
This
comma
n d
repla
ces
the
)lo
ad
sys
t em
command
tha
t
w
as
a
v
aila
b le
in
Axiom
r
elea
ses
2
54
CH AP
TER
8
.
AXIOM
SYSTEM
COMMAN DS
Since
th is
co
mmand
is
only
useful
fo
r
ev
alua
t ing
s
in g
le
expre
ssions
,
t he
)
fin
co
mm a
nd
ma
y
8
.19
.
)QUIT
25
5
8.19
)qui t
Use
r
Lev
e
l
R
equired:
i n
t er
preter
Co
m
m
a nd
Syn
tax:
)q
ui
t
)s
et
qu
it
pr
ot
et
ed
|
un
pr
ote
t
ed
Co
m
m
a nd
Des
ripti
on:
This
comma
n d
2
56
CH AP
TER
8
.
AXIOM
SYSTEM
COMMAN DS
will
rea
d
the
c
on
ten
ts
of
t he
le
m
a trix.
input
in
t o
Axio
m.
T
h e
\
.i nput"
le
extensio
n
is
o
pti o
nal.
This
co
mmand
remem
b
er
s
the
previo
us
le
y
o
u
edited,
rea
d
or
compiled.
If
y
o
u
do
not
sp
ecify
a
le
na
me,
th e
prev
ious
le
will
b
e
rea
d.
The
)
if
the
re
option
c
hec
ks
t o
see
whet her
the
8
.22
2
58
CH AP
TER
8
.
AXIOM
SYSTEM
COMMAN DS
)
sho
w
POL
Y
INT
)
op
era
ti
ons
)
sho
w
Pol
yn
omi
al
I
nte
ge
r
)
sho
w
Pol
yn
omi
al
I
nte
ge
r
)
op
er
ati
on
s
a
re
a
mong
th e
com
rm2 T
8
.25
.
)SYS TE
M
25
9
This
command
is
used
to
cr
ea
t e
shor
t
synon
yrrop9e624i
2
60
CH AP
TER
8
.
AXIOM
SYSTEM
COMMAN DS
W
e
do
no
t
r
8
.26
.
)TRA
CE
26
1
8
.26
.
8
.28
.
)WH A
T
26
2
66
CH AP
TER
8
.
AXIOM
SYSTEM
COMMAN DS
)ap
ro
pos
p
att
ern1
[p
a t
tern2
.. .]
Com
m
and
Desriptio
n:
This
co
m ma
nd
is
used
8
.29
.
MAKE
FILE
26
7267
Bi
bliograph
y
[1]
Jenks
,
R.J.
and
S uto
r,
R .S.
\Axio
m
{
The
S c-1000u33i 0 Td[eS
I
n dex
Mult iplica
t io
n,
39
E
xp
onen
tiatio
n ,
39
+
Ad dition,
39
Nu mer
ical
N eg
atio
n ,
39
Sub tr
action,
39
=
D iv
ision,
3
9
<
less
t ha
n,
39
<
=
les
s
than
o
r
equal,
39
=
>
blo
c
k
exit,
7
8,
80
,
81
>
gr
eater
than,
3
9
>
=
g
rea
ter
tha
n
or
equa
l,
39
~
L
og
i c
al
Neg
ation,
3
9
)abb,
1
80
)abbr
eviation,
18
0,
2
44
)b
o ot,
2
36
,
25
4,
2
60
,
2
64
)cd,
2
36
,
25
3,
258
)clear
,
5
0,
2
388 Td(+)Tj11.04 0 Td[4)cd,
INDEX
27
3
s
t a
rt-up
pr
ole,
2
22
n,
2
47
r
st,
5
5,
9
8
rst
,
97
r
stD eno
m,
46
r
stN umer
,
4
6
Flexible
A r
ra
ys
,
69
FlexibleAr
ra
y
,
6
7
ex
ib leAr
ra
y
,
1
01
Flo
at,
9
1,
1
69
,
2
01
o
ating
p
o
in
t,
91
fo
n
t ,
21
8
fo
r,
8
5
fo
r
b
y
,
88
fo
r
list,
8
5
fo
r
s
egmen
t,
8
5
F
OR
TRA N,
1
3
F
OR
TRA N
o
utp ut
for
mat,
2
26
a
rr
a
ys,
2
30
br
eak
i ng
in
to
m
ultiple
s
t a
t e
m en
t s
,
2
27
da
t a
t
yp
es,
22
8
in
t eg
ers
vs.
o
ats,
22
8
line
l e
n g
th,
22
7
o
pt imiza
t io
n
lev
el,
22
8
pr
ecisio
n ,
22
9
F
r
action,
18
,
17
4,
17
6,
191
,
2
01
,
2
04
fr
action
pa
rtial,
93
F
r
action(Complex(In
teg
er)),
17
4
F
r
action(In
tege
r),
1
74
fr
actionP
ar
t,
35
fr
ame,
2
09
,
24
7
ex
p
o
sure
and,
2
09
fr
ame
dr
op,
2
48
fr
ame
f1.88 Td(fr)Tj6f1.88 Tdn6 0 Td(24)Tj9.96 0 T(48)T73-88.56 12 Td(fr)Tj6.84 0 Td(ame)Tj21.12 0 Tla(in)Tj16.68 0 Td(st,)Tj13.92 0 Td(24)Tj9.96 0 T(48)T59-92.52 11.88 Td(fr)Tj6.84 0 Td(ame)Tj21.12 0 (n(da)Tj10.44 0 Tdm[(l)-1000(e)]Tj81.72 0 Ts(r,)Tj10.08 0 Td(2)Tj4.92 0 Td(48)Tj-66.12 12 Td(fr)Tj6.84 0 Td(ame)Tj21.12 0 (newame,)Tj16 0 Td(24)Tj9.96 0 T(48)Tj-13.08 12 Td(fr)Tj6.84 0 Td(ame)Tj21.12 0 (nex(mat,)T522.2 0 Td(2)Tj4.92 0 Td(48)Tj-058.8 11.88 Tdun(action,)j-41.92 0 Td(1)Tj4.92 0 T0(27)Tj6-41.76 12 Tca(dr)Tj9.36 0 Tld(gial,)Tj9.96 0 Td(3)Tj4.92 0 T0(30)Tj912.08 12 Tdiece-al,)Tj6.84 0 Twis(ame)Tj81.72 0 Tdeni(action,)j7j8.04 0 Td(1)Tj4.92 0 T0(27)T1Tj10.Tj6. 0 TGalev)Tj81.72 0 Tussiaametege
ra
74
,t,
o
1ar
,48
INDEX
27
5
a
daptiv
e,
14
1
a
xes
colo
r,
1
42
clip
p
o
in
t s
,
14
2
line
colo
r,
1
42
ma
x
p
oin
ts,
14
2
min
p
oin
ts,
14
2
p
oin
t
colo
r,
1
42
p
oin
t
size,
1
42
r
eset
v
i e
wp
o
rt,
1
42
s
creen
reso
lut io
n,
1
42
to
sca
le,
1
42
units
co
lor
,
1
42
v
i e
wp
o
rt
p
os
INDEX
27
7
P
o
stScript,
13
2,
14
1,
1
62
,
