<s>
In	O
programming	O
language	O
theory	O
,	O
subtyping	O
(	O
also	O
subtype	O
polymorphism	B-Application
or	O
inclusion	O
polymorphism	B-Application
)	O
is	O
a	O
form	O
of	O
type	B-Application
polymorphism	I-Application
in	O
which	O
a	O
subtype	O
is	O
a	O
datatype	O
that	O
is	O
related	O
to	O
another	O
datatype	O
(	O
the	O
supertype	O
)	O
by	O
some	O
notion	O
of	O
substitutability	B-Application
,	O
meaning	O
that	O
program	O
elements	O
,	O
typically	O
subroutines	O
or	O
functions	O
,	O
written	O
to	O
operate	O
on	O
elements	O
of	O
the	O
supertype	O
can	O
also	O
operate	O
on	O
elements	O
of	O
the	O
subtype	O
.	O
</s>
<s>
Subtyping	O
is	O
therefore	O
a	O
form	O
of	O
type	B-Application
polymorphism	I-Application
.	O
</s>
<s>
In	O
object-oriented	B-Language
programming	I-Language
the	O
term	O
'	O
polymorphism	B-Application
 '	O
is	O
commonly	O
used	O
to	O
refer	O
solely	O
to	O
this	O
subtype	O
polymorphism	B-Application
,	O
while	O
the	O
techniques	O
of	O
parametric	O
polymorphism	B-Application
would	O
be	O
considered	O
generic	B-Language
programming	I-Language
.	O
</s>
<s>
Functional	B-Language
programming	I-Language
languages	I-Language
often	O
allow	O
the	O
subtyping	O
of	O
records	O
.	O
</s>
<s>
Consequently	O
,	O
simply	O
typed	O
lambda	B-General_Concept
calculus	O
extended	O
with	O
record	O
types	O
is	O
perhaps	O
the	O
simplest	O
theoretical	O
setting	O
in	O
which	O
a	O
useful	O
notion	O
of	O
subtyping	O
may	O
be	O
defined	O
and	O
studied	O
.	O
</s>
<s>
Since	O
functional	B-Language
programming	I-Language
languages	I-Language
,	O
by	O
definition	O
,	O
support	O
function	B-General_Concept
literals	I-General_Concept
,	O
which	O
can	O
also	O
be	O
stored	O
in	O
records	O
,	O
records	O
types	O
with	O
subtyping	O
provide	O
some	O
of	O
the	O
features	O
of	O
object-oriented	B-Language
programming	I-Language
.	O
</s>
<s>
Typically	O
,	O
functional	B-Language
programming	I-Language
languages	I-Language
also	O
provide	O
some	O
,	O
usually	O
restricted	O
,	O
form	O
of	O
parametric	O
polymorphism	B-Application
.	O
</s>
<s>
Various	O
calculi	O
that	O
attempt	O
to	O
capture	O
the	O
theoretical	O
properties	O
of	O
object-oriented	B-Language
programming	I-Language
may	O
be	O
derived	O
from	O
system	O
F	O
<	O
:	O
.	O
</s>
<s>
Subtyping	O
should	O
not	O
be	O
confused	O
with	O
the	O
notion	O
of	O
(	O
class	O
or	O
object	O
)	O
inheritance	B-Language
from	O
object-oriented	B-Language
languages	I-Language
;	O
subtyping	O
is	O
a	O
relation	O
between	O
types	O
(	O
interfaces	O
in	O
object-oriented	B-Language
parlance	O
)	O
whereas	O
inheritance	B-Language
is	O
a	O
relation	O
between	O
implementations	O
stemming	O
from	O
a	O
language	O
feature	O
that	O
allows	O
new	O
objects	O
to	O
be	O
created	O
from	O
existing	O
ones	O
.	O
</s>
<s>
In	O
a	O
number	O
of	O
object-oriented	B-Language
languages	I-Language
,	O
subtyping	O
is	O
called	O
interface	B-Application
inheritance	I-Application
,	O
with	O
inheritance	B-Language
referred	O
to	O
as	O
implementation	B-Language
inheritance	I-Language
.	O
</s>
<s>
The	O
notion	O
of	O
subtyping	O
in	O
programming	O
languages	O
dates	O
back	O
to	O
the	O
1960s	O
;	O
it	O
was	O
introduced	O
in	O
Simula	B-Language
derivatives	O
.	O
</s>
<s>
The	O
concept	O
of	O
subtyping	O
has	O
gained	O
visibility	O
(	O
and	O
synonymy	O
with	O
polymorphism	B-Application
in	O
some	O
circles	O
)	O
with	O
the	O
mainstream	O
adoption	O
of	O
object-oriented	B-Language
programming	I-Language
.	O
</s>
<s>
In	O
this	O
context	O
,	O
the	O
principle	O
of	O
safe	O
substitution	O
is	O
often	O
called	O
the	O
Liskov	B-Application
substitution	I-Application
principle	I-Application
,	O
after	O
Barbara	O
Liskov	O
who	O
popularized	O
it	O
in	O
a	O
keynote	B-Protocol
address	O
at	O
a	O
conference	O
on	O
object-oriented	B-Language
programming	I-Language
in	O
1987	O
.	O
</s>
<s>
Because	O
it	O
must	O
consider	O
mutable	B-Application
objects	I-Application
,	O
the	O
ideal	O
notion	O
of	O
subtyping	O
defined	O
by	O
Liskov	O
and	O
Jeannette	O
Wing	O
,	O
called	O
behavioral	B-Application
subtyping	I-Application
is	O
considerably	O
stronger	O
than	O
what	O
can	O
be	O
implemented	O
in	O
a	O
type	O
checker	O
.	O
</s>
<s>
Conceptually	O
,	O
each	O
of	O
these	O
is	O
a	O
variety	O
of	O
the	O
basic	O
type	O
"	O
bird	O
"	O
that	O
inherits	B-Language
many	O
"	O
bird	O
"	O
characteristics	O
but	O
has	O
some	O
specific	O
differences	O
.	O
</s>
<s>
The	O
UML	B-Language
notation	O
is	O
used	O
in	O
this	O
diagram	O
,	O
with	O
open-headed	O
arrows	O
showing	O
the	O
direction	O
and	O
type	O
of	O
the	O
relationship	O
between	O
the	O
supertype	O
and	O
its	O
subtypes	O
.	O
</s>
<s>
As	O
a	O
more	O
practical	O
example	O
,	O
a	O
language	O
might	O
allow	O
integer	O
values	O
to	O
be	O
used	O
wherever	O
floating	O
point	O
values	O
are	O
expected	O
(	O
Integer	O
<	O
:	O
Float	O
)	O
,	O
or	O
it	O
might	O
define	O
a	O
generic	B-Language
type	I-Language
Number	O
as	O
a	O
common	O
supertype	O
of	O
integers	O
and	O
the	O
reals	O
.	O
</s>
<s>
Rewriting	O
this	O
function	O
so	O
that	O
it	O
would	O
only	O
accept	O
'	O
x	O
 '	O
and	O
'	O
y	O
 '	O
of	O
the	O
same	O
type	O
requires	O
bounded	O
polymorphism	B-Application
.	O
</s>
<s>
User-defined	B-Language
types	I-Language
like	O
records	O
(	O
structs	O
,	O
interfaces	O
)	O
or	O
classes	O
are	O
defined	O
intensionally	O
by	O
an	O
explicit	O
type	O
declaration	O
or	O
by	O
using	O
an	O
existing	O
value	O
,	O
which	O
encodes	O
type	O
information	O
,	O
as	O
a	O
prototype	O
to	O
be	O
copied	O
or	O
extended	O
.	O
</s>
<s>
In	O
the	O
context	O
of	O
subsumption	O
,	O
the	O
type	O
definitions	O
can	O
be	O
expressed	O
using	O
Set-builder	B-Language
notation	I-Language
,	O
which	O
uses	O
a	O
predicate	O
to	O
define	O
a	O
set	O
.	O
</s>
<s>
The	O
class-based	B-Application
object-oriented	B-Language
subtyping	O
described	O
above	O
is	O
nominal	O
;	O
a	O
structural	O
subtyping	O
rule	O
for	O
an	O
object-oriented	B-Language
language	I-Language
might	O
say	O
that	O
if	O
objects	O
of	O
type	O
A	O
can	O
handle	O
all	O
of	O
the	O
messages	O
that	O
objects	O
of	O
type	O
B	O
can	O
handle	O
(	O
that	O
is	O
,	O
if	O
they	O
define	O
all	O
the	O
same	O
methods	O
)	O
,	O
then	O
A	O
is	O
a	O
subtype	O
of	O
B	O
regardless	O
of	O
whether	O
either	O
inherits	B-Language
from	O
the	O
other	O
.	O
</s>
<s>
This	O
so-called	O
duck	B-Application
typing	I-Application
is	O
common	O
in	O
dynamically	O
typed	O
object-oriented	B-Language
languages	I-Language
.	O
</s>
<s>
The	O
subtyping	O
induced	O
by	O
subclassing	O
in	O
an	O
object-oriented	B-Language
language	I-Language
is	O
usually	O
inclusive	O
;	O
subtyping	O
relations	O
that	O
relate	O
integers	O
and	O
floating-point	O
numbers	O
,	O
which	O
are	O
represented	O
differently	O
,	O
are	O
usually	O
coercive	O
.	O
</s>
<s>
The	O
second	O
method	B-Language
,	O
called	O
depth	O
subtyping	O
,	O
replaces	O
the	O
various	O
fields	O
with	O
their	O
subtypes	O
.	O
</s>
<s>
Depth	O
subtyping	O
only	O
makes	O
sense	O
for	O
immutable	B-Application
records	O
:	O
for	O
example	O
,	O
you	O
can	O
assign	O
1.5	O
to	O
the	O
'	O
x	O
 '	O
field	O
of	O
a	O
real	O
point	O
(	O
a	O
record	O
with	O
two	O
real	O
fields	O
)	O
,	O
but	O
you	O
ca	O
n't	O
do	O
the	O
same	O
to	O
the	O
'	O
x	O
 '	O
field	O
of	O
an	O
integer	O
point	O
(	O
which	O
,	O
however	O
,	O
is	O
a	O
deep	O
subtype	O
of	O
the	O
real	O
point	O
type	O
)	O
because	O
1.5	O
is	O
not	O
an	O
integer	O
(	O
see	O
Variance	O
)	O
.	O
</s>
<s>
Subtyping	O
of	O
records	O
can	O
be	O
defined	O
in	O
System	O
F	O
<	O
:	O
,	O
which	O
combines	O
parametric	O
polymorphism	B-Application
with	O
subtyping	O
of	O
record	O
types	O
and	O
is	O
a	O
theoretical	O
basis	O
for	O
many	O
functional	B-Language
programming	I-Language
languages	I-Language
that	O
support	O
both	O
features	O
.	O
</s>
<s>
This	O
is	O
what	O
exactly	O
works	O
in	O
Scala	B-Application
:	O
a	O
n-ary	O
function	O
is	O
internally	O
a	O
class	O
that	O
inherits	B-Language
the	O
trait	O
(	O
which	O
can	O
be	O
seen	O
as	O
a	O
general	O
interface	B-Application
in	O
Java-like	O
languages	O
)	O
,	O
where	O
are	O
the	O
parameter	O
types	O
,	O
and	O
is	O
its	O
return	O
type	O
;	O
"	O
−	O
"	O
before	O
the	O
type	O
means	O
the	O
type	O
is	O
contravariant	O
while	O
"	O
+	O
"	O
means	O
covariant	O
.	O
</s>
<s>
In	O
languages	O
that	O
allow	O
side	O
effects	O
,	O
like	O
most	O
object-oriented	B-Language
languages	I-Language
,	O
subtyping	O
is	O
generally	O
not	O
sufficient	O
to	O
guarantee	O
that	O
a	O
function	O
can	O
be	O
safely	O
used	O
in	O
the	O
context	O
of	O
another	O
.	O
</s>
<s>
Liskov	O
's	O
work	O
in	O
this	O
area	O
focused	O
on	O
behavioral	B-Application
subtyping	I-Application
,	O
which	O
besides	O
the	O
type	O
system	O
safety	O
discussed	O
in	O
this	O
article	O
also	O
requires	O
that	O
subtypes	O
preserve	O
all	O
invariants	B-Application
guaranteed	O
by	O
the	O
supertypes	O
in	O
some	O
contract	B-Application
.	O
</s>
<s>
The	O
subtyping	O
of	O
mutable	B-Application
references	O
is	O
similar	O
to	O
the	O
treatment	O
of	O
parameter	O
values	O
and	O
return	O
values	O
.	O
</s>
<s>
Mutable	B-Application
references	O
which	O
act	O
as	O
both	O
sources	O
and	O
sinks	O
are	O
invariant	B-Application
.	O
</s>
<s>
Subtyping	O
and	O
inheritance	B-Language
are	O
independent	O
(	O
orthogonal	O
)	O
relationships	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
between	O
two	O
types	O
S	O
and	O
T	O
,	O
all	O
combinations	O
of	O
subtyping	O
and	O
inheritance	B-Language
are	O
possible	O
:	O
</s>
<s>
In	O
most	O
object	B-Language
oriented	I-Language
programming	I-Language
languages	I-Language
,	O
Int64	O
are	O
unrelated	O
by	O
inheritance	B-Language
to	O
Int32	O
.	O
</s>
<s>
Assume	O
a	O
super	O
class	O
of	O
type	O
T	O
having	O
a	O
method	B-Language
m	O
returning	O
an	O
object	O
of	O
the	O
same	O
type	O
(	O
i.e.	O
</s>
<s>
the	O
type	O
of	O
m	O
is	O
T	O
→	O
T	O
,	O
also	O
note	O
that	O
the	O
first	O
parameter	O
of	O
m	O
is	O
this/self	O
)	O
and	O
a	O
derived	O
class	O
type	O
S	O
from	O
T	O
.	O
By	O
inheritance	B-Language
,	O
the	O
type	O
of	O
m	O
in	O
S	O
is	O
S	O
→	O
S	O
.	O
In	O
order	O
for	O
S	O
to	O
be	O
a	O
subtype	O
of	O
T	O
the	O
type	O
of	O
m	O
in	O
S	O
must	O
be	O
a	O
subtype	O
of	O
the	O
type	O
of	O
m	O
in	O
T	O
,	O
in	O
other	O
words	O
:	O
S	O
→	O
S	O
≤	O
:	O
T	O
→	O
T	O
.	O
By	O
bottom-up	O
application	O
of	O
the	O
function	O
subtyping	O
rule	O
,	O
this	O
means	O
:	O
S	O
≤	O
:	O
T	O
and	O
T	O
≤	O
:	O
S	O
,	O
which	O
is	O
only	O
possible	O
if	O
S	O
and	O
T	O
are	O
the	O
same	O
.	O
</s>
<s>
Since	O
inheritance	B-Language
is	O
an	O
irreflexive	O
relation	O
,	O
S	O
ca	O
n't	O
be	O
a	O
subtype	O
of	O
T	O
.	O
</s>
<s>
Subtyping	O
and	O
inheritance	B-Language
are	O
compatible	O
when	O
all	O
inherited	O
fields	O
and	O
methods	O
of	O
the	O
derived	O
type	O
have	O
types	O
which	O
are	O
subtypes	O
of	O
the	O
corresponding	O
fields	O
and	O
methods	O
from	O
the	O
inherited	O
type	O
.	O
</s>
<s>
C	O
.	O
Szyperski	O
,	O
D	O
.	O
Gruntz	O
,	O
S	O
.	O
Murer	O
,	O
Component	O
software	O
:	O
beyond	O
object-oriented	B-Language
programming	I-Language
,	O
2nd	O
ed.	O
,	O
Pearson	O
Education	O
,	O
2002	O
,	O
,	O
pp	O
.	O
</s>
<s>
Also	O
in	O
Carl	O
A	O
.	O
Gunter	O
and	O
John	O
C	O
.	O
Mitchell	O
,	O
editors	O
,	O
Theoretical	O
Aspects	O
of	O
Object-Oriented	B-Language
Programming	I-Language
:	O
Types	O
,	O
Semantics	O
,	O
and	O
Language	O
Design	O
(	O
MIT	O
Press	O
,	O
1994	O
)	O
.	O
</s>
