<s>
In	O
computer	B-General_Concept
science	I-General_Concept
and	O
logic	O
,	O
a	O
dependent	O
type	O
is	O
a	O
type	O
whose	O
definition	O
depends	O
on	O
a	O
value	O
.	O
</s>
<s>
In	O
functional	B-Language
programming	I-Language
languages	I-Language
like	O
Agda	B-Language
,	O
ATS	B-Language
,	O
Coq	B-Application
,	O
F*	B-Application
,	O
Epigram	B-Language
,	O
and	O
Idris	B-Language
,	O
dependent	O
types	O
help	O
reduce	O
bugs	O
by	O
enabling	O
the	O
programmer	O
to	O
assign	O
types	O
that	O
further	O
restrain	O
the	O
set	O
of	O
possible	O
implementations	O
.	O
</s>
<s>
(	O
Note	O
that	O
this	O
is	O
different	O
from	O
polymorphism	B-Application
and	O
generic	B-Language
programming	I-Language
,	O
both	O
of	O
which	O
include	O
the	O
type	O
as	O
an	O
argument	O
.	O
)	O
</s>
<s>
In	O
1934	O
,	O
Haskell	B-Language
Curry	O
noticed	O
that	O
the	O
types	O
used	O
in	O
typed	O
lambda	O
calculus	O
,	O
and	O
in	O
its	O
combinatory	B-Application
logic	I-Application
counterpart	O
,	O
followed	O
the	O
same	O
pattern	O
as	O
axioms	O
in	O
propositional	O
logic	O
.	O
</s>
<s>
Predicate	B-Algorithm
logic	O
is	O
an	O
extension	O
of	O
propositional	O
logic	O
,	O
adding	O
quantifiers	O
.	O
</s>
<s>
there	O
is	O
no	O
fixed	O
codomain	B-Algorithm
)	O
is	O
a	O
dependent	O
function	O
and	O
the	O
type	O
of	O
this	O
function	O
is	O
called	O
dependent	O
product	O
type	O
,	O
pi-type	O
(	O
type	O
)	O
or	O
dependent	O
function	O
type	O
.	O
</s>
<s>
For	O
a	O
more	O
concrete	O
example	O
,	O
taking	O
to	O
be	O
equal	O
to	O
the	O
family	O
of	O
unsigned	O
integers	O
from	O
0	O
to	O
255	O
(	O
the	O
ones	O
that	O
fit	O
into	O
8	O
bits	O
or	O
1	O
byte	O
)	O
and	O
for	O
,	O
then	O
devolves	O
into	O
the	O
product	O
of	O
precisely	O
because	O
the	O
finite	O
set	O
of	O
integers	O
from	O
0	O
to	O
255	O
would	O
ultimately	O
stop	O
at	O
the	O
bounds	O
just	O
mentioned	O
,	O
resulting	O
in	O
a	O
finite	O
codomain	B-Algorithm
of	O
the	O
dependent	O
function	O
.	O
</s>
<s>
For	O
a	O
more	O
concrete	O
example	O
,	O
taking	O
to	O
again	O
be	O
equal	O
to	O
the	O
family	O
of	O
unsigned	O
integers	O
from	O
0	O
to	O
255	O
,	O
and	O
to	O
again	O
be	O
equal	O
to	O
for	O
256	O
more	O
arbitrary	O
,	O
then	O
devolves	O
into	O
the	O
sum	O
for	O
the	O
same	O
reasons	O
as	O
to	O
what	O
happened	O
to	O
the	O
codomain	B-Algorithm
of	O
the	O
dependent	O
function	O
.	O
</s>
<s>
By	O
the	O
Curry	O
–	O
Howard	O
correspondence	O
,	O
can	O
be	O
interpreted	O
as	O
a	O
logical	B-Algorithm
predicate	I-Algorithm
on	O
terms	O
of	O
.	O
</s>
<s>
For	O
a	O
given	O
,	O
whether	O
the	O
type	O
is	O
inhabited	O
indicates	O
whether	O
satisfies	O
this	O
predicate	B-Algorithm
.	O
</s>
<s>
Henk	O
Barendregt	O
developed	O
the	O
lambda	B-Application
cube	I-Application
as	O
a	O
means	O
of	O
classifying	O
type	O
systems	O
along	O
three	O
axes	O
.	O
</s>
<s>
The	O
eight	O
corners	O
of	O
the	O
resulting	O
cube-shaped	O
diagram	O
each	O
correspond	O
to	O
a	O
type	O
system	O
,	O
with	O
simply	O
typed	O
lambda	O
calculus	O
in	O
the	O
least	O
expressive	O
corner	O
,	O
and	O
calculus	B-Application
of	I-Application
constructions	I-Application
in	O
the	O
most	O
expressive	O
.	O
</s>
<s>
The	O
three	O
axes	O
of	O
the	O
cube	O
correspond	O
to	O
three	O
different	O
augmentations	O
of	O
the	O
simply	O
typed	O
lambda	O
calculus	O
:	O
the	O
addition	O
of	O
dependent	O
types	O
,	O
the	O
addition	O
of	O
polymorphism	B-Application
,	O
and	O
the	O
addition	O
of	O
higher	O
kinded	O
type	O
constructors	O
(	O
functions	O
from	O
types	O
to	O
types	O
,	O
for	O
example	O
)	O
.	O
</s>
<s>
The	O
lambda	B-Application
cube	I-Application
is	O
generalized	O
further	O
by	O
pure	O
type	O
systems	O
.	O
</s>
<s>
The	O
higher	O
order	O
system	O
extends	O
to	O
all	O
four	O
forms	O
of	O
abstraction	O
from	O
the	O
lambda	B-Application
cube	I-Application
:	O
functions	O
from	O
terms	O
to	O
terms	O
,	O
types	O
to	O
types	O
,	O
terms	O
to	O
types	O
and	O
types	O
to	O
terms	O
.	O
</s>
<s>
The	O
system	O
corresponds	O
to	O
the	O
calculus	B-Application
of	I-Application
constructions	I-Application
whose	O
derivative	O
,	O
the	O
calculus	B-Application
of	I-Application
inductive	I-Application
constructions	I-Application
is	O
the	O
underlying	O
system	O
of	O
the	B-Application
Coq	I-Application
proof	I-Application
assistant	I-Application
.	O
</s>
<s>
The	O
code-generation	O
aspect	O
provides	O
a	O
powerful	O
approach	O
to	O
formal	O
program	O
verification	O
and	O
proof-carrying	B-Application
code	I-Application
,	O
since	O
the	O
code	O
is	O
derived	O
directly	O
from	O
a	O
mechanically	O
verified	O
mathematical	O
proof	O
.	O
</s>
