<s>
In	O
functional	B-Language
programming	I-Language
,	O
a	O
generalized	B-Application
algebraic	I-Application
data	I-Application
type	I-Application
(	O
GADT	B-Application
,	O
also	O
first-class	B-Application
phantom	I-Application
type	I-Application
,	O
guarded	B-Application
recursive	I-Application
datatype	I-Application
,	O
or	O
equality-qualified	B-Application
type	I-Application
)	O
is	O
a	O
generalization	O
of	O
parametric	O
algebraic	O
data	O
types	O
.	O
</s>
<s>
In	O
a	O
GADT	B-Application
,	O
the	O
product	O
constructors	O
(	O
called	O
data	O
constructors	O
in	O
Haskell	B-Language
)	O
can	O
provide	O
an	O
explicit	O
instantiation	O
of	O
the	O
ADT	O
as	O
the	O
type	O
instantiation	O
of	O
their	O
return	O
value	O
.	O
</s>
<s>
For	O
a	O
data	O
constructor	O
of	O
Haskell	B-Language
2010	O
,	O
the	O
return	O
value	O
has	O
the	O
type	O
instantiation	O
implied	O
by	O
the	O
instantiation	O
of	O
the	O
ADT	O
parameters	O
at	O
the	O
constructor	O
's	O
application	O
.	O
</s>
<s>
They	O
are	O
currently	O
implemented	O
in	O
the	O
GHC	B-Application
compiler	I-Application
as	O
a	O
non-standard	O
extension	O
,	O
used	O
by	O
,	O
among	O
others	O
,	O
Pugs	B-Language
and	O
Darcs	B-Protocol
.	O
</s>
<s>
OCaml	B-Language
supports	O
GADT	B-Application
natively	O
since	O
version	O
4.00	O
.	O
</s>
<s>
The	O
GHC	B-Application
implementation	O
provides	O
support	O
for	O
existentially	O
quantified	O
type	O
parameters	O
and	O
for	O
local	O
constraints	O
.	O
</s>
<s>
An	O
early	O
version	O
of	O
generalized	B-Application
algebraic	I-Application
data	I-Application
types	I-Application
were	O
described	O
by	O
and	O
based	O
on	O
pattern	B-Language
matching	I-Language
in	O
ALF	B-Language
.	O
</s>
<s>
Generalized	B-Application
algebraic	I-Application
data	I-Application
types	I-Application
were	O
introduced	O
independently	O
by	O
and	O
prior	O
by	O
as	O
extensions	O
to	O
ML	B-Language
's	O
and	O
Haskell	B-Language
's	O
algebraic	O
data	O
types	O
.	O
</s>
<s>
They	O
are	O
similar	O
to	O
the	O
inductive	O
families	O
of	O
data	O
types	O
(	O
or	O
inductive	O
datatypes	O
)	O
found	O
in	O
Coq	B-Application
's	O
Calculus	B-Application
of	I-Application
Inductive	I-Application
Constructions	I-Application
and	O
other	O
dependently	O
typed	O
languages	O
,	O
modulo	O
the	O
dependent	O
types	O
and	O
except	O
that	O
the	O
latter	O
have	O
an	O
additional	O
positivity	O
restriction	O
which	O
is	O
not	O
enforced	O
in	O
GADTs	B-Application
.	O
</s>
<s>
introduced	O
extended	O
algebraic	O
data	O
types	O
which	O
combine	O
GADTs	B-Application
together	O
with	O
the	O
existential	O
data	O
types	O
and	O
type	O
class	O
constraints	O
.	O
</s>
<s>
Type	O
inference	O
in	O
the	O
absence	O
of	O
any	O
programmer	O
supplied	O
type	O
annotations	O
is	O
undecidable	O
and	O
functions	O
defined	O
over	O
GADTs	B-Application
do	O
not	O
admit	O
principal	O
types	O
in	O
general	O
.	O
</s>
<s>
In	O
spring	O
2021	O
,	O
Scala	B-Application
3.0	O
is	O
released	O
.	O
</s>
<s>
This	O
major	O
update	O
of	O
Scala	B-Application
introduce	O
the	O
possibility	O
to	O
write	O
GADTs	B-Application
with	O
the	O
same	O
syntax	O
as	O
ADTs	O
,	O
which	O
is	O
not	O
the	O
case	O
in	O
other	O
programming	O
languages	O
according	O
to	O
Martin	O
Odersky	O
.	O
</s>
<s>
Applications	O
of	O
GADTs	B-Application
include	O
generic	B-Language
programming	I-Language
,	O
modelling	O
programming	O
languages	O
(	O
higher-order	B-Application
abstract	I-Application
syntax	I-Application
)	O
,	O
maintaining	O
invariants	B-Application
in	O
data	B-General_Concept
structures	I-General_Concept
,	O
expressing	O
constraints	O
in	O
embedded	O
domain-specific	O
languages	O
,	O
and	O
modelling	O
objects	O
.	O
</s>
<s>
An	O
important	O
application	O
of	O
GADTs	B-Application
is	O
to	O
embed	O
higher-order	B-Application
abstract	I-Application
syntax	I-Application
in	O
a	O
type	B-Language
safe	I-Language
fashion	O
.	O
</s>
<s>
Here	O
is	O
an	O
embedding	O
of	O
the	O
simply	O
typed	O
lambda	O
calculus	O
with	O
an	O
arbitrary	O
collection	O
of	O
base	O
types	O
,	O
tuples	O
and	O
a	O
fixed	B-Application
point	I-Application
combinator	I-Application
:	O
</s>
<s>
And	O
a	O
type	B-Language
safe	I-Language
evaluation	O
function	O
:	O
</s>
<s>
Furthermore	O
,	O
ill-formed	O
expressions	O
such	O
as	O
App	O
(	O
Lam	O
(	O
\x	O
->	O
Lam	O
(	O
\y	O
->	O
App	O
x	O
y	O
)	O
)	O
)	O
(	O
Lift	O
True	O
)	O
would	O
have	O
been	O
possible	O
to	O
construct	O
,	O
while	O
they	O
are	O
type	O
incorrect	O
using	O
the	O
GADT	B-Application
.	O
</s>
