6.4.14. Visible type application¶

TypeApplications
¶ Since: 8.0.1 Allow the use of type application syntax.
The TypeApplications
extension allows you to use
visible type application in expressions. Here is an
example: show (read @Int "5")
. The @Int
is the visible type application; it specifies the value of the type variable
in read
‘s type.
A visible type application is preceded with an @
sign. (To disambiguate the syntax, the @
must be
preceded with a nonidentifier letter, usually a space. For example,
read@Int 5
would not parse.) It can be used whenever
the full polymorphic type of the function is known. If the function
is an identifier (the common case), its type is considered known only when
the identifier has been given a type signature. If the identifier does
not have a type signature, visible type application cannot be used.
GHC also permits visible kind application, where users can declare the kind arguments to be instantiated in kindpolymorphic cases. Its usage parallels visible type application in the term level, as specified above.
In addition to visible type application in terms and types, the type application syntax can be used in patterns matching a data constructor to bind type variables in that constructor’s type.
6.4.14.1. Inferred vs. specified type variables¶
GHC tracks a distinction between what we call inferred and specified type variables. Only specified type variables are available for instantiation with visible type application. An example illustrates this well:
f :: (Eq b, Eq a) => a > b > Bool
f x y = (x == x) && (y == y)
g x y = (x == x) && (y == y)
The functions f
and g
have the same body, but only f
is given
a type signature. When GHC is figuring out how to process a visible type application,
it must know what variable to instantiate. It thus must be able to provide
an ordering to the type variables in a function’s type.
If the user has supplied a type signature, as in f
, then this is easy:
we just take the ordering from the type signature, going left to right and
using the first occurrence of a variable to choose its position within the
ordering. Thus, the variables in f
will be b
, then a
.
In contrast, there is no reliable way to do this for g
; we will not know
whether Eq a
or Eq b
will be listed first in the constraint in g
‘s
type. In order to have visible type application be robust between releases of
GHC, we thus forbid its use with g
.
We say that the type variables in f
are specified, while those in
g
are inferred. The general rule is this: if the user has written
a type variable in the source program, it is specified; if not, it is
inferred.
This rule applies in datatype declarations, too. For example, if we have
data Proxy a = Proxy
(and PolyKinds
is enabled), then
a
will be assigned kind k
, where k
is a fresh kind variable.
Because k
was not written by the user, it will be unavailable for
type application in the type of the constructor Proxy
; only the a
will be available.
Inferred variables are printed in braces. Thus, the type of the data
constructor Proxy
from the previous example is
forall {k} (a :: k). Proxy a
.
We can observe this behavior in a GHCi session:
> :set XTypeApplications fprintexplicitforalls
> let myLength1 :: Foldable f => f a > Int; myLength1 = length
> :type +v myLength1
myLength1 :: forall (f :: * > *) a. Foldable f => f a > Int
> let myLength2 = length
> :type +v myLength2
myLength2 :: forall {a} {t :: * > *}. Foldable t => t a > Int
> :type +v myLength2 @[]
<interactive>:1:1: error:
• Cannot apply expression of type ‘t0 a0 > Int’
to a visible type argument ‘[]’
• In the expression: myLength2 @[]
Notice that since myLength1
was defined with an explicit type signature,
:type +v
reports that all of its type variables are available
for type application. On the other hand, myLength2
was not given a type
signature. As a result, all of its type variables are surrounded with braces,
and trying to use visible type application with myLength2
fails.
Also note the use of :type +v
in the GHCi session above instead
of :type
. This is because :type
gives you the type
that would be inferred for a variable assigned to the expression provided
(that is, the type of x
in let x = <expr>
). As we saw above with
myLength2
, this type will have no variables available to visible type
application. On the other hand, :type +v
gives you the actual
type of the expression provided. To illustrate this:
> :type myLength1
myLength1 :: forall {a} {f :: * > *}. Foldable f => f a > Int
> :type myLength2
myLength2 :: forall {a} {t :: * > *}. Foldable t => t a > Int
Using :type
might lead one to conclude that none of the type
variables in myLength1
‘s type signature are available for type
application. This isn’t true, however! Be sure to use :type +v
if you want the most accurate information with respect to visible type
application properties.
6.4.14.2. Ordering of specified variables¶
In the simple case of the previous section, we can say that specified variables appear in lefttoright order. However, not all cases are so simple. Here are the rules in the subtler cases:
If an identifier’s type has a
forall
, then the order of type variables as written in theforall
is retained.If any of the variables depend on other variables (that is, if some of the variables are kind variables), the variables are reordered so that kind variables come before type variables, preserving the lefttoright order as much as possible. That is, GHC performs a stable topological sort on the variables. Example:
h :: Proxy (a :: (j, k)) > Proxy (b :: Proxy a) > ()  as if h :: forall j k a b. ...
In this example,
a
depends onj
andk
, andb
depends ona
. Even thougha
appears lexically beforej
andk
,j
andk
are quantified first, becausea
depends onj
andk
. Note further thatj
andk
are not reordered with respect to each other, even though doing so would not violate dependency conditions.A “stable topological sort” here, we mean that we perform this algorithm (which we call ScopedSort):
 Work lefttoright through the input list of type variables, with a cursor.
 If variable
v
at the cursor is depended on by any earlier variablew
, movev
immediately before the leftmost suchw
.
Class methods’ type arguments include the class type variables, followed by any variables an individual method is polymorphic in. So,
class Monad m where return :: a > m a
means thatreturn
‘s type arguments arem, a
.With the
RankNTypes
extension (Lexically scoped type variables), it is possible to declare type arguments somewhere other than the beginning of a type. For example, we can havepair :: forall a. a > forall b. b > (a, b)
and then saypair @Bool True @Char
which would have typeChar > (Bool, Char)
.Partial type signatures (Partial Type Signatures) work nicely with visible type application. If you want to specify only the second type argument to
wurble
, then you can saywurble @_ @Int
. The first argument is a wildcard, just like in a partial type signature. However, if used in a visible type application/visible kind application, it is not necessary to specifyPartialTypeSignatures
and your code will not generate a warning informing you of the omitted type.
The section in this manual on kind polymorphism describes how variables in type and class declarations are ordered (Inferring the order of variables in a type/class declaration).
6.4.14.3. Manually defining inferred variables¶
Since the 9.0.1 release, GHC permits labelling the userwritten
type or kind variables as inferred, in contrast
to the default of specified. By writing the type variable binder in
braces as {tyvar}
or {tyvar :: kind}
, the new variable will be
classified as inferred, not specified. Doing so gives the programmer control
over which variables can be manually instantiated and which can’t.
Note that the braces do not influence scoping: variables in braces are still
brought into scope just the same.
Consider for example:
myConst :: forall {a} b. a > b > a
myConst x _ = x
In this example, despite both variables appearing in a type signature, a
is
an inferred variable while b
is specified. This means that the expression
myConst @Int
has type forall {a}. a > Int > a
.
The braces are allowed in the following places:
In the type signatures of functions, variables, class methods, as well as type annotations on expressions. Consider the example above.
In data constructor declarations, using the GADT syntax. Consider:
data T a where MkT :: forall {k} (a :: k). Proxy a > T a
The constructor
MkT
defined in this example is kind polymorphic, which is emphasized to the reader by explicitly abstracting over thek
variable. As this variable is marked as inferred, it can not be manually instantiated.In existential variable quantifications, e.g.:
data HList = HNil  forall {a}. HCons a HList
In pattern synonym signatures. Consider for instance:
data T a where MkT :: forall a b. a > b > T a pattern Pat :: forall {c}. () => forall {d}. c > d > T c pattern Pat x y = MkT x y
Note that in this example,
a
is a universal variable in the data typeT
, whereb
is existential. When writing the pattern synonym, both types are allowed to be specified or inferred.On the righthand side of a type synonym, e.g.:
type Foo = forall a {b}. Either a b
In type signatures on variables bound in RULES, e.g.:
{# RULES "parametricity" forall (f :: forall {a}. a > a). map f = id #}
The braces are not allowed in the following places:
In visible dependent quantifiers. Consider:
data T :: forall {k} > k > Type
This example is rejected, as a visible argument should by definition be explicitly applied. Making them inferred (and thus not appliable) would be conflicting.
In SPECIALISE pragmas or in instance declaration heads, e.g.:
instance forall {a}. Eq (Maybe a) where ...
The reason for this is, essentially, that none of these define a new construct. This means that no new type is being defined where specificity could play a role.
On the lefthand sides of type declarations, such as classes, data types, etc.
Note that while specified and inferred type variables have different properties visàvis visible type application, they do not otherwise affect GHC’s notion of equality over types. For example, given the following definitions:
id1 :: forall a. a > a
id1 x = x
id2 :: forall {a}. a > a
id2 x = x
app1 :: (forall a. a > a) > b > b
app1 g x = g x
app2 :: (forall {a}. a > a) > b > b
app2 g x = g x
GHC will deem all of app1 id1
, app1 id2
, app2 id1
, and app2 id2
to be well typed.
6.4.14.4. Type Applications in Patterns¶
The type application syntax can be used in patterns that match a data constructor. The syntax can’t be used with record patterns or infix patterns. This is useful in particular to bind existential type variables associated with a GADT data constructor as in the following example:
{# LANGUAGE AllowAmbiguousTypes #}
{# LANGUAGE GADTs #}
{# LANGUAGE RankNTypes #}
{# LANGUAGE TypeApplications #}
import Data.Proxy
data Foo where
Foo :: forall a. (Show a, Num a) => Foo
test :: Foo > String
test x = case x of
Foo @t > show @t 0
main :: IO ()
main = print $ test (Foo @Float)
In this example, the case in test`
is binding an existential variable introduced
by Foo
that otherwise could not be named and used.
It’s possible to bind variables to any part of the type arguments to a constructor; there is no need for them to be existential. In addition, it’s possible to “match” against part of the type argument using type constructors.
For a somewhatcontrived example:
foo :: (Num a) => Maybe [a] > String
foo (Nothing @[t]) = show (0 :: t)
foo (Just @[t] xs) = show (sum xs :: t)
Here, we’re binding the type variable t to be the type of the elements of the list type which is itself the argument to Maybe.
The order of the type arguments specified by the type applications in a pattern is the same
as that for an expression: either the order as given by the user in an explicit forall
in the
definition of the data constructor, or if that is not present, the order in which the type
variables appear in its type signature from left to right.
For example if we have the following declaration in GADT syntax:
data Foo :: * > * where
A :: forall s t. [(t,s)] > Foo (t,s)
B :: (t,s) > Foo (t,s)
Then the type arguments to A
will match first s
and then t
, while the type arguments
to B
will match first t
and then s
.
Type arguments appearing in patterns can influence the inferred type of a definition:
foo (Nothing @Int) = 0
foo (Just x) = x
will have inferred type:
foo :: Maybe Int > Int
which is more restricted than what it would be without the application:
foo :: Num a => Maybe a > a
For more information and detail regarding type applications in patterns, see the paper
Type variables in patterns by Eisenberg, Breitner
and Peyton Jones. Relative to that paper, the implementation in GHC for now at least makes one
additional conservative restriction, that type variables occurring in patterns must not
already be in scope, and so are always new variables that only bind whatever type is
matched, rather than ever referring to a variable from an outer scope. Type wildcards
_
may be used in any place where no new variable needs binding.