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 non-identifier 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 kind-polymorphic cases. Its usage parallels visible type application in the term level, as specified above.

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 -fprint-explicit-foralls
> 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 left-to-right 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 the forall 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 left-to-right 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 on j and k, and b depends on a. Even though a appears lexically before j and k, j and k are quantified first, because a depends on j and k. Note further that j and k 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 left-to-right through the input list of type variables, with a cursor.

    • If variable v at the cursor is depended on by any earlier variable w, move v immediately before the leftmost such w.

  • 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 that return’s type arguments are m, 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 have pair :: forall a. a -> forall b. b -> (a, b) and then say pair @Bool True @Char which would have type Char -> (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 say wurble @_ @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 specify PartialTypeSignatures 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

While user-written type or kind variables are specified by default, GHC permits labelling these variables as inferred. 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 the k 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 type T, where b is existential. When writing the pattern synonym, both types are allowed to be specified or inferred.

  • On the right-hand 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 default type signatures for class methods, 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 left-hand sides of type declarations, such as classes, data types, etc.