6.4.17. Type abstractions

TypeAbstractions
Since:

9.8.1

Status:

Experimental

Allow the use of type abstraction syntax.

The TypeAbstractions extension provides a way to explicitly bind scoped type or kind variables using the @a syntax. The feature is only partially implemented, and this text covers only the implemented parts, whereas the full specification can be found in GHC Proposals #448 and #425.

6.4.17.1. Type Abstractions in Patterns

Since: GHC 9.2

The type abstraction 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 somewhat-contrived 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.

6.4.17.2. Type Abstractions in Functions

Since: GHC 9.10

Type abstraction syntax can be used in lambdas and function left-hand sides to bring into scope type variables associated with invisible forall. For example:

id :: forall a. a -> a
id @t x = x :: t

Here type variables t and a stand for the same type, i.e. the first and only type argument of id. In a call id @Int we have a = Int, t = Int. The difference is that a is in scope in the type signature, while t is in scope in the function equation.

The scope of a can be extended to cover the function equation as well by enabling ScopedTypeVariables. Using a separate binder like @t is the modern and more flexible alternative for that, capable of handling higher-rank scenarios (see the higherRank example below).

When multiple variables are bound with @-binders, they are matched left-to-right with the corresponding forall-bound variables in the type signature:

const :: forall a. forall b. a -> b -> a
const @ta @tb x  = x

In this example, @ta corresponds to forall a. and @tb to forall b.. It is also possible to use @-binders in combination with implicit quantification (i.e. no explicit forall in the signature):

const :: a -> b -> a
const @ta @tb x  = x

In such cases, type variables in the signature are considered to be quantified with an implicit forall in the order in which they appear in the signature, c.f. TypeApplications.

It is not possible to match against a specific type (such as Maybe or Int) in an @-binder. The binder must be irrefutable, i.e. it may take one of the following forms:

  • type variable pattern @a

  • type variable pattern with a kind annotation @(f :: Type -> Type)

  • wildcard @_, with or without a kind annotation

The main advantage to using @-binders over ScopedTypeVariables is the ability to use them in lambdas passed to higher-rank functions:

higherRank :: (forall a. (Num a, Bounded a) => a -> a) -> (Int8, Int16)
higherRank f = (f 42, f 42)

ex :: (Int8, Int16)
ex = higherRank (\ @a x -> maxBound @a - x )
                   -- @a-binder in a lambda pattern in an argument
                   -- to a higher-order function

At the moment, an @-binder is valid only in a limited set of circumstances:

  • In a function left-hand side, where the function must have an explicit type signature:

    f1 :: forall a. a -> forall b. b -> (a, b)
f1 @a x @b y = (x :: a, y :: b)        -- OK

    It would be illegal to omit the type signature for f, nor is it possible to move the binder to a lambda on the RHS:

    f2 :: forall a. a -> forall b. b -> (a, b)
f2 = \ @a x @b y -> (x :: a, y :: b)   -- ILLEGAL

  • In a lambda annotated with an inline type signature:

    f3 = (\ @a x @b y -> (x :: a, y :: b) )      -- OK
    :: forall a. a -> forall b. b -> (a, b)

  • In a lambda used as an argument to a higher-rank function or data constructor:

    h :: (forall a. a -> forall b. b -> (a, b)) -> (Int, Bool)
h = ...

f4 = h (\ @a x @b y -> (x :: a, y :: b))     -- OK

  • In a lambda used as a field of a data structure (e.g. a list item), whose type is impredicative (see ImpredicativeTypes):

    f5 :: [forall a. a -> a -> a]
f5 = [ \ @a x _ -> x :: a,
       \ @a _ y -> y :: a ]

  • In a lambda of multiple arguments, where the first argument is visible, and only if DeepSubsumption is off:

    {-# LANGUAGE NoDeepSubsumption #-}
f6 :: () -> forall a. a -> (a, a)
f6 = \ _ @a x -> (x :: a, x)   -- OK

6.4.17.3. Invisible Binders in Type Declarations

Since: GHC 9.8

6.4.17.3.1. Syntax

The type abstraction syntax can be used in type declaration headers, including type, data, newtype, class, type family, and data family declarations. Here are a few examples:

type C :: forall k. k -> Constraint
class C @k a where ...
        ^^

type D :: forall k j. k -> j -> Type
data D @k @j (a :: k) (b :: j) = ...
       ^^ ^^

type F :: forall p q. p -> q -> (p, q)
type family F @p @q a b where ...
              ^^ ^^

Just as ordinary type parameters, invisible type variable binders may have kind annotations:

type F :: forall p q. p -> q -> (p, q)
type family F @(p :: Type) @(q :: Type) (a :: p) (b :: q) where ...

6.4.17.3.2. Scope

The @k-binders scope over the body of the declaration and can be used to bring implicit type or kind variables into scope. Consider:

type C :: forall i. (i -> i -> i) -> Constraint
class C @i a where
    p :: P a i

Without the @i binder in C @i a, the i in P a i would no longer refer to the class variable i and would be implicitly quantified in the method signature instead.

6.4.17.3.3. Type checking

Invisible type variable binders require either a standalone kind signature or a complete user-supplied kind.

If a standalone kind signature is given, GHC will match up @k-binders with the corresponding forall k. quantifiers in the signature:

type B :: forall k. k -> forall j. j -> Type
data B @k (a :: k) @j (b :: j)

Quantifier-binder pairs of B

forall k.

@k

k ->

(a :: k)

forall j.

@j

j ->

(b :: j)

The matching is done left-to-right. Consider:

type S :: forall a b. a -> b -> Type
type S @k x y = ...

In this example, @k is matched with forall a., not forall b.:

Quantifier-binder pairs of S

forall a.

@k

forall b.

a ->

x

b ->

y

When a standalone kind signature is absent but the definition has a complete user-supplied kind (and the CUSKs extension is enabled), a @k-binder gives rise to a forall k. quantifier in the inferred kind signature. The inferred forall k. does not float to the left; the order of quantifiers continues to match the order of binders in the header:

-- Inferred kind: forall k. k -> forall j. j -> Type
data B @(k :: Type) (a :: k) @(j :: Type) (b :: j)

6.4.17.4. Wildcard Binders in Type Declarations

Since: GHC 9.12

Unused type variable binders may be replaced with wildcards:

type UConst a b = a   -- unused named binder `b`
type WConst a _ = a   -- wildcard binder

Just like a named binder, a wildcard binder _ can be:

  • plain: _

  • kinded: (_ :: k -> Type)

  • invisible, plain: @_

  • invisible, kinded: @(_ :: k -> Type)

Wildcard binders and type abstractions are, in principle, separate features. However, both are included in the TypeAbstractions extension for administrative convenience. The two features can be used together to skip over a prefix of invisible type variables:

type D :: forall k1 k2 k3. k1 -> k2 -> k3 -> Type
data D @_ @_ @k3 a b c = ...

In this example we are interested in k3 and use wildcards @_ to avoid binding k1 and k2.