6.4.8. Generalised Algebraic Data Types (GADTs)

Implies:MonoLocalBinds, GADTSyntax

Allow use of Generalised Algebraic Data Types (GADTs).

Generalised Algebraic Data Types generalise ordinary algebraic data types by allowing constructors to have richer return types. Here is an example:

data Term a where
    Lit    :: Int -> Term Int
    Succ   :: Term Int -> Term Int
    IsZero :: Term Int -> Term Bool
    If     :: Term Bool -> Term a -> Term a -> Term a
    Pair   :: Term a -> Term b -> Term (a,b)

Notice that the return type of the constructors is not always Term a, as is the case with ordinary data types. This generality allows us to write a well-typed eval function for these Terms:

eval :: Term a -> a
eval (Lit i)      = i
eval (Succ t)     = 1 + eval t
eval (IsZero t)   = eval t == 0
eval (If b e1 e2) = if eval b then eval e1 else eval e2
eval (Pair e1 e2) = (eval e1, eval e2)

The key point about GADTs is that pattern matching causes type refinement. For example, in the right hand side of the equation

eval :: Term a -> a
eval (Lit i) =  ...

the type a is refined to Int. That’s the whole point! A precise specification of the type rules is beyond what this user manual aspires to, but the design closely follows that described in the paper Simple unification-based type inference for GADTs, (ICFP 2006). The general principle is this: type refinement is only carried out based on user-supplied type annotations. So if no type signature is supplied for eval, no type refinement happens, and lots of obscure error messages will occur. However, the refinement is quite general. For example, if we had:

eval :: Term a -> a -> a
eval (Lit i) j =  i+j

the pattern match causes the type a to be refined to Int (because of the type of the constructor Lit), and that refinement also applies to the type of j, and the result type of the case expression. Hence the addition i+j is legal.

These and many other examples are given in papers by Hongwei Xi, and Tim Sheard. There is a longer introduction on the wiki, and Ralf Hinze’s Fun with phantom types also has a number of examples. Note that papers may use different notation to that implemented in GHC.

The rest of this section outlines the extensions to GHC that support GADTs. The extension is enabled with GADTs. The GADTs extension also sets GADTSyntax and MonoLocalBinds.

  • A GADT can only be declared using GADT-style syntax (Declaring data types with explicit constructor signatures); the old Haskell 98 syntax for data declarations always declares an ordinary data type. The result type of each constructor must begin with the type constructor being defined, but for a GADT the arguments to the type constructor can be arbitrary monotypes. For example, in the Term data type above, the type of each constructor must end with Term ty, but the ty need not be a type variable (e.g. the Lit constructor).

  • It is permitted to declare an ordinary algebraic data type using GADT-style syntax. What makes a GADT into a GADT is not the syntax, but rather the presence of data constructors whose result type is not just T a b.

  • You cannot use a deriving clause for a GADT; only for an ordinary data type.

  • As mentioned in Declaring data types with explicit constructor signatures, record syntax is supported. For example:

    data Term a where
        Lit    :: { val  :: Int }      -> Term Int
        Succ   :: { num  :: Term Int } -> Term Int
        Pred   :: { num  :: Term Int } -> Term Int
        IsZero :: { arg  :: Term Int } -> Term Bool
        Pair   :: { arg1 :: Term a
                  , arg2 :: Term b
                  }                    -> Term (a,b)
        If     :: { cnd  :: Term Bool
                  , tru  :: Term a
                  , fls  :: Term a
                  }                    -> Term a

    However, for GADTs there is the following additional constraint: every constructor that has a field f must have the same result type (modulo alpha conversion) Hence, in the above example, we cannot merge the num and arg fields above into a single name. Although their field types are both Term Int, their selector functions actually have different types:

    num :: Term Int -> Term Int
    arg :: Term Bool -> Term Int

    See Field selectors and TypeApplications for a full description of how the types of top-level field selectors are determined.

  • When pattern-matching against data constructors drawn from a GADT, for example in a case expression, the following rules apply:

    • The type of the scrutinee must be rigid.
    • The type of the entire case expression must be rigid.
    • The type of any free variable mentioned in any of the case alternatives must be rigid.

    A type is “rigid” if it is completely known to the compiler at its binding site. The easiest way to ensure that a variable a rigid type is to give it a type signature. For more precise details see Simple unification-based type inference for GADTs. The criteria implemented by GHC are given in the Appendix.

  • When GHC typechecks multiple patterns in a function clause, it typechecks each pattern in order from left to right. This has consequences for patterns that match on GADTs, such as in this example:

    data U a where
      MkU :: U ()
    v1 :: U a -> a -> a
    v1 MkU () = ()
    v2 :: a -> U a -> a
    v2 () MkU = ()

    Although v1 and v2 may appear to be the same function but with differently ordered arguments, GHC will only typecheck v1. This is because in v1, GHC will first typecheck the MkU pattern, which causes a to be refined to (). This refinement is what allows the subsequent () pattern to typecheck at type a. In v2, however, GHC first tries to typecheck the () pattern, and because a has not been refined to () yet, GHC concludes that () is not of type a. v2 can be made to typecheck by matching on MkU before (), like so:

    v2 :: a -> U a -> a
    v2 x MkU = case x of () -> ()
  • Not only does GHC typecheck patterns from left to right, it also typechecks them from the outside in. This can be seen in this example:

    data F x y where
      MkF :: y -> F (Maybe z) y
    g :: F a a -> a
    g (MkF Nothing) = Nothing

    In the function clause for g, GHC first checks MkF, the outermost pattern, followed by the inner Nothing pattern. This outside-in order can interact somewhat counterintuitively with Pattern type signatures. Consider the following variation of g:

    g2 :: F a a -> a
    g2 (MkF Nothing :: F (Maybe z) (Maybe z)) = Nothing @z

    The g2 function attempts to use the pattern type signature F (Maybe z) (Maybe z) to bring the type variable z into scope so that it can be used on the right-hand side of the definition with Visible type application. However, GHC will reject the pattern type signature in g2:

    • Couldn't match type ‘a’ with ‘Maybe z’
      Expected: F a a
        Actual: F (Maybe z) (Maybe z)

    Again, this is because of the outside-in order GHC uses when typechecking patterns. GHC first tries to check the pattern type signature F (Maybe z) (Maybe z), but at that point, GHC has not refined a to be Maybe z, so GHC is unable to conclude that F a a is equal to F (Maybe z) (Maybe z). Here, the MkF pattern is considered to be inside of the pattern type signature, so GHC cannot use the type refinement from the MkF pattern when typechecking the pattern type signature.

    There are two possible ways to repair g2. One way is to use a case expression to write a pattern signature after matching on MkF, like so:

    g3 :: F a a -> a
    g3 f@(MkF Nothing) =
      case f of
        (_ :: F (Maybe z) (Maybe z)) -> Nothing @z

    Another way is to use Type Applications in Patterns instead of a pattern type signature:

    g4 :: F a a -> a
    g4 (MkF @(Maybe z) Nothing) = Nothing @z

    Here, the visible type argument @(Maybe z) indicates that the y in the type of MkF :: y -> F (Maybe z) y should be instantiated to Maybe z. In addition, @(Maybe z) also brings z into scope. Although g4 no longer uses a pattern type signature, it accomplishes the same end result, as the right-hand side Nothing @z will typecheck successfully.