6.4.8. Generalised Algebraic Data Types (GADTs)¶

GADTs
¶ Implies: MonoLocalBinds
,GADTSyntax
Since: 6.8.1 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 welltyped 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
unificationbased type inference for
GADTs, (ICFP
2006). The general principle is this: type refinement is only carried
out based on usersupplied 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 GADTstyle 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 withTerm ty
, but thety
need not be a type variable (e.g. theLit
constructor).It is permitted to declare an ordinary algebraic data type using GADTstyle 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 thenum
andarg
fields above into a single name. Although their field types are bothTerm Int
, their selector functions actually have different types:num :: Term Int > Term Int arg :: Term Bool > Term Int
When patternmatching 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 unificationbased type inference for GADTs. The criteria implemented by GHC are given in the Appendix.