6.6.4. Deriving instances of extra classes (Data, etc.)
Haskell 98 allows the programmer to add “deriving( Eq, Ord )” to a
data type declaration, to generate a standard instance declaration for
classes specified in the deriving clause. In Haskell 98, the only
classes that may appear in the deriving clause are the standard
classes Eq, Ord, Enum, Ix, Bounded, Read, and
Show.
GHC extends this list with several more classes that may be automatically derived:
With
DeriveGeneric, you can derive instances of the classesGenericandGeneric1, defined inGHC.Generics. You can use these to define generic functions, as described in Generic programming.With
DeriveFunctor, you can derive instances of the classFunctor, defined inGHC.Base.With
DeriveDataTypeable, you can derive instances of the classData, defined inData.Data.With
DeriveFoldable, you can derive instances of the classFoldable, defined inData.Foldable.With
DeriveTraversable, you can derive instances of the classTraversable, defined inData.Traversable. Since theTraversableinstance dictates the instances ofFunctorandFoldable, you’ll probably want to derive them too, soDeriveTraversableimpliesDeriveFunctorandDeriveFoldable.With
DeriveLift, you can derive instances of the classLift, defined in theLanguage.Haskell.TH.Syntaxmodule of thetemplate-haskellpackage.
You can also use a standalone deriving declaration instead (see Stand-alone deriving declarations).
In each case the appropriate class must be in scope before it can be
mentioned in the deriving clause.
6.6.4.1. Deriving Functor instances
- DeriveFunctor
- Since:
7.10.1
Allow automatic deriving of instances for the
Functortypeclass.
With DeriveFunctor, one can derive Functor instances for data types
of kind Type -> Type. For example, this declaration:
data Example a = Ex a Char (Example a) (Example Char)
deriving Functor
would generate the following instance:
instance Functor Example where
fmap f (Ex a1 a2 a3 a4) = Ex (f a1) a2 (fmap f a3) a4
The basic algorithm for DeriveFunctor walks the arguments of each
constructor of a data type, applying a mapping function depending on the type
of each argument. If a plain type variable is found that is syntactically
equivalent to the last type parameter of the data type (a in the above
example), then we apply the function f directly to it. If a type is
encountered that is not syntactically equivalent to the last type parameter
but does mention the last type parameter somewhere in it, then a recursive
call to fmap is made. If a type is found which doesn’t mention the last
type parameter at all, then it is left alone.
The second of those cases, in which a type is unequal to the type parameter but does contain the type parameter, can be surprisingly tricky. For example, the following example compiles:
newtype Right a = Right (Either Int a) deriving Functor
Modifying the code slightly, however, produces code which will not compile:
newtype Wrong a = Wrong (Either a Int) deriving Functor
The difference involves the placement of the last type parameter, a. In the
Right case, a occurs within the type Either Int a, and moreover, it
appears as the last type argument of Either. In the Wrong case,
however, a is not the last type argument to Either; rather, Int is.
This distinction is important because of the way DeriveFunctor works. The
derived Functor Right instance would be:
instance Functor Right where
fmap f (Right a) = Right (fmap f a)
Given a value of type Right a, GHC must produce a value of type
Right b. Since the argument to the Right constructor has type
Either Int a, the code recursively calls fmap on it to produce a value
of type Either Int b, which is used in turn to construct a final value of
type Right b.
The generated code for the Functor Wrong instance would look exactly the
same, except with Wrong replacing every occurrence of Right. The
problem is now that fmap is being applied recursively to a value of type
Either a Int. This cannot possibly produce a value of type
Either b Int, as fmap can only change the last type parameter! This
causes the generated code to be ill-typed.
As a general rule, if a data type has a derived Functor instance and its
last type parameter occurs on the right-hand side of the data declaration, then
either it must (1) occur bare (e.g., newtype Id a = Id a), or (2) occur as the
last argument of a type constructor (as in Right above).
There are two exceptions to this rule:
Tuple types. When a non-unit tuple is used on the right-hand side of a data declaration,
DeriveFunctortreats it as a product of distinct types. In other words, the following code:newtype Triple a = Triple (a, Int, [a]) deriving Functor
Would result in a generated
Functorinstance like so:instance Functor Triple where fmap f (Triple a) = Triple (case a of (a1, a2, a3) -> (f a1, a2, fmap f a3))
That is,
DeriveFunctorpattern-matches its way into tuples and maps over each type that constitutes the tuple. The generated code is reminiscent of what would be generated fromdata Triple a = Triple a Int [a], except with extra machinery to handle the tuple.Function types. The last type parameter can appear anywhere in a function type as long as it occurs in a covariant position. To illustrate what this means, consider the following three examples:
newtype CovFun1 a = CovFun1 (Int -> a) deriving Functor newtype CovFun2 a = CovFun2 ((a -> Int) -> a) deriving Functor newtype CovFun3 a = CovFun3 (((Int -> a) -> Int) -> a) deriving Functor
All three of these examples would compile without issue. On the other hand:
newtype ContraFun1 a = ContraFun1 (a -> Int) deriving Functor newtype ContraFun2 a = ContraFun2 ((Int -> a) -> Int) deriving Functor newtype ContraFun3 a = ContraFun3 (((a -> Int) -> a) -> Int) deriving Functor
While these examples look similar, none of them would successfully compile. This is because all occurrences of the last type parameter
aoccur in contravariant positions, not covariant ones.Intuitively, a covariant type is produced, and a contravariant type is consumed. Most types in Haskell are covariant, but the function type is special in that the lefthand side of a function arrow reverses variance. If a function type
a -> bappears in a covariant position (e.g.,CovFun1above), thenais in a contravariant position andbis in a covariant position. Similarly, ifa -> bappears in a contravariant position (e.g.,CovFun2above), thenais inacovariant position andbis in a contravariant position.To see why a data type with a contravariant occurrence of its last type parameter cannot have a derived
Functorinstance, let’s suppose that aFunctor ContraFun1instance exists. The implementation would look something like this:instance Functor ContraFun1 where fmap f (ContraFun g) = ContraFun (\x -> _)
We have
f :: a -> b,g :: a -> Int, andx :: b. Using these, we must somehow fill in the hole (denoted with an underscore) with a value of typeInt. What are our options?We could try applying
gtox. This won’t work though, asgexpects an argument of typea, andx :: b. Even worse, we can’t turnxinto something of typea, sincefalso needs an argument of typea! In short, there’s no good way to make this work.On the other hand, a derived
Functorinstances for theCovFuns are within the realm of possibility:instance Functor CovFun1 where fmap f (CovFun1 g) = CovFun1 (\x -> f (g x)) instance Functor CovFun2 where fmap f (CovFun2 g) = CovFun2 (\h -> f (g (\x -> h (f x)))) instance Functor CovFun3 where fmap f (CovFun3 g) = CovFun3 (\h -> f (g (\k -> h (\x -> f (k x)))))
There are some other scenarios in which a derived Functor instance will
fail to compile:
A data type has no type parameters (e.g.,
data Nothing = Nothing).A data type’s last type variable is used in a
DatatypeContextsconstraint (e.g.,data Ord a => O a = O a).A data type’s last type variable is used in an
ExistentialQuantificationconstraint, or is refined in a GADT. For example,data T a b where T4 :: Ord b => b -> T a b T5 :: b -> T b b T6 :: T a (b,b) deriving instance Functor (T a)
would not compile successfully due to the way in which
bis constrained.
When the last type parameter has a phantom role (see Roles), the derived
Functor instance will not be produced using the usual algorithm. Instead,
the entire value will be coerced.
data Phantom a = Z | S (Phantom a) deriving Functor
will produce the following instance:
instance Functor Phantom where
fmap _ = coerce
When a type has no constructors, the derived Functor instance will
simply force the (bottom) value of the argument using
EmptyCase.
data V a deriving Functor
type role V nominal
will produce
instance Functor V where
fmap _ z = case z of
6.6.4.2. Deriving Foldable instances
- DeriveFoldable
- Since:
7.10.1
Allow automatic deriving of instances for the
Foldabletypeclass.
With DeriveFoldable, one can derive Foldable instances for data types
of kind Type -> Type. For example, this declaration:
data Example a = Ex a Char (Example a) (Example Char)
deriving Foldable
would generate the following instance:
instance Foldable Example where
foldr f z (Ex a1 a2 a3 a4) = f a1 (foldr f z a3)
foldMap f (Ex a1 a2 a3 a4) = mappend (f a1) (foldMap f a3)
The algorithm for DeriveFoldable is adapted from the
DeriveFunctor algorithm, but it generates definitions for
foldMap, foldr, and null instead of fmap. In addition,
DeriveFoldable filters out all constructor arguments on the RHS
expression whose types do not mention the last type parameter, since those
arguments do not need to be folded over.
When the type parameter has a phantom role (see Roles),
DeriveFoldable derives a trivial instance. For example, this
declaration:
data Phantom a = Z | S (Phantom a)
will generate the following instance.
instance Foldable Phantom where
foldMap _ _ = mempty
Similarly, when the type has no constructors, DeriveFoldable will
derive a trivial instance:
data V a deriving Foldable
type role V nominal
will generate the following.
instance Foldable V where
foldMap _ _ = mempty
Here are the differences between the generated code for Functor and
Foldable:
#. When a bare type variable a is encountered, DeriveFunctor
would generate f a for an fmap definition. DeriveFoldable
would generate f a z for foldr, f a for foldMap, and False
for null.
When a type that is not syntactically equivalent to
a, but which does containa, is encountered,DeriveFunctorrecursively callsfmapon it. Similarly,DeriveFoldablewould recursively callfoldrandfoldMap. Depending on the context,nullmay recursively callnullorall null. For example, givendata F a = F (P a) data G a = G (P (a, Int)) data H a = H (P (Q a))
Foldablederiving will producenull (F x) = null x null (G x) = null x null (H x) = all null x
DeriveFunctorputs everything back together again at the end by invoking the constructor.DeriveFoldable, however, builds up a value of some type. Forfoldr, this is accomplished by chaining applications offand recursivefoldrcalls on the state valuez. ForfoldMap, this happens by combining all values withmappend. Fornull, the values are usually combined with&&. However, if any of the values is known to beFalse, all the rest will be dropped. For example,data SnocList a = Nil | Snoc (SnocList a) a
will not produce
null (Snoc xs _) = null xs && False
(which would walk the whole list), but rather
null (Snoc _ _) = False
There are some other differences regarding what data types can have derived
Foldable instances:
Data types containing function types on the right-hand side cannot have derived
Foldableinstances.Foldableinstances can be derived for data types in which the last type parameter is existentially constrained or refined in a GADT. For example, this data type:data E a where E1 :: (a ~ Int) => a -> E a E2 :: Int -> E Int E3 :: (a ~ Int) => a -> E Int E4 :: (a ~ Int) => Int -> E a deriving instance Foldable E
would have the following generated
Foldableinstance:instance Foldable E where foldr f z (E1 e) = f e z foldr f z (E2 e) = z foldr f z (E3 e) = z foldr f z (E4 e) = z foldMap f (E1 e) = f e foldMap f (E2 e) = mempty foldMap f (E3 e) = mempty foldMap f (E4 e) = mempty
Notice how every constructor of
Eutilizes some sort of existential quantification, but only the argument ofE1is actually “folded over”. This is because we make a deliberate choice to only fold over universally polymorphic types that are syntactically equivalent to the last type parameter. In particular:
We don’t fold over the arguments of
E1orE4because even though(a ~ Int),Intis not syntactically equivalent toa.We don’t fold over the argument of
E3becauseais not universally polymorphic. TheainE3is (implicitly) existentially quantified, so it is not the same as the last type parameter ofE.
6.6.4.3. Deriving Traversable instances
- DeriveTraversable
- Implies:
- Since:
7.10.1
Allow automatic deriving of instances for the
Traversabletypeclass.
With DeriveTraversable, one can derive Traversable instances for data
types of kind Type -> Type. For example, this declaration:
data Example a = Ex a Char (Example a) (Example Char)
deriving (Functor, Foldable, Traversable)
would generate the following Traversable instance:
instance Traversable Example where
traverse f (Ex a1 a2 a3 a4)
= fmap (\b1 b3 -> Ex b1 a2 b3 a4) (f a1) <*> traverse f a3
The algorithm for DeriveTraversable is adapted from the
DeriveFunctor algorithm, but it generates a definition for traverse
instead of fmap. In addition, DeriveTraversable filters out
all constructor arguments on the RHS expression whose types do not mention the
last type parameter, since those arguments do not produce any effects in a
traversal.
When the type parameter has a phantom role (see Roles),
DeriveTraversable coerces its argument. For example, this
declaration:
data Phantom a = Z | S (Phantom a) deriving Traversable
will generate the following instance:
instance Traversable Phantom where
traverse _ z = pure (coerce z)
When the type has no constructors, DeriveTraversable will
derive the laziest instance it can.
data V a deriving Traversable
type role V nominal
will generate the following, using EmptyCase:
instance Traversable V where
traverse _ z = pure (case z of)
Here are the differences between the generated code in each extension:
When a bare type variable
ais encountered, bothDeriveFunctorandDeriveTraversablewould generatef afor anfmapandtraversedefinition, respectively.When a type that is not syntactically equivalent to
a, but which does containa, is encountered,DeriveFunctorrecursively callsfmapon it. Similarly,DeriveTraversablewould recursively calltraverse.DeriveFunctorputs everything back together again at the end by invoking the constructor.DeriveTraversabledoes something similar, but it works in anApplicativecontext by chaining everything together with(<*>).
Unlike DeriveFunctor, DeriveTraversable cannot be used on data
types containing a function type on the right-hand side.
For a full specification of the algorithms used in DeriveFunctor,
DeriveFoldable, and DeriveTraversable, see
this wiki page.
6.6.4.4. Deriving Data instances
- DeriveDataTypeable
- Since:
6.8.1
Enable automatic deriving of instances for the
Datatypeclass
6.6.4.5. Deriving Typeable instances
The class Typeable is very special:
Typeableis kind-polymorphic (see Kind polymorphism).GHC has a custom solver for discharging constraints that involve class
Typeable, and handwritten instances are forbidden. This ensures that the programmer cannot subvert the type system by writing bogus instances.Derived instances of
Typeablemay be declared if theDeriveDataTypeableextension is enabled, but they are ignored, and they may be reported as an error in a later version of the compiler.The rules for solving
Typeableconstraints are as follows:A concrete type constructor applied to some types.
instance (Typeable t1, .., Typeable t_n) => Typeable (T t1 .. t_n)
This rule works for any concrete type constructor, including type constructors with polymorphic kinds. The only restriction is that if the type constructor has a polymorphic kind, then it has to be applied to all of its kinds parameters, and these kinds need to be concrete (i.e., they cannot mention kind variables).
A type variable applied to some types:
instance (Typeable f, Typeable t1, .., Typeable t_n) => Typeable (f t1 .. t_n)
A concrete type literal.:
instance Typeable 0 -- Type natural literals instance Typeable "Hello" -- Type-level symbols
6.6.4.6. Deriving Lift instances
- DeriveLift
- Since:
8.0.1
Enable automatic deriving of instances for the
Lifttypeclass for Template Haskell.
The class Lift, unlike other derivable classes, lives in
template-haskell instead of base. Having a data type be an instance of
Lift permits its values to be promoted to Template Haskell expressions (of
type ExpQ and Code Q a), which can then be spliced into Haskell source
code.
Here is an example of how one can derive Lift:
{-# LANGUAGE DeriveLift #-}
module Bar where
import Language.Haskell.TH.Syntax
data Foo a = Foo a | a :^: a deriving Lift
{-
instance (Lift a) => Lift (Foo a) where
lift (Foo a) = [| Foo a |]
lift ((:^:) u v) = [| (:^:) u v |]
liftTyped (Foo a) = [|| Foo a ||]
liftTyped ((:^:) u v) = [|| (:^:) u v ||]
-}
-----
{-# LANGUAGE TemplateHaskell #-}
module Baz where
import Bar
import Language.Haskell.TH.Lift
foo :: Foo String
foo = $(lift $ Foo "foo")
fooExp :: Lift a => Foo a -> Q Exp
fooExp f = [| f |]
Note that the Lift typeclass takes advantage of Representation polymorphism in order
to support instances involving unboxed types. This means DeriveLift
also works for these types:
{-# LANGUAGE DeriveLift, MagicHash #-}
module Unboxed where
import GHC.Exts
import Language.Haskell.TH.Syntax
data IntHash = IntHash Int# deriving Lift
{-
instance Lift IntHash where
lift (IntHash i) = [| IntHash i |]
liftTyped (IntHash i) = [|| IntHash i ||]
-}