Data.CZipWith

Lifted versions of the Functor, Pointed, Apply and Traversable classes, plus template-haskell magic to automatically derive instances. "Lifted" because these classes are about datatypes parameterized over a constructor (i.e. of kind (* -> *) -> *). For example fmap :: (a -> b) -> f a -> f b becomes cMap :: (forall a . f a -> g a) -> c f -> c g.

For the lifted version of Applicative, we focus on liftA2 instead of \<*\> as this is the only way to make the lifted version work. As a consequence, the class and method are named after zipWith because of the similarity of the signatures and the semantics.

liftA2   :: Applicative f =>         (g   -> h   -> i  ) -> f g -> f h -> f i
zipWith  ::                          (g   -> h   -> i  ) -> [g] -> [h] -> [i]
cZipWith :: CZipWith k => (forall a . g a -> h a -> i a) -> k g -> k h -> k i

Types of the corresponding kind occur for example when handling program configuration: When we define our an example configuration type like

data MyConfig f = MyConfig
  { flag_foo       :: f Bool
  , flag_bar       :: f Bool
  , flag_someLimit :: f Int
  }

then

• MyConfig Maybe can be used as the result-type of parsing the commandline or a configuration file; it includes the option that some field was not specified;
• MyConfig Identity can be used to represent both the default configuration and the actual configuration derived from defaults and the user input;
• MyConfig (Const Text) type to represent documentation for our config, to be displayed to the user.

This has the advantage that our configuration is defined in one place only, so that changes are easy to make and we do not ever run into any internal desynchonization of different datatypes. And once we obtained the final config :: MyConfig Identity, we don't have to think about Nothing cases anymore.

cPointed can initialize such polymorphic containers, and CZipWith further helps with this use-case, more specifically the merging of input and default config: we can express the merging of user/default config :: MyConfig Maybe -> MyConfig Identity -> MyConfig Identity in terms of cZipWith. The instances are simple boilerplate and thus can be realized using the provided template-haskell.

As an example for such usage, the brittany package uses this approach together with using automatically-derived Semigroup-instances that allow merging of config values (for example when commandline args do not override, but are added to those settings read from config file). See the module containing the config type.

Documentation

class CFunctor c where

The "lifted Functor" class

Methods

cMap :: (forall a. f a -> g a) -> c f -> c g

cMap :: CZipWith c => (forall a. f a -> g a) -> c f -> c g

class CPointed c where

The "lifted Apply" class

Minimal complete definition

cPoint

Methods

cPoint :: (forall a. f a) -> c f

class CZipWith k where

laws:

• cZipWith (\x _ -> x) g _ = g

• cZipWith (\_ y -> y) _ h = h

This class seems to be some kind of "lifted" version of Applicative (or rather: of Apply), but it also seems to share an important property with the Distributive class from the distributive package, even when Distributive and CZipWith methods don't appear all that similar. From the corresponding docs:

To be distributable a container will need to have a way to consistently
zip a potentially infinite number of copies of itself. This effectively
means that the holes in all values of that type, must have the same
cardinality, fixed sized vectors, infinite streams, functions, etc.
and no extra information to try to merge together.

Especially "all values of that type must have the same cardinality" is true for instances of CZipWith, the only difference being that the "holes" are instantiations of the f :: * -> * to some type, where they are simply a :: * for Distributive.

For many Distributive instances there are corresponding datatypes that are instances of CZipWith (although they do not seem particularly useful..), for example:

newtype CUnit a f = CUnit (f a)      -- corresponding to Identity
data CPair a b f = CPair (f a) (f b) -- corresponding to 'data MonoPair a = MonoPair a a'
                                     -- (Pair being a trivial fixed-size vector example)
data CStream a f = CStream (f a) (CStream a f) -- corresponding to an infinite stream

Minimal complete definition

cZipWith

Methods

cZipWith :: (forall a. g a -> h a -> i a) -> k g -> k h -> k i

zipWith on constructors instead of values.

class CZipWith c => CZipWithM c where

Where CZipWith is a "lifted Apply", this is a "lifted Traversable".

laws:

naturality

t . cTraverse f = cTraverse (t . f) for every applicative transformation t

identity

cTraverse Identity = Identity

composition

cTraverse (Compose . fmap g . f) = Compose . fmap (cTraverse g) . cTraverse f

and cZipWithM f k l must behave like cTraverse getCompose (cZipWith (x y -> Compose (f x y)) k l)

Minimal complete definition

cTraverse | cZipWithM

Methods

cTraverse :: Applicative m => (forall a. f a -> m (g a)) -> c f -> m (c g)

cZipWithM :: Applicative m => (forall a. f a -> g a -> m (h a)) -> c f -> c g -> m (c h)

cSequence :: Applicative m => CZipWithM c => c (Compose m f) -> m (c f)

The equivalent of Traversable's sequence/sequenceA

deriveCPointed :: Name -> DecsQ

Derives a cPointed instance for a datatype of kind (* -> *) -> *.

Requires that for this datatype (we shall call its argument f :: * -> * here)

• there is exactly one constructor;
• all fields in the one constructor are either of the form f x for some x or of the form X f for some type X where there is an instance cPointed X.

For example, the following would be valid usage:

data A f = A
  { a_str  :: f String
  , a_bool :: f Bool
  }

data B f = B
  { b_int   :: f Int
  , b_float :: f Float
  , b_a     :: A f
  }

derivecPointed ''A
derivecPointed ''B

This produces the following instances:

instance cPointed A where
  cPoint f = A f f

instance cPointed B where
  cPoint f = B f f (cPoint f f)

deriveCZipWith :: Name -> DecsQ

Derives a CZipWith instance for a datatype of kind (* -> *) -> *.

Requires that for this datatype (we shall call its argument f :: * -> * here)

• there is exactly one constructor;
• all fields in the one constructor are either of the form f x for some x or of the form X f for some type X where there is an instance CZipWith X.

For example, the following would be valid usage:

data A f = A
  { a_str  :: f String
  , a_bool :: f Bool
  }

data B f = B
  { b_int   :: f Int
  , b_float :: f Float
  , b_a     :: A f
  }

deriveCZipWith ''A
deriveCZipWith ''B

This produces the following instances:

instance CZipWith A where
  cZipWith f (A x1 x2) (A y1 y2) = A (f x1 y1) (f x2 y2)

instance CZipWith B where
  cZipWith f (B x1 x2 x3) (B y1 y2 y3) =
    B (f x1 y1) (f x2 y2) (cZipWith f x3 y3)

deriveCZipWithM :: Name -> DecsQ

Derives a CZipWithM instance for a datatype of kind (* -> *) -> *.

Requires that for this datatype (we shall call its argument f :: * -> * here)

• there is exactly one constructor;
• all fields in the one constructor are either of the form f x for some x or of the form X f for some type X where there is an instance CZipWithM X.

For example, the following would be valid usage:

data A f = A
  { a_str  :: f String
  , a_bool :: f Bool
  }

data B f = B
  { b_int   :: f Int
  , b_float :: f Float
  , b_a     :: A f
  }

deriveCZipWithM ''A
deriveCZipWithM ''B

This produces the following instances:

instance CZipWithM A where
  cZipWithM f (A x1 x2) (A y1 y2) = A <$> f x1 y1 <*> f x2 y2

instance CZipWith B where
  cZipWithM f (B x1 x2 x3) (B y1 y2 y3) =
    B <$> f x1 y1 <*> f x2 y2 <*> cZipWithM f x3 y3