Examples of Type Functions

The map example from Associated Types with Class in the new form:

data family Map k :: * -> *

data instance Map ()     v = MapUnit (Maybe v)
data instance Map (a, b) v = MapPair (Map a (Map b v))

We can define operations on indexed maps using a type class whose instances corresponds to the type indexes. Note that a declaration, such as

data instance Map Int Char = Nonsense

is not acceptable, as it constraints the second argument of Map, which is not a type index, but a parametric argument.

Generic finite maps:

  class GMapKey k where
    data GMap k :: * -> *
    empty       :: GMap k v
    lookup      :: k -> GMap k v -> Maybe v
    insert      :: k -> v -> GMap k v -> GMap k v
  
  instance GMapKey Int where
    data GMap Int v        = GMapInt (Map.Map Int v)
    empty                  = GMapInt Map.empty
    lookup k (GMapInt m)   = Map.lookup k m
    insert k v (GMapInt m) = GMapInt (Map.insert k v m)
  
  instance GMapKey Char where
    data GMap Char v        = GMapChar (GMap Int v)
    empty                   = GMapChar empty
    lookup k (GMapChar m)   = lookup (ord k) m
    insert k v (GMapChar m) = GMapChar (insert (ord k) v m)
  
  instance GMapKey () where
    data GMap () v           = GMapUnit (Maybe v)
    empty                    = GMapUnit Nothing
    lookup () (GMapUnit v)   = v
    insert () v (GMapUnit _) = GMapUnit $ Just v
  
  instance (GMapKey a, GMapKey b) => GMapKey (a, b) where
    data GMap (a, b) v            = GMapPair (GMap a (GMap b v))
    empty                               = GMapPair empty
    lookup (a, b) (GMapPair gm)   = lookup a gm >>= lookup b 
    insert (a, b) v (GMapPair gm) = GMapPair $ case lookup a gm of
                                    Nothing  -> insert a (insert b v empty) gm
                                    Just gm2 -> insert a (insert b v gm2  ) gm
  
  instance (GMapKey a, GMapKey b) => GMapKey (Either a b) where
    data GMap (Either a b) v                = GMapEither (GMap a v) (GMap b v)
    empty                                   = GMapEither empty empty
    lookup (Left  a) (GMapEither gm1  _gm2) = lookup a gm1
    lookup (Right b) (GMapEither _gm1 gm2 ) = lookup b gm2
    insert (Left  a) v (GMapEither gm1 gm2) = GMapEither (insert a v gm1) gm2
    insert (Right a) v (GMapEither gm1 gm2) = GMapEither gm1 (insert a v gm2)