Documentation

Bitraversable identifies bifunctorial data structures whose elements can be traversed in order, performing Applicative or Monad actions at each element, and collecting a result structure with the same shape.

As opposed to Traversable data structures, which have one variety of element on which an action can be performed, Bitraversable data structures have two such varieties of elements.

A definition of bitraverse must satisfy the following laws:

Naturality

bitraverse (t . f) (t . g) ≡ t . bitraverse f g for every applicative transformation t

Identity

bitraverse Identity Identity ≡ Identity

Composition

Compose . fmap (bitraverse g1 g2) . bitraverse f1 f2 ≡ bitraverse (Compose . fmap g1 . f1) (Compose . fmap g2 . f2)

where an applicative transformation is a function

t :: (Applicative f, Applicative g) => f a -> g a

preserving the Applicative operations:

t (pure x) ≡ pure x
t (f <*> x) ≡ t f <*> t x

and the identity functor Identity and composition functors Compose are from Data.Functor.Identity and Data.Functor.Compose.

Some simple examples are Either and (,):

instance Bitraversable Either where
  bitraverse f _ (Left x) = Left <$> f x
  bitraverse _ g (Right y) = Right <$> g y

instance Bitraversable (,) where
  bitraverse f g (x, y) = (,) <$> f x <*> g y

Bitraversable relates to its superclasses in the following ways:

bimap f g ≡ runIdentity . bitraverse (Identity . f) (Identity . g)
bifoldMap f g ≡ getConst . bitraverse (Const . f) (Const . g)

These are available as bimapDefault and bifoldMapDefault respectively.

If the type is also an instance of Traversable, then it must satisfy (up to laziness):

traverse ≡ bitraverse pure

Since: base-4.10.0.0

Alias for bisequence.

Since: base-4.10.0.0

Sequences all the actions in a structure, building a new structure with the same shape using the results of the actions. For a version that ignores the results, see bisequence_.

bisequence ≡ bitraverse id id

Examples

>>> bisequence (Just 4, Nothing)
Nothing

>>> bisequence (Just 4, Just 5)
Just (4,5)

>>> bisequence ([1, 2, 3], [4, 5])
[(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)]

Since: base-4.10.0.0

Alias for bitraverse.

Since: base-4.10.0.0

bifor is bitraverse with the structure as the first argument. For a version that ignores the results, see bifor_.

Examples

>>> bifor (Left []) listToMaybe (find even)
Nothing

>>> bifor (Left [1, 2, 3]) listToMaybe (find even)
Just (Left 1)

>>> bifor (Right [4, 5]) listToMaybe (find even)
Just (Right 4)

>>> bifor ([1, 2, 3], [4, 5]) listToMaybe (find even)
Just (1,4)

>>> bifor ([], [4, 5]) listToMaybe (find even)
Nothing

Since: base-4.10.0.0

Alias for bifor.

Since: base-4.10.0.0

The bimapAccumL function behaves like a combination of bimap and bifoldl; it traverses a structure from left to right, threading a state of type a and using the given actions to compute new elements for the structure.

Examples

>>> bimapAccumL (\acc bool -> (acc + 1, show bool)) (\acc string -> (acc * 2, reverse string)) 3 (True, "foo")
(8,("True","oof"))

Since: base-4.10.0.0

The bimapAccumR function behaves like a combination of bimap and bifoldr; it traverses a structure from right to left, threading a state of type a and using the given actions to compute new elements for the structure.

Examples

>>> bimapAccumR (\acc bool -> (acc + 1, show bool)) (\acc string -> (acc * 2, reverse string)) 3 (True, "foo")
(7,("True","oof"))

Since: base-4.10.0.0

A default definition of bimap in terms of the Bitraversable operations.

bimapDefault f g ≡
     runIdentity . bitraverse (Identity . f) (Identity . g)

Since: base-4.10.0.0

A default definition of bifoldMap in terms of the Bitraversable operations.

bifoldMapDefault f g ≡
    getConst . bitraverse (Const . f) (Const . g)

Since: base-4.10.0.0