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

As opposed toTraversable 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 ofbitraverse must satisfy the following laws:

naturality

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

identity

bitraverseIdentityIdentity ≡Identity

composition

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

where anapplicative transformation is a function

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

preserving theApplicative operations:

t (pure x) =pure xt (f<*> x) = t f<*> t x

and the identity functorIdentity and composition functorsCompose are defined as

newtype Identity a = Identity { runIdentity :: a }instance Functor Identity where  fmap f (Identity x) = Identity (f x)instance Applicative Identity where  pure = Identity  Identity f <*> Identity x = Identity (f x)newtype Compose f g a = Compose (f (g a))instance (Functor f, Functor g) => Functor (Compose f g) where  fmap f (Compose x) = Compose (fmap (fmap f) x)instance (Applicative f, Applicative g) => Applicative (Compose f g) where  pure = Compose . pure . pure  Compose f <*> Compose x = Compose ((<*>) <$> f <*> x)

Some simple examples areEither and '(,)':

instance Bitraversable Either where  bitraverse f _ (Left x) = Left <$> f x  bitraverse _ g (Right y) = Right <$> g yinstance 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 asbimapDefault andbifoldMapDefault respectively.

Since: 4.10.0.0

Minimal complete definition

Nothing

Methods

bitraverse ::Applicative f => (a -> f c) -> (b -> f d) -> t a b -> f (t c d)Source#

bitraverse f g ≡bisequenceA .bimap f g

Alias forbisequence.

Since: 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, seebisequence_.

bisequence ≡bitraverseidid

Since: 4.10.0.0

Alias forbitraverse.

Since: 4.10.0.0

bifor isbitraverse with the structure as the first argument. For a version that ignores the results, seebifor_.

Since: 4.10.0.0

Alias forbifor.

Since: 4.10.0.0

ThebimapAccumL function behaves like a combination ofbimap andbifoldl; it traverses a structure from left to right, threading a state of typea and using the given actions to compute new elements for the structure.

Since: 4.10.0.0

ThebimapAccumR function behaves like a combination ofbimap andbifoldl; it traverses a structure from right to left, threading a state of typea and using the given actions to compute new elements for the structure.

Since: 4.10.0.0

A default definition ofbimap in terms of theBitraversable operations.

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

Since: 4.10.0.0

A default definition ofbifoldMap in terms of theBitraversable operations.

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

Since: 4.10.0.0