Copyright | Conor McBride and Ross Paterson 2005 |
---|---|

License | BSD-style (see the LICENSE file in the distribution) |

Maintainer | libraries@haskell.org |

Stability | experimental |

Portability | portable |

Safe Haskell | Trustworthy |

Language | Haskell2010 |

Class of data structures that can be traversed from left to right, performing an action on each element.

See also

- "Applicative Programming with Effects", by Conor McBride and Ross Paterson,
*Journal of Functional Programming*18:1 (2008) 1-13, online at http://www.soi.city.ac.uk/~ross/papers/Applicative.html. - "The Essence of the Iterator Pattern", by Jeremy Gibbons and Bruno Oliveira, in
*Mathematically-Structured Functional Programming*, 2006, online at http://web.comlab.ox.ac.uk/oucl/work/jeremy.gibbons/publications/#iterator. - "An Investigation of the Laws of Traversals", by Mauro Jaskelioff and Ondrej Rypacek, in
*Mathematically-Structured Functional Programming*, 2012, online at http://arxiv.org/pdf/1202.2919.

class (Functor t, Foldable t) => Traversable t where Source

Functors representing data structures that can be traversed from left to right.

A definition of `traverse`

must satisfy the following laws:

*naturality*-
`t . traverse f = traverse (t . f)`

for every applicative transformation`t`

*identity*`traverse Identity = Identity`

*composition*`traverse (Compose . fmap g . f) = Compose . fmap (traverse g) . traverse f`

A definition of `sequenceA`

must satisfy the following laws:

*naturality*-
`t . sequenceA = sequenceA . fmap t`

for every applicative transformation`t`

*identity*`sequenceA . fmap Identity = Identity`

*composition*`sequenceA . fmap Compose = Compose . fmap sequenceA . sequenceA`

where an *applicative transformation* is a function

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

preserving the `Applicative`

operations, i.e.

and the identity functor `Identity`

and composition of functors `Compose`

are defined as

newtype Identity a = Identity a instance Functor Identity where fmap f (Identity x) = Identity (f x) instance Applicative Identity where pure x = Identity x 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 x = Compose (pure (pure x)) Compose f <*> Compose x = Compose ((<*>) <$> f <*> x)

(The naturality law is implied by parametricity.)

Instances are similar to `Functor`

, e.g. given a data type

data Tree a = Empty | Leaf a | Node (Tree a) a (Tree a)

a suitable instance would be

instance Traversable Tree where traverse f Empty = pure Empty traverse f (Leaf x) = Leaf <$> f x traverse f (Node l k r) = Node <$> traverse f l <*> f k <*> traverse f r

This is suitable even for abstract types, as the laws for `<*>`

imply a form of associativity.

The superclass instances should satisfy the following:

- In the
`Functor`

instance,`fmap`

should be equivalent to traversal with the identity applicative functor (`fmapDefault`

). - In the
`Foldable`

instance,`foldMap`

should be equivalent to traversal with a constant applicative functor (`foldMapDefault`

).

traverse :: Applicative f => (a -> f b) -> t a -> f (t b) Source

Map each element of a structure to an action, evaluate these actions from left to right, and collect the results. For a version that ignores the results see `traverse_`

.

sequenceA :: Applicative f => t (f a) -> f (t a) Source

Evaluate each action in the structure from left to right, and and collect the results. For a version that ignores the results see `sequenceA_`

.

mapM :: Monad m => (a -> m b) -> t a -> m (t b) Source

Map each element of a structure to a monadic action, evaluate these actions from left to right, and collect the results. For a version that ignores the results see `mapM_`

.

sequence :: Monad m => t (m a) -> m (t a) Source

Evaluate each monadic action in the structure from left to right, and collect the results. For a version that ignores the results see `sequence_`

.

for :: (Traversable t, Applicative f) => t a -> (a -> f b) -> f (t b) Source

`for`

is `traverse`

with its arguments flipped. For a version that ignores the results see `for_`

.

forM :: (Traversable t, Monad m) => t a -> (a -> m b) -> m (t b) Source

`forM`

is `mapM`

with its arguments flipped. For a version that ignores the results see `forM_`

.

mapAccumL :: Traversable t => (a -> b -> (a, c)) -> a -> t b -> (a, t c) Source

The `mapAccumL`

function behaves like a combination of `fmap`

and `foldl`

; it applies a function to each element of a structure, passing an accumulating parameter from left to right, and returning a final value of this accumulator together with the new structure.

mapAccumR :: Traversable t => (a -> b -> (a, c)) -> a -> t b -> (a, t c) Source

The `mapAccumR`

function behaves like a combination of `fmap`

and `foldr`

; it applies a function to each element of a structure, passing an accumulating parameter from right to left, and returning a final value of this accumulator together with the new structure.

fmapDefault :: forall t a b. Traversable t => (a -> b) -> t a -> t b Source

This function may be used as a value for `fmap`

in a `Functor`

instance, provided that `traverse`

is defined. (Using `fmapDefault`

with a `Traversable`

instance defined only by `sequenceA`

will result in infinite recursion.)

`fmapDefault`

f ≡`runIdentity`

.`traverse`

(`Identity`

. f)

foldMapDefault :: forall t m a. (Traversable t, Monoid m) => (a -> m) -> t a -> m Source

This function may be used as a value for `foldMap`

in a `Foldable`

instance.

`foldMapDefault`

f ≡`getConst`

.`traverse`

(`Const`

. f)

© The University of Glasgow and others

Licensed under a BSD-style license (see top of the page).

https://downloads.haskell.org/~ghc/8.2.1/docs/html/libraries/base-4.10.0.0/Data-Traversable.html