Safe Haskell	Safe
Language	Haskell2010

Papa.Base.Export.Data.Traversable

Synopsis

Documentation

class (Functor t, Foldable t) => Traversable (t :: * -> *) where #

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.

```
t (pure x) = pure x
```
```
t (x <*> y) = t x <*> t y
```

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).

Minimal complete definition

traverse | sequenceA

Methods

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

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) #

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

Instances

Traversable []	Since: 2.1
Methods traverse :: Applicative f => (a -> f b) -> [a] -> f [b] # sequenceA :: Applicative f => [f a] -> f [a] # mapM :: Monad m => (a -> m b) -> [a] -> m [b] # sequence :: Monad m => [m a] -> m [a] #
Traversable Maybe	Since: 2.1
Methods traverse :: Applicative f => (a -> f b) -> Maybe a -> f (Maybe b) # sequenceA :: Applicative f => Maybe (f a) -> f (Maybe a) # mapM :: Monad m => (a -> m b) -> Maybe a -> m (Maybe b) # sequence :: Monad m => Maybe (m a) -> m (Maybe a) #
Traversable Par1
Methods traverse :: Applicative f => (a -> f b) -> Par1 a -> f (Par1 b) # sequenceA :: Applicative f => Par1 (f a) -> f (Par1 a) # mapM :: Monad m => (a -> m b) -> Par1 a -> m (Par1 b) # sequence :: Monad m => Par1 (m a) -> m (Par1 a) #
Traversable Min	Since: 4.9.0.0
Methods traverse :: Applicative f => (a -> f b) -> Min a -> f (Min b) # sequenceA :: Applicative f => Min (f a) -> f (Min a) # mapM :: Monad m => (a -> m b) -> Min a -> m (Min b) # sequence :: Monad m => Min (m a) -> m (Min a) #
Traversable Max	Since: 4.9.0.0
Methods traverse :: Applicative f => (a -> f b) -> Max a -> f (Max b) # sequenceA :: Applicative f => Max (f a) -> f (Max a) # mapM :: Monad m => (a -> m b) -> Max a -> m (Max b) # sequence :: Monad m => Max (m a) -> m (Max a) #
Traversable First	Since: 4.9.0.0
Methods traverse :: Applicative f => (a -> f b) -> First a -> f (First b) # sequenceA :: Applicative f => First (f a) -> f (First a) # mapM :: Monad m => (a -> m b) -> First a -> m (First b) # sequence :: Monad m => First (m a) -> m (First a) #
Traversable Last	Since: 4.9.0.0
Methods traverse :: Applicative f => (a -> f b) -> Last a -> f (Last b) # sequenceA :: Applicative f => Last (f a) -> f (Last a) # mapM :: Monad m => (a -> m b) -> Last a -> m (Last b) # sequence :: Monad m => Last (m a) -> m (Last a) #
Traversable Option	Since: 4.9.0.0
Methods traverse :: Applicative f => (a -> f b) -> Option a -> f (Option b) # sequenceA :: Applicative f => Option (f a) -> f (Option a) # mapM :: Monad m => (a -> m b) -> Option a -> m (Option b) # sequence :: Monad m => Option (m a) -> m (Option a) #
Traversable NonEmpty	Since: 4.9.0.0
Methods traverse :: Applicative f => (a -> f b) -> NonEmpty a -> f (NonEmpty b) # sequenceA :: Applicative f => NonEmpty (f a) -> f (NonEmpty a) # mapM :: Monad m => (a -> m b) -> NonEmpty a -> m (NonEmpty b) # sequence :: Monad m => NonEmpty (m a) -> m (NonEmpty a) #
Traversable ZipList	Since: 4.9.0.0
Methods traverse :: Applicative f => (a -> f b) -> ZipList a -> f (ZipList b) # sequenceA :: Applicative f => ZipList (f a) -> f (ZipList a) # mapM :: Monad m => (a -> m b) -> ZipList a -> m (ZipList b) # sequence :: Monad m => ZipList (m a) -> m (ZipList a) #
Traversable Identity
Methods traverse :: Applicative f => (a -> f b) -> Identity a -> f (Identity b) # sequenceA :: Applicative f => Identity (f a) -> f (Identity a) # mapM :: Monad m => (a -> m b) -> Identity a -> m (Identity b) # sequence :: Monad m => Identity (m a) -> m (Identity a) #
Traversable Dual	Since: 4.8.0.0
Methods traverse :: Applicative f => (a -> f b) -> Dual a -> f (Dual b) # sequenceA :: Applicative f => Dual (f a) -> f (Dual a) # mapM :: Monad m => (a -> m b) -> Dual a -> m (Dual b) # sequence :: Monad m => Dual (m a) -> m (Dual a) #
Traversable Sum	Since: 4.8.0.0
Methods traverse :: Applicative f => (a -> f b) -> Sum a -> f (Sum b) # sequenceA :: Applicative f => Sum (f a) -> f (Sum a) # mapM :: Monad m => (a -> m b) -> Sum a -> m (Sum b) # sequence :: Monad m => Sum (m a) -> m (Sum a) #
Traversable Product	Since: 4.8.0.0
Methods traverse :: Applicative f => (a -> f b) -> Product a -> f (Product b) # sequenceA :: Applicative f => Product (f a) -> f (Product a) # mapM :: Monad m => (a -> m b) -> Product a -> m (Product b) # sequence :: Monad m => Product (m a) -> m (Product a) #
Traversable First	Since: 4.8.0.0
Methods traverse :: Applicative f => (a -> f b) -> First a -> f (First b) # sequenceA :: Applicative f => First (f a) -> f (First a) # mapM :: Monad m => (a -> m b) -> First a -> m (First b) # sequence :: Monad m => First (m a) -> m (First a) #
Traversable Last	Since: 4.8.0.0
Methods traverse :: Applicative f => (a -> f b) -> Last a -> f (Last b) # sequenceA :: Applicative f => Last (f a) -> f (Last a) # mapM :: Monad m => (a -> m b) -> Last a -> m (Last b) # sequence :: Monad m => Last (m a) -> m (Last a) #
Traversable (Either a)	Since: 4.7.0.0
Methods traverse :: Applicative f => (a -> f b) -> Either a a -> f (Either a b) # sequenceA :: Applicative f => Either a (f a) -> f (Either a a) # mapM :: Monad m => (a -> m b) -> Either a a -> m (Either a b) # sequence :: Monad m => Either a (m a) -> m (Either a a) #
Traversable (V1 *)
Methods traverse :: Applicative f => (a -> f b) -> V1 * a -> f (V1 * b) # sequenceA :: Applicative f => V1 * (f a) -> f (V1 * a) # mapM :: Monad m => (a -> m b) -> V1 * a -> m (V1 * b) # sequence :: Monad m => V1 * (m a) -> m (V1 * a) #
Traversable (U1 *)	Since: 4.9.0.0
Methods traverse :: Applicative f => (a -> f b) -> U1 * a -> f (U1 * b) # sequenceA :: Applicative f => U1 * (f a) -> f (U1 * a) # mapM :: Monad m => (a -> m b) -> U1 * a -> m (U1 * b) # sequence :: Monad m => U1 * (m a) -> m (U1 * a) #
Traversable ((,) a)	Since: 4.7.0.0
Methods traverse :: Applicative f => (a -> f b) -> (a, a) -> f (a, b) # sequenceA :: Applicative f => (a, f a) -> f (a, a) # mapM :: Monad m => (a -> m b) -> (a, a) -> m (a, b) # sequence :: Monad m => (a, m a) -> m (a, a) #
Ix i => Traversable (Array i)	Since: 2.1
Methods traverse :: Applicative f => (a -> f b) -> Array i a -> f (Array i b) # sequenceA :: Applicative f => Array i (f a) -> f (Array i a) # mapM :: Monad m => (a -> m b) -> Array i a -> m (Array i b) # sequence :: Monad m => Array i (m a) -> m (Array i a) #
Traversable (Arg a)	Since: 4.9.0.0
Methods traverse :: Applicative f => (a -> f b) -> Arg a a -> f (Arg a b) # sequenceA :: Applicative f => Arg a (f a) -> f (Arg a a) # mapM :: Monad m => (a -> m b) -> Arg a a -> m (Arg a b) # sequence :: Monad m => Arg a (m a) -> m (Arg a a) #
Traversable (Proxy *)	Since: 4.7.0.0
Methods traverse :: Applicative f => (a -> f b) -> Proxy * a -> f (Proxy * b) # sequenceA :: Applicative f => Proxy * (f a) -> f (Proxy * a) # mapM :: Monad m => (a -> m b) -> Proxy * a -> m (Proxy * b) # sequence :: Monad m => Proxy * (m a) -> m (Proxy * a) #
Traversable f => Traversable (Rec1 * f)
Methods traverse :: Applicative f => (a -> f b) -> Rec1 * f a -> f (Rec1 * f b) # sequenceA :: Applicative f => Rec1 * f (f a) -> f (Rec1 * f a) # mapM :: Monad m => (a -> m b) -> Rec1 * f a -> m (Rec1 * f b) # sequence :: Monad m => Rec1 * f (m a) -> m (Rec1 * f a) #
Traversable (URec * Char)
Methods traverse :: Applicative f => (a -> f b) -> URec * Char a -> f (URec * Char b) # sequenceA :: Applicative f => URec * Char (f a) -> f (URec * Char a) # mapM :: Monad m => (a -> m b) -> URec * Char a -> m (URec * Char b) # sequence :: Monad m => URec * Char (m a) -> m (URec * Char a) #
Traversable (URec * Double)
Methods traverse :: Applicative f => (a -> f b) -> URec * Double a -> f (URec * Double b) # sequenceA :: Applicative f => URec * Double (f a) -> f (URec * Double a) # mapM :: Monad m => (a -> m b) -> URec * Double a -> m (URec * Double b) # sequence :: Monad m => URec * Double (m a) -> m (URec * Double a) #
Traversable (URec * Float)
Methods traverse :: Applicative f => (a -> f b) -> URec * Float a -> f (URec * Float b) # sequenceA :: Applicative f => URec * Float (f a) -> f (URec * Float a) # mapM :: Monad m => (a -> m b) -> URec * Float a -> m (URec * Float b) # sequence :: Monad m => URec * Float (m a) -> m (URec * Float a) #
Traversable (URec * Int)
Methods traverse :: Applicative f => (a -> f b) -> URec * Int a -> f (URec * Int b) # sequenceA :: Applicative f => URec * Int (f a) -> f (URec * Int a) # mapM :: Monad m => (a -> m b) -> URec * Int a -> m (URec * Int b) # sequence :: Monad m => URec * Int (m a) -> m (URec * Int a) #
Traversable (URec * Word)
Methods traverse :: Applicative f => (a -> f b) -> URec * Word a -> f (URec * Word b) # sequenceA :: Applicative f => URec * Word (f a) -> f (URec * Word a) # mapM :: Monad m => (a -> m b) -> URec * Word a -> m (URec * Word b) # sequence :: Monad m => URec * Word (m a) -> m (URec * Word a) #
Traversable (URec * (Ptr ()))
Methods traverse :: Applicative f => (a -> f b) -> URec * (Ptr ()) a -> f (URec * (Ptr ()) b) # sequenceA :: Applicative f => URec * (Ptr ()) (f a) -> f (URec * (Ptr ()) a) # mapM :: Monad m => (a -> m b) -> URec * (Ptr ()) a -> m (URec * (Ptr ()) b) # sequence :: Monad m => URec * (Ptr ()) (m a) -> m (URec * (Ptr ()) a) #
Traversable (Const * m)	Since: 4.7.0.0
Methods traverse :: Applicative f => (a -> f b) -> Const * m a -> f (Const * m b) # sequenceA :: Applicative f => Const * m (f a) -> f (Const * m a) # mapM :: Monad m => (a -> m b) -> Const * m a -> m (Const * m b) # sequence :: Monad m => Const * m (m a) -> m (Const * m a) #
Traversable (K1 * i c)
Methods traverse :: Applicative f => (a -> f b) -> K1 * i c a -> f (K1 * i c b) # sequenceA :: Applicative f => K1 * i c (f a) -> f (K1 * i c a) # mapM :: Monad m => (a -> m b) -> K1 * i c a -> m (K1 * i c b) # sequence :: Monad m => K1 * i c (m a) -> m (K1 * i c a) #
(Traversable f, Traversable g) => Traversable ((:+:) * f g)
Methods traverse :: Applicative f => (a -> f b) -> (* :+: f) g a -> f ((* :+: f) g b) # sequenceA :: Applicative f => (* :+: f) g (f a) -> f ((* :+: f) g a) # mapM :: Monad m => (a -> m b) -> (* :+: f) g a -> m ((* :+: f) g b) # sequence :: Monad m => (* :+: f) g (m a) -> m ((* :+: f) g a) #
(Traversable f, Traversable g) => Traversable ((::) f g)
Methods traverse :: Applicative f => (a -> f b) -> (* :: f) g a -> f (( :: f) g b) # sequenceA :: Applicative f => ( :: f) g (f a) -> f (( :: f) g a) # mapM :: Monad m => (a -> m b) -> ( :: f) g a -> m (( :: f) g b) # sequence :: Monad m => ( :: f) g (m a) -> m (( :*: f) g a) #
Traversable f => Traversable (M1 * i c f)
Methods traverse :: Applicative f => (a -> f b) -> M1 * i c f a -> f (M1 * i c f b) # sequenceA :: Applicative f => M1 * i c f (f a) -> f (M1 * i c f a) # mapM :: Monad m => (a -> m b) -> M1 * i c f a -> m (M1 * i c f b) # sequence :: Monad m => M1 * i c f (m a) -> m (M1 * i c f a) #
(Traversable f, Traversable g) => Traversable ((:.:) * * f g)
Methods traverse :: Applicative f => (a -> f b) -> (* :.: ) f g a -> f (( :.: ) f g b) # sequenceA :: Applicative f => ( :.: ) f g (f a) -> f (( :.: ) f g a) # mapM :: Monad m => (a -> m b) -> ( :.: ) f g a -> m (( :.: ) f g b) # sequence :: Monad m => ( :.: ) f g (m a) -> m (( :.: *) f g a) #

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

for is traverse with its arguments flipped. For a version that ignores the results see for_.

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

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.

fmapDefault :: Traversable t => (a -> b) -> t a -> t b #

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 :: (Traversable t, Monoid m) => (a -> m) -> t a -> m #

This function may be used as a value for foldMap in a Foldable instance.

foldMapDefault f ≡ getConst . traverse (Const . f)