{-# LANGUAGE Trustworthy #-} ----------------------------------------------------------------------------- -- | -- Module : Data.Traversable -- 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 -- -- 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>. -- -- Note that the functions 'mapM' and 'sequence' generalize "Prelude" -- functions of the same names from lists to any 'Traversable' functor. -- To avoid ambiguity, either import the "Prelude" hiding these names -- or qualify uses of these function names with an alias for this module. -- ----------------------------------------------------------------------------- module Data.Traversable ( -- * The 'Traversable' class Traversable(..), -- * Utility functions for, forM, mapAccumL, mapAccumR, -- * General definitions for superclass methods fmapDefault, foldMapDefault, ) where import Prelude hiding (mapM, sequence, foldr) import qualified Prelude (mapM, foldr) import Control.Applicative import Data.Foldable (Foldable()) import Data.Monoid (Monoid) import Data.Proxy import GHC.Arr -- | Functors representing data structures that can be traversed from -- left to right. -- -- Minimal complete definition: 'traverse' or 'sequenceA'. -- -- 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 Indentity 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 <$> traverse f l <*> f k <*> traverse f r -- -- This is suitable even for abstract types, as the laws for '<*>' -- imply a form of associativity. -- -- The superclass instances should satisfy the following: -- -- * In the 'Functor' instance, 'fmap' should be equivalent to traversal -- with the identity applicative functor ('fmapDefault'). -- -- * In the 'Foldable' instance, 'Data.Foldable.foldMap' should be -- equivalent to traversal with a constant applicative functor -- ('foldMapDefault'). -- class (Functor t, Foldable t) => Traversable t where -- | Map each element of a structure to an action, evaluate -- these actions from left to right, and collect the results. traverse :: Applicative f => (a -> f b) -> t a -> f (t b) traverse f = sequenceA . fmap f -- | Evaluate each action in the structure from left to right, -- and collect the results. sequenceA :: Applicative f => t (f a) -> f (t a) sequenceA = traverse id -- | Map each element of a structure to a monadic action, evaluate -- these actions from left to right, and collect the results. mapM :: Monad m => (a -> m b) -> t a -> m (t b) mapM f = unwrapMonad . traverse (WrapMonad . f) -- | Evaluate each monadic action in the structure from left to right, -- and collect the results. sequence :: Monad m => t (m a) -> m (t a) sequence = mapM id {-# MINIMAL traverse | sequenceA #-} -- instances for Prelude types instance Traversable Maybe where traverse _ Nothing = pure Nothing traverse f (Just x) = Just <$> f x instance Traversable [] where {-# INLINE traverse #-} -- so that traverse can fuse traverse f = Prelude.foldr cons_f (pure []) where cons_f x ys = (:) <$> f x <*> ys mapM = Prelude.mapM instance Traversable (Either a) where traverse _ (Left x) = pure (Left x) traverse f (Right y) = Right <$> f y instance Traversable ((,) a) where traverse f (x, y) = (,) x <$> f y instance Ix i => Traversable (Array i) where traverse f arr = listArray (bounds arr) `fmap` traverse f (elems arr) instance Traversable Proxy where traverse _ _ = pure Proxy {-# INLINE traverse #-} sequenceA _ = pure Proxy {-# INLINE sequenceA #-} mapM _ _ = return Proxy {-# INLINE mapM #-} sequence _ = return Proxy {-# INLINE sequence #-} instance Traversable (Const m) where traverse _ (Const m) = pure $ Const m -- general functions -- | 'for' is 'traverse' with its arguments flipped. for :: (Traversable t, Applicative f) => t a -> (a -> f b) -> f (t b) {-# INLINE for #-} for = flip traverse -- | 'forM' is 'mapM' with its arguments flipped. forM :: (Traversable t, Monad m) => t a -> (a -> m b) -> m (t b) {-# INLINE forM #-} forM = flip mapM -- left-to-right state transformer newtype StateL s a = StateL { runStateL :: s -> (s, a) } instance Functor (StateL s) where fmap f (StateL k) = StateL $ \ s -> let (s', v) = k s in (s', f v) instance Applicative (StateL s) where pure x = StateL (\ s -> (s, x)) StateL kf <*> StateL kv = StateL $ \ s -> let (s', f) = kf s (s'', v) = kv s' in (s'', f v) -- |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. mapAccumL :: Traversable t => (a -> b -> (a, c)) -> a -> t b -> (a, t c) mapAccumL f s t = runStateL (traverse (StateL . flip f) t) s -- right-to-left state transformer newtype StateR s a = StateR { runStateR :: s -> (s, a) } instance Functor (StateR s) where fmap f (StateR k) = StateR $ \ s -> let (s', v) = k s in (s', f v) instance Applicative (StateR s) where pure x = StateR (\ s -> (s, x)) StateR kf <*> StateR kv = StateR $ \ s -> let (s', v) = kv s (s'', f) = kf s' in (s'', f v) -- |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. mapAccumR :: Traversable t => (a -> b -> (a, c)) -> a -> t b -> (a, t c) mapAccumR f s t = runStateR (traverse (StateR . flip f) t) s -- | 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 :: Traversable t => (a -> b) -> t a -> t b {-# INLINE fmapDefault #-} fmapDefault f = getId . traverse (Id . f) -- | This function may be used as a value for `Data.Foldable.foldMap` -- in a `Foldable` instance. foldMapDefault :: (Traversable t, Monoid m) => (a -> m) -> t a -> m foldMapDefault f = getConst . traverse (Const . f) -- local instances newtype Id a = Id { getId :: a } instance Functor Id where fmap f (Id x) = Id (f x) instance Applicative Id where pure = Id Id f <*> Id x = Id (f x)