Foldable and Traversable
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Data.Sequence is recommended as an efficient alternative to [list]s, with a more symmetric feel and better complexity on various operations.
When you've been using it for a little while, there seem to be some baffling omissions from the API. The first couple you are likely to notice are the absence of "map
toList
The answer to these lies in the long list of instances which Sequence has:
 The Sequence version of map is "", which comes from the Functor class.fmap
 The Sequence version of is in thetoListclass.Foldable
Sequence
Foldable
Traversable
Functor
Contents 
1 What do these classes all mean? A brief tour:
1.1 Functor
A functor is simply a container. Given a container, and a function which works on the elements, we can apply that function to each element. For lists, the familiar "Functor
map
Note that the function can produce elements of a different type, so we may have a different type at the end.
Examples:
Prelude Data.Sequence> map (\n > replicate n 'a') [1,3,5] ["a","aaa","aaaaa"] Prelude Data.Sequence> fmap (\n > replicate n 'a') (1 < 3 < 5 < empty) fromList ["a","aaa","aaaaa"]
1.2 Foldable
AFoldable
Functor
Foldable
Functor
foldr
foldr
toList = foldr (:) []
Foldable
Foldable
Functor
toList
fmap
Data.Sequence
toList
mapM_
(>>)
Foldable
sequence_
1.3 Traversable
ATraversable
Foldable
Foldable
foldr
Traversable
Traversable
mapM
sequence
2 Some trickier functions: concatMap and filter
NeitherTraversable
Foldable
concatMap
filter
Foldable
Traversable
concatMap
filter
concatMap
Sequence
concatMap :: (a > Seq b) > Seq a > Seq b concatMap = foldMap
Monoid
concatMap :: (Foldable f, Monoid (f b)) => (a > f b) > f a > f b concatMap = foldMap
toList (f `mappend` g) = toList f ++ toList g
filter
Sequence
\a > [a]
pure
Applicative
Applicative
filter :: (Applicative f, Foldable f, Monoid (f a)) => (a > Bool) > f a > f a filter p = foldMap (\a > if p a then pure a else mempty)
Foldable
Monad
concatMap
>>=
pure
return
3 Generalising zipWith
Another really useful list combinator that doesn't appear in the interfaces forSequence
Foldable
Traversable
zipWith
zipWith
Traversable
Traversable
Foldable
Monad
Applicative
import Prelude hiding (sequence) import Data.Sequence import Data.Foldable import Data.Traversable import Control.Applicative data Supply s v = Supply { unSupply :: [s] > ([s],v) } instance Functor (Supply s) where fmap f av = Supply (\l > let (l',v) = unSupply av l in (l',f v)) instance Applicative (Supply s) where pure v = Supply (\l > (l,v)) af <*> av = Supply (\l > let (l',f) = unSupply af l (l'',v) = unSupply av l' in (l'',f v)) runSupply :: (Supply s v) > [s] > v runSupply av l = snd $ unSupply av l supply :: Supply s s supply = Supply (\(x:xs) > (xs,x)) zipTF :: (Traversable t, Foldable f) => t a > f b > t (a,b) zipTF t f = runSupply (traverse (\a > (,) a <$> supply) t) (toList f) zipWithTF :: (Traversable t,Foldable f) => (a > b > c) > t a > f b > t c zipWithTF g t f = runSupply (traverse (\a > g a <$> supply) t) (toList f) zipWithTFM :: (Traversable t,Foldable f,Monad m) => (a > b > m c) > t a > f b > m (t c) zipWithTFM g t f = sequence (zipWithTF g t f) zipWithTFA :: (Traversable t,Foldable f,Applicative m) => (a > b > m c) > t a > f b > m (t c) zipWithTFA g t f = sequenceA (zipWithTF g t f)
Foldable
State
module GenericZip (zipWithTF, zipTF, zipWithTFA, zipWithTFM) where import Data.Foldable import Data.Traversable import qualified Data.Traversable as T import Control.Applicative import Control.Monad.State   The state contains the list of values obtained form the foldable container  and a String indicating the name of the function currectly being executed data ZipState a = ZipState {fName :: String, list :: [a]}   State monad containing ZipState type ZipM l a = State (ZipState l) a   pops the first element of the list inside the state pop :: ZipM l l pop = do st < get let xs = list st n = fName st case xs of (a:as) > do put st{list=as} return a [] > error $ n ++ ": insufficient input"   pop a value form the state and supply it to the second  argument of a binary function supplySecond :: (a > b > c) > a > ZipM b c supplySecond f a = do b < pop return $ f a b zipWithTFError :: (Traversable t,Foldable f) => String > (a > b > c) > t a > f b > t c zipWithTFError str g t f = evalState (T.mapM (supplySecond g) t) (ZipState str (toList f)) zipWithTF :: (Traversable t,Foldable f) => (a > b > c) > t a > f b > t c zipWithTF = zipWithTFError "GenericZip.zipWithTF" zipTF :: (Traversable t, Foldable f) => t a > f b > t (a,b) zipTF = zipWithTFError "GenericZip.zipTF" (,) zipWithTFM :: (Traversable t,Foldable f,Monad m) => (a > b > m c) > t a > f b > m (t c) zipWithTFM g t f = T.sequence (zipWithTFError "GenericZip.zipWithTFM" g t f) zipWithTFA :: (Traversable t,Foldable f,Applicative m) => (a > b > m c) > t a > f b > m (t c) zipWithTFA g t f = sequenceA (zipWithTFError "GenericZip.zipWithTFA" g t f)
Data.Traversable
mapAccumL
mapAccumR
mapAccumL :: Traversable t => (a > b > (a, c)) > a > t b > (a, t c) mapAccumR :: Traversable t => (a > b > (a, c)) > a > t b > (a, t c)
Using these, the first version above can be written as
zipWithTF :: (Traversable t, Foldable f) => (a > b > c) > t a > f b > t c zipWithTF g t f = snd (mapAccumL map_one (toList f) t) where map_one (x:xs) y = (xs, g y x)
mapAccumL
mapAccumR
reverse
reverseT :: (Traversable t) => t a > t a reverseT t = snd (mapAccumR (\ (x:xs) _ > (xs, x)) (toList t) t)