Control.Lens.Fold

A Fold a c is a generalization of something Foldable. It allows you to extract multiple results from a container. A Foldable container can be characterized by the behavior of foldMap :: (Foldable t, Monoid m) => (c -> m) -> t c -> m. Since we want to be able to work with monomorphic containers, we could generalize this signature to forall m. Monoid m => (c -> m) -> a -> m, and then decorate it with Accessor to obtain

type Fold a c = forall m b d. Monoid m => Getting m a b c d

Every Getter is a valid Fold that simply doesn't use the Monoid it is passed.

But in practice the type we use is slightly more complicated to allow for better error messages and for it to be transformed by certain Applicative transformers.

Everything you can do with a Foldable container, you can with with a Fold and there are combinators that generalize the usual Foldable operations here.

Folds

type Fold a c = forall f b d. (Gettable f, Applicative f) => (c -> f d) -> a -> f b

A Fold describes how to retrieve multiple values in a way that can be composed with other lens-like constructions.

A Fold a c provides a structure with operations very similar to those of the Foldable typeclass, see foldMapOf and the other Fold combinators.

By convention, if there exists a foo method that expects a Foldable (f c), then there should be a fooOf method that takes a Fold a c and a value of type a.

A Getter is a legal Fold that just ignores the supplied Monoid

Unlike a Traversal a Fold is read-only. Since a Fold cannot be used to write back there are no lens laws that apply.

Building Folds

folding :: (Foldable f, Applicative g, Gettable g) => (a -> f c) -> LensLike g a b c d

Obtain a Fold by lifting an operation that returns a foldable result.

This can be useful to lift operations from Data.List and elsewhere into a Fold.

folded :: Foldable f => Fold (f c) c

Obtain a Fold from any Foldable.

unfolded :: (b -> Maybe (a, b)) -> Fold b a

Build a fold that unfolds its values from a seed.

unfoldr = toListOf . unfolded

iterated :: (a -> a) -> Fold a a

x ^. iterated f Return an infinite fold of repeated applications of f to x.

 toListOf (iterated f) a = iterate f a

filtered :: (Gettable f, Applicative f) => (c -> Bool) -> LensLike f a b c d -> LensLike f a b c d

Obtain a Fold by filtering a Lens, Iso, Getter, Fold or Traversal.

backwards :: LensLike (Backwards f) a b c d -> LensLike f a b c d

This allows you to traverse the elements of a Traversal or Fold in the opposite order.

Note: backwards should have no impact on a Getter Setter, Lens or Iso.

To change the direction of an Iso, use from.

repeated :: Fold a a

Fold by repeating the input forever.

repeat = toListOf repeated

replicated :: Int -> Fold a a

A fold that replicates its input n times.

replicate n = toListOf (replicated n)

cycled :: (Applicative f, Gettable f) => LensLike f a b c d -> LensLike f a b c d

Transform a fold into a fold that loops over its elements over and over.

>>> import Control.Lens
>>> take 6 $ toListOf (cycled traverse) [1,2,3]
[1,2,3,1,2,3]

takingWhile :: (Gettable f, Applicative f) => (c -> Bool) -> Getting (Endo (f b)) a b c d -> LensLike f a b c d

Obtain a Fold by taking elements from another Fold, Lens, Iso, Getter or Traversal while a predicate holds.

takeWhile p = toListOf (takingWhile p folded)

>>> toListOf (takingWhile (<=3) folded) [1..]
[1,2,3]

droppingWhile :: (Gettable f, Applicative f) => (c -> Bool) -> Getting (Endo (f b)) a b c d -> LensLike f a b c d

Obtain a Fold by dropping elements from another Fold, Lens, Iso, Getter or Traversal while a predicate holds.

dropWhile p = toListOf (droppingWhile p folded)

>>> toListOf (droppingWhile (<=3) folded) [1..6]
[4,5,6]

Folding

foldMapOf :: Getting r a b c d -> (c -> r) -> a -> r

foldMap = foldMapOf folded

foldMapOf = views

 foldMapOf ::             Getter a c        -> (c -> r) -> a -> r
 foldMapOf :: Monoid r => Fold a c          -> (c -> r) -> a -> r
 foldMapOf ::             Lens a b c d      -> (c -> r) -> a -> r
 foldMapOf ::             Iso a b c d       -> (c -> r) -> a -> r
 foldMapOf :: Monoid r => Traversal a b c d -> (c -> r) -> a -> r

foldOf :: Getting c a b c d -> a -> c

fold = foldOf folded

foldOf = view

 foldOf ::             Getter a m        -> a -> m
 foldOf :: Monoid m => Fold a m          -> a -> m
 foldOf ::             Lens a b m d      -> a -> m
 foldOf ::             Iso a b m d       -> a -> m
 foldOf :: Monoid m => Traversal a b m d -> a -> m

foldrOf :: Getting (Endo e) a b c d -> (c -> e -> e) -> e -> a -> e

Right-associative fold of parts of a structure that are viewed through a Lens, Getter, Fold or Traversal.

foldr = foldrOf folded

 foldrOf :: Getter a c        -> (c -> e -> e) -> e -> a -> e
 foldrOf :: Fold a c          -> (c -> e -> e) -> e -> a -> e
 foldrOf :: Lens a b c d      -> (c -> e -> e) -> e -> a -> e
 foldrOf :: Iso a b c d       -> (c -> e -> e) -> e -> a -> e
 foldrOf :: Traversal a b c d -> (c -> e -> e) -> e -> a -> e

foldlOf :: Getting (Dual (Endo e)) a b c d -> (e -> c -> e) -> e -> a -> e

Left-associative fold of the parts of a structure that are viewed through a Lens, Getter, Fold or Traversal.

foldl = foldlOf folded

 foldlOf :: Getter a c        -> (e -> c -> e) -> e -> a -> e
 foldlOf :: Fold a c          -> (e -> c -> e) -> e -> a -> e
 foldlOf :: Lens a b c d      -> (e -> c -> e) -> e -> a -> e
 foldlOf :: Iso a b c d       -> (e -> c -> e) -> e -> a -> e
 foldlOf :: Traversal a b c d -> (e -> c -> e) -> e -> a -> e

toListOf :: Getting [c] a b c d -> a -> [c]

toList = toListOf folded

 toListOf :: Getter a c        -> a -> [c]
 toListOf :: Fold a c          -> a -> [c]
 toListOf :: Lens a b c d      -> a -> [c]
 toListOf :: Iso a b c d       -> a -> [c]
 toListOf :: Traversal a b c d -> a -> [c]

anyOf :: Getting Any a b c d -> (c -> Bool) -> a -> Bool

any = anyOf folded

 anyOf :: Getter a c        -> (c -> Bool) -> a -> Bool
 anyOf :: Fold a c          -> (c -> Bool) -> a -> Bool
 anyOf :: Lens a b c d      -> (c -> Bool) -> a -> Bool
 anyOf :: Iso a b c d       -> (c -> Bool) -> a -> Bool
 anyOf :: Traversal a b c d -> (c -> Bool) -> a -> Bool

allOf :: Getting All a b c d -> (c -> Bool) -> a -> Bool

all = allOf folded

 allOf :: Getter a c        -> (c -> Bool) -> a -> Bool
 allOf :: Fold a c          -> (c -> Bool) -> a -> Bool
 allOf :: Lens a b c d      -> (c -> Bool) -> a -> Bool
 allOf :: Iso a b c d       -> (c -> Bool) -> a -> Bool
 allOf :: Traversal a b c d -> (c -> Bool) -> a -> Bool

andOf :: Getting All a b Bool d -> a -> Bool

and = andOf folded

 andOf :: Getter a Bool       -> a -> Bool
 andOf :: Fold a Bool         -> a -> Bool
 andOf :: Lens a b Bool d     -> a -> Bool
 andOf :: Iso a b Bool d      -> a -> Bool
 andOf :: Traversl a b Bool d -> a -> Bool

orOf :: Getting Any a b Bool d -> a -> Bool

or = orOf folded

 orOf :: Getter a Bool        -> a -> Bool
 orOf :: Fold a Bool          -> a -> Bool
 orOf :: Lens a b Bool d      -> a -> Bool
 orOf :: Iso a b Bool d       -> a -> Bool
 orOf :: Traversal a b Bool d -> a -> Bool

productOf :: Getting (Product c) a b c d -> a -> c

product = productOf folded

 productOf ::          Getter a c        -> a -> c
 productOf :: Num c => Fold a c          -> a -> c
 productOf ::          Lens a b c d      -> a -> c
 productOf ::          Iso a b c d       -> a -> c
 productOf :: Num c => Traversal a b c d -> a -> c

sumOf :: Getting (Sum c) a b c d -> a -> c

sum = sumOf folded

sumOf _1 :: (a, b) -> a

sumOf (folded . _1) :: (Foldable f, Num a) => f (a, b) -> a

 sumOf ::          Getter a c        -> a -> c
 sumOf :: Num c => Fold a c          -> a -> c
 sumOf ::          Lens a b c d      -> a -> c
 sumOf ::          Iso a b c d       -> a -> c
 sumOf :: Num c => Traversal a b c d -> a -> c

traverseOf_ :: Functor f => Getting (Traversed f) a b c d -> (c -> f e) -> a -> f ()

When passed a Getter, traverseOf_ can work over a Functor.

When passed a Fold, traverseOf_ requires an Applicative.

traverse_ = traverseOf_ folded

traverseOf_ _2 :: Functor f => (c -> f e) -> (c1, c) -> f ()

traverseOf_ traverseLeft :: Applicative f => (a -> f b) -> Either a c -> f ()

The rather specific signature of traverseOf_ allows it to be used as if the signature was either:

 traverseOf_ :: Functor f     => Getter a c        -> (c -> f e) -> a -> f ()
 traverseOf_ :: Applicative f => Fold a c          -> (c -> f e) -> a -> f ()
 traverseOf_ :: Functor f     => Lens a b c d      -> (c -> f e) -> a -> f ()
 traverseOf_ :: Functor f     => Iso a b c d       -> (c -> f e) -> a -> f ()
 traverseOf_ :: Applicative f => Traversal a b c d -> (c -> f e) -> a -> f ()

forOf_ :: Functor f => Getting (Traversed f) a b c d -> a -> (c -> f e) -> f ()

for_ = forOf_ folded

 forOf_ :: Functor f     => Getter a c        -> a -> (c -> f e) -> f ()
 forOf_ :: Applicative f => Fold a c          -> a -> (c -> f e) -> f ()
 forOf_ :: Functor f     => Lens a b c d      -> a -> (c -> f e) -> f ()
 forOf_ :: Functor f     => Iso a b c d       -> a -> (c -> f e) -> f ()
 forOf_ :: Applicative f => Traversal a b c d -> a -> (c -> f e) -> f ()

sequenceAOf_ :: Functor f => Getting (Traversed f) a b (f ()) d -> a -> f ()

sequenceA_ = sequenceAOf_ folded

 sequenceAOf_ :: Functor f     => Getter a (f ())        -> a -> f ()
 sequenceAOf_ :: Applicative f => Fold a (f ())          -> a -> f ()
 sequenceAOf_ :: Functor f     => Lens a b (f ()) d      -> a -> f ()
 sequenceAOf_ :: Functor f     => Iso a b (f ()) d       -> a -> f ()
 sequenceAOf_ :: Applicative f => Traversal a b (f ()) d -> a -> f ()

mapMOf_ :: Monad m => Getting (Sequenced m) a b c d -> (c -> m e) -> a -> m ()

mapM_ = mapMOf_ folded

 mapMOf_ :: Monad m => Getter a c        -> (c -> m e) -> a -> m ()
 mapMOf_ :: Monad m => Fold a c          -> (c -> m e) -> a -> m ()
 mapMOf_ :: Monad m => Lens a b c d      -> (c -> m e) -> a -> m ()
 mapMOf_ :: Monad m => Iso a b c d       -> (c -> m e) -> a -> m ()
 mapMOf_ :: Monad m => Traversal a b c d -> (c -> m e) -> a -> m ()

forMOf_ :: Monad m => Getting (Sequenced m) a b c d -> a -> (c -> m e) -> m ()

forM_ = forMOf_ folded

 forMOf_ :: Monad m => Getter a c        -> a -> (c -> m e) -> m ()
 forMOf_ :: Monad m => Fold a c          -> a -> (c -> m e) -> m ()
 forMOf_ :: Monad m => Lens a b c d      -> a -> (c -> m e) -> m ()
 forMOf_ :: Monad m => Iso a b c d       -> a -> (c -> m e) -> m ()
 forMOf_ :: Monad m => Traversal a b c d -> a -> (c -> m e) -> m ()

sequenceOf_ :: Monad m => Getting (Sequenced m) a b (m c) d -> a -> m ()

sequence_ = sequenceOf_ folded

 sequenceOf_ :: Monad m => Getter a (m b)        -> a -> m ()
 sequenceOf_ :: Monad m => Fold a (m b)          -> a -> m ()
 sequenceOf_ :: Monad m => Lens a b (m b) d      -> a -> m ()
 sequenceOf_ :: Monad m => Iso a b (m b) d       -> a -> m ()
 sequenceOf_ :: Monad m => Traversal a b (m b) d -> a -> m ()

asumOf :: Alternative f => Getting (Endo (f c)) a b (f c) d -> a -> f c

The sum of a collection of actions, generalizing concatOf.

asum = asumOf folded

 asumOf :: Alternative f => Getter a c        -> a -> f c
 asumOf :: Alternative f => Fold a c          -> a -> f c
 asumOf :: Alternative f => Lens a b c d      -> a -> f c
 asumOf :: Alternative f => Iso a b c d       -> a -> f c
 asumOf :: Alternative f => Traversal a b c d -> a -> f c

msumOf :: MonadPlus m => Getting (Endo (m c)) a b (m c) d -> a -> m c

The sum of a collection of actions, generalizing concatOf.

msum = msumOf folded

 msumOf :: MonadPlus m => Getter a c        -> a -> m c
 msumOf :: MonadPlus m => Fold a c          -> a -> m c
 msumOf :: MonadPlus m => Lens a b c d      -> a -> m c
 msumOf :: MonadPlus m => Iso a b c d       -> a -> m c
 msumOf :: MonadPlus m => Traversal a b c d -> a -> m c

concatMapOf :: Getting [e] a b c d -> (c -> [e]) -> a -> [e]

concatMap = concatMapOf folded

 concatMapOf :: Getter a c        -> (c -> [e]) -> a -> [e]
 concatMapOf :: Fold a c          -> (c -> [e]) -> a -> [e]
 concatMapOf :: Lens a b c d      -> (c -> [e]) -> a -> [e]
 concatMapOf :: Iso a b c d       -> (c -> [e]) -> a -> [e]
 concatMapOf :: Traversal a b c d -> (c -> [e]) -> a -> [e]

concatOf :: Getting [e] a b [e] d -> a -> [e]

concat = concatOf folded

 concatOf :: Getter a [e]        -> a -> [e]
 concatOf :: Fold a [e]          -> a -> [e]
 concatOf :: Iso a b [e] d       -> a -> [e]
 concatOf :: Lens a b [e] d      -> a -> [e]
 concatOf :: Traversal a b [e] d -> a -> [e]

elemOf :: Eq c => Getting Any a b c d -> c -> a -> Bool

elem = elemOf folded

 elemOf :: Eq c => Getter a c        -> c -> a -> Bool
 elemOf :: Eq c => Fold a c          -> c -> a -> Bool
 elemOf :: Eq c => Lens a b c d      -> c -> a -> Bool
 elemOf :: Eq c => Iso a b c d       -> c -> a -> Bool
 elemOf :: Eq c => Traversal a b c d -> c -> a -> Bool

notElemOf :: Eq c => Getting All a b c d -> c -> a -> Bool

notElem = notElemOf folded

 notElemOf :: Eq c => Getter a c        -> c -> a -> Bool
 notElemOf :: Eq c => Fold a c          -> c -> a -> Bool
 notElemOf :: Eq c => Iso a b c d       -> c -> a -> Bool
 notElemOf :: Eq c => Lens a b c d      -> c -> a -> Bool
 notElemOf :: Eq c => Traversal a b c d -> c -> a -> Bool

lengthOf :: Getting (Sum Int) a b c d -> a -> Int

Note: this can be rather inefficient for large containers.

length = lengthOf folded

>>> lengthOf _1 ("hello",())
1

lengthOf (folded . folded) :: Foldable f => f (g a) -> Int

 lengthOf :: Getter a c        -> a -> Int
 lengthOf :: Fold a c          -> a -> Int
 lengthOf :: Lens a b c d      -> a -> Int
 lengthOf :: Iso a b c d       -> a -> Int
 lengthOf :: Traversal a b c d -> a -> Int

nullOf :: Getting All a b c d -> a -> Bool

Returns True if this Fold or Traversal has no targets in the given container.

Note: nullOf on a valid Iso, Lens or Getter should always return False

null = nullOf folded

This may be rather inefficient compared to the null check of many containers.

>>> nullOf _1 (1,2)
False

nullOf (folded . _1 . folded) :: Foldable f => f (g a, b) -> Bool

 nullOf :: Getter a c        -> a -> Bool
 nullOf :: Fold a c          -> a -> Bool
 nullOf :: Iso a b c d       -> a -> Bool
 nullOf :: Lens a b c d      -> a -> Bool
 nullOf :: Traversal a b c d -> a -> Bool

headOf :: Getting (First c) a b c d -> a -> Maybe c

Perform a safe head of a Fold or Traversal or retrieve Just the result from a Getter or Lens.

listToMaybe . toList = headOf folded

 headOf :: Getter a c        -> a -> Maybe c
 headOf :: Fold a c          -> a -> Maybe c
 headOf :: Lens a b c d      -> a -> Maybe c
 headOf :: Iso a b c d       -> a -> Maybe c
 headOf :: Traversal a b c d -> a -> Maybe c

lastOf :: Getting (Last c) a b c d -> a -> Maybe c

Perform a safe last of a Fold or Traversal or retrieve Just the result from a Getter or Lens.

 lastOf :: Getter a c        -> a -> Maybe c
 lastOf :: Fold a c          -> a -> Maybe c
 lastOf :: Lens a b c d      -> a -> Maybe c
 lastOf :: Iso a b c d       -> a -> Maybe c
 lastOf :: Traversal a b c d -> a -> Maybe c

maximumOf :: Getting (Max c) a b c d -> a -> Maybe c

Obtain the maximum element (if any) targeted by a Fold or Traversal

Note: maximumOf on a valid Iso, Lens or Getter will always return Just a value.

maximum = fromMaybe (error empty) . maximumOf folded

 maximumOf ::          Getter a c        -> a -> Maybe c
 maximumOf :: Ord c => Fold a c          -> a -> Maybe c
 maximumOf ::          Iso a b c d       -> a -> Maybe c
 maximumOf ::          Lens a b c d      -> a -> Maybe c
 maximumOf :: Ord c => Traversal a b c d -> a -> Maybe c

minimumOf :: Getting (Min c) a b c d -> a -> Maybe c

Obtain the minimum element (if any) targeted by a Fold or Traversal

Note: minimumOf on a valid Iso, Lens or Getter will always return Just a value.

minimum = fromMaybe (error empty) . minimumOf folded

 minimumOf ::          Getter a c        -> a -> Maybe c
 minimumOf :: Ord c => Fold a c          -> a -> Maybe c
 minimumOf ::          Iso a b c d       -> a -> Maybe c
 minimumOf ::          Lens a b c d      -> a -> Maybe c
 minimumOf :: Ord c => Traversal a b c d -> a -> Maybe c

maximumByOf :: Getting (Endo (Maybe c)) a b c d -> (c -> c -> Ordering) -> a -> Maybe c

Obtain the maximum element (if any) targeted by a Fold, Traversal, Lens, Iso, or Getter according to a user supplied ordering.

maximumBy cmp = fromMaybe (error empty) . maximumByOf folded cmp

 maximumByOf :: Getter a c        -> (c -> c -> Ordering) -> a -> Maybe c
 maximumByOf :: Fold a c          -> (c -> c -> Ordering) -> a -> Maybe c
 maximumByOf :: Iso a b c d       -> (c -> c -> Ordering) -> a -> Maybe c
 maximumByOf :: Lens a b c d      -> (c -> c -> Ordering) -> a -> Maybe c
 maximumByOf :: Traversal a b c d -> (c -> c -> Ordering) -> a -> Maybe c

minimumByOf :: Getting (Endo (Maybe c)) a b c d -> (c -> c -> Ordering) -> a -> Maybe c

Obtain the minimum element (if any) targeted by a Fold, Traversal, Lens, Iso or Getter according to a user supplied ordering.

 minimumBy cmp = fromMaybe (error "empty") . minimumByOf folded cmp

 minimumByOf :: Getter a c        -> (c -> c -> Ordering) -> a -> Maybe c
 minimumByOf :: Fold a c          -> (c -> c -> Ordering) -> a -> Maybe c
 minimumByOf :: Iso a b c d       -> (c -> c -> Ordering) -> a -> Maybe c
 minimumByOf :: Lens a b c d      -> (c -> c -> Ordering) -> a -> Maybe c
 minimumByOf :: Traversal a b c d -> (c -> c -> Ordering) -> a -> Maybe c

findOf :: Getting (First c) a b c d -> (c -> Bool) -> a -> Maybe c

The findOf function takes a Lens (or Getter, Iso, Fold, or Traversal), a predicate and a structure and returns the leftmost element of the structure matching the predicate, or Nothing if there is no such element.

 findOf :: Getter a c        -> (c -> Bool) -> a -> Maybe c
 findOf :: Fold a c          -> (c -> Bool) -> a -> Maybe c
 findOf :: Iso a b c d       -> (c -> Bool) -> a -> Maybe c
 findOf :: Lens a b c d      -> (c -> Bool) -> a -> Maybe c
 findOf :: Traversal a b c d -> (c -> Bool) -> a -> Maybe c

foldrOf' :: Getting (Dual (Endo (e -> e))) a b c d -> (c -> e -> e) -> e -> a -> e

Strictly fold right over the elements of a structure.

foldr' = foldrOf' folded

 foldrOf' :: Getter a c        -> (c -> e -> e) -> e -> a -> e
 foldrOf' :: Fold a c          -> (c -> e -> e) -> e -> a -> e
 foldrOf' :: Iso a b c d       -> (c -> e -> e) -> e -> a -> e
 foldrOf' :: Lens a b c d      -> (c -> e -> e) -> e -> a -> e
 foldrOf' :: Traversal a b c d -> (c -> e -> e) -> e -> a -> e

foldlOf' :: Getting (Endo (e -> e)) a b c d -> (e -> c -> e) -> e -> a -> e

Fold over the elements of a structure, associating to the left, but strictly.

foldl' = foldlOf' folded

 foldlOf' :: Getter a c          -> (e -> c -> e) -> e -> a -> e
 foldlOf' :: Fold a c            -> (e -> c -> e) -> e -> a -> e
 foldlOf' :: Iso a b c d         -> (e -> c -> e) -> e -> a -> e
 foldlOf' :: Lens a b c d        -> (e -> c -> e) -> e -> a -> e
 foldlOf' :: Traversal a b c d   -> (e -> c -> e) -> e -> a -> e

foldr1Of :: Getting (Endo (Maybe c)) a b c d -> (c -> c -> c) -> a -> c

A variant of foldrOf that has no base case and thus may only be applied to lenses and structures such that the lens views at least one element of the structure.

foldr1Of l f = foldr1 f . toListOf l

foldr1 = foldr1Of folded

 foldr1Of :: Getter a c        -> (c -> c -> c) -> a -> c
 foldr1Of :: Fold a c          -> (c -> c -> c) -> a -> c
 foldr1Of :: Iso a b c d       -> (c -> c -> c) -> a -> c
 foldr1Of :: Lens a b c d      -> (c -> c -> c) -> a -> c
 foldr1Of :: Traversal a b c d -> (c -> c -> c) -> a -> c

foldl1Of :: Getting (Dual (Endo (Maybe c))) a b c d -> (c -> c -> c) -> a -> c

A variant of foldlOf that has no base case and thus may only be applied to lenses and strutures such that the lens views at least one element of the structure.

foldl1Of l f = foldl1Of l f . toList

foldl1 = foldl1Of folded

 foldl1Of :: Getter a c        -> (c -> c -> c) -> a -> c
 foldl1Of :: Fold a c          -> (c -> c -> c) -> a -> c
 foldl1Of :: Iso a b c d       -> (c -> c -> c) -> a -> c
 foldl1Of :: Lens a b c d      -> (c -> c -> c) -> a -> c
 foldl1Of :: Traversal a b c d -> (c -> c -> c) -> a -> c

foldrMOf :: Monad m => Getting (Dual (Endo (e -> m e))) a b c d -> (c -> e -> m e) -> e -> a -> m e

Monadic fold over the elements of a structure, associating to the right, i.e. from right to left.

foldrM = foldrMOf folded

 foldrMOf :: Monad m => Getter a c        -> (c -> e -> m e) -> e -> a -> m e
 foldrMOf :: Monad m => Fold a c          -> (c -> e -> m e) -> e -> a -> m e
 foldrMOf :: Monad m => Iso a b c d       -> (c -> e -> m e) -> e -> a -> m e
 foldrMOf :: Monad m => Lens a b c d      -> (c -> e -> m e) -> e -> a -> m e
 foldrMOf :: Monad m => Traversal a b c d -> (c -> e -> m e) -> e -> a -> m e

foldlMOf :: Monad m => Getting (Endo (e -> m e)) a b c d -> (e -> c -> m e) -> e -> a -> m e

Monadic fold over the elements of a structure, associating to the left, i.e. from left to right.

foldlM = foldlMOf folded

 foldlMOf :: Monad m => Getter a c        -> (e -> c -> m e) -> e -> a -> m e
 foldlMOf :: Monad m => Fold a c          -> (e -> c -> m e) -> e -> a -> m e
 foldlMOf :: Monad m => Iso a b c d       -> (e -> c -> m e) -> e -> a -> m e
 foldlMOf :: Monad m => Lens a b c d      -> (e -> c -> m e) -> e -> a -> m e
 foldlMOf :: Monad m => Traversal a b c d -> (e -> c -> m e) -> e -> a -> m e