Control.Lens.Fold

A Fold s a 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) => (a -> m) -> t a -> m. Since we want to be able to work with monomorphic containers, we could generalize this signature to forall m. Monoid m => (a -> m) -> s -> m, and then decorate it with Accessor to obtain

type Fold s a = forall m. Monoid m => Getting m s a

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

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 s a = forall f. (Contravariant f, Applicative f) => (a -> f a) -> s -> f s

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

A Fold s a 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 a), then there should be a fooOf method that takes a Fold s a and a value of type s.

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.

type IndexedFold i s a = forall p f. (Indexable i p, Contravariant f, Applicative f) => p a (f a) -> s -> f s

Every IndexedFold is a valid Fold and can be used for Getting.

Getting Started

(^..) :: s -> Getting (Endo [a]) s a -> [a]

A convenient infix (flipped) version of toListOf.

>>> [[1,2],[3]]^..traverse.traverse
[1,2,3]

>>> (1,2)^..both
[1,2]

 toList xs ≡ xs ^.. folded
 (^..) ≡ flip toListOf

 (^..) :: s -> Getter s a     -> [a]
 (^..) :: s -> Fold s a       -> [a]
 (^..) :: s -> Lens' s a      -> [a]
 (^..) :: s -> Iso' s a       -> [a]
 (^..) :: s -> Traversal' s a -> [a]
 (^..) :: s -> Prism' s a     -> [a]

(^?) :: s -> Getting (First a) s a -> Maybe a

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

When using a Traversal as a partial Lens, or a Fold as a partial Getter this can be a convenient way to extract the optional value.

Note: if you get stack overflows due to this, you may want to use firstOf instead, which can deal more gracefully with heavily left-biased trees.

>>> Left 4 ^?_Left
Just 4

>>> Right 4 ^?_Left
Nothing

>>> "world" ^? ix 3
Just 'l'

>>> "world" ^? ix 20
Nothing

 (^?) ≡ flip preview

 (^?) :: s -> Getter s a     -> Maybe a
 (^?) :: s -> Fold s a       -> Maybe a
 (^?) :: s -> Lens' s a      -> Maybe a
 (^?) :: s -> Iso' s a       -> Maybe a
 (^?) :: s -> Traversal' s a -> Maybe a

(^?!) :: s -> Getting (Endo a) s a -> a

Perform an *UNSAFE* head of a Fold or Traversal assuming that it is there.

>>> Left 4 ^?! _Left
4

>>> "world" ^?! ix 3
'l'

 (^?!) :: s -> Getter s a     -> a
 (^?!) :: s -> Fold s a       -> a
 (^?!) :: s -> Lens' s a      -> a
 (^?!) :: s -> Iso' s a       -> a
 (^?!) :: s -> Traversal' s a -> a

pre :: Getting (First a) s a -> IndexPreservingGetter s (Maybe a)

This converts a Fold to a IndexPreservingGetter that returns the first element, if it exists, as a Maybe.

 pre :: Getter s a           -> IndexPreservingGetter s (Maybe a)
 pre :: Fold s a             -> IndexPreservingGetter s (Maybe a)
 pre :: Simple Traversal s a -> IndexPreservingGetter s (Maybe a)
 pre :: Simple Lens s a      -> IndexPreservingGetter s (Maybe a)
 pre :: Simple Iso s a       -> IndexPreservingGetter s (Maybe a)
 pre :: Simple Prism s a     -> IndexPreservingGetter s (Maybe a)

ipre :: IndexedGetting i (First (i, a)) s a -> IndexPreservingGetter s (Maybe (i, a))

This converts an IndexedFold to an IndexPreservingGetter that returns the first index and element, if they exist, as a Maybe.

 ipre :: IndexedGetter i s a             -> IndexPreservingGetter s (Maybe (i, a))
 ipre :: IndexedFold i s a               -> IndexPreservingGetter s (Maybe (i, a))
 ipre :: Simple (IndexedTraversal i) s a -> IndexPreservingGetter s (Maybe (i, a))
 ipre :: Simple (IndexedLens i) s a      -> IndexPreservingGetter s (Maybe (i, a))

preview :: MonadReader s m => Getting (First a) s a -> m (Maybe a)

Retrieve the first value targeted by a Fold or Traversal (or Just the result from a Getter or Lens). See also (^?).

 listToMaybe . toList ≡ preview folded

This is usually applied in the Reader Monad (->) s.

 preview = view . pre

 preview :: Getter s a     -> s -> Maybe a
 preview :: Fold s a       -> s -> Maybe a
 preview :: Lens' s a      -> s -> Maybe a
 preview :: Iso' s a       -> s -> Maybe a
 preview :: Traversal' s a -> s -> Maybe a

However, it may be useful to think of its full generality when working with a Monad transformer stack:

 preview :: MonadReader s m => Getter s a     -> m (Maybe a)
 preview :: MonadReader s m => Fold s a       -> m (Maybe a)
 preview :: MonadReader s m => Lens' s a      -> m (Maybe a)
 preview :: MonadReader s m => Iso' s a       -> m (Maybe a)
 preview :: MonadReader s m => Traversal' s a -> m (Maybe a)

previews :: MonadReader s m => Getting (First r) s a -> (a -> r) -> m (Maybe r)

Retrieve a function of the first value targeted by a Fold or Traversal (or Just the result from a Getter or Lens).

This is usually applied in the Reader Monad (->) s.

ipreview :: MonadReader s m => IndexedGetting i (First (i, a)) s a -> m (Maybe (i, a))

Retrieve the first index and value targeted by a Fold or Traversal (or Just the result from a Getter or Lens). See also (^@?).

 ipreview = view . ipre

This is usually applied in the Reader Monad (->) s.

 ipreview :: IndexedGetter i s a     -> s -> Maybe (i, a)
 ipreview :: IndexedFold i s a       -> s -> Maybe (i, a)
 ipreview :: IndexedLens' i s a      -> s -> Maybe (i, a)
 ipreview :: IndexedTraversal' i s a -> s -> Maybe (i, a)

However, it may be useful to think of its full generality when working with a Monad transformer stack:

 ipreview :: MonadReader s m => IndexedGetter s a     -> m (Maybe (i, a))
 ipreview :: MonadReader s m => IndexedFold s a       -> m (Maybe (i, a))
 ipreview :: MonadReader s m => IndexedLens' s a      -> m (Maybe (i, a))
 ipreview :: MonadReader s m => IndexedTraversal' s a -> m (Maybe (i, a))

ipreviews :: MonadReader s m => IndexedGetting i (First r) s a -> (i -> a -> r) -> m (Maybe r)

Retrieve a function of the first index and value targeted by an IndexedFold or IndexedTraversal (or Just the result from an IndexedGetter or IndexedLens). See also (^@?).

 ipreviews = views . ipre

This is usually applied in the Reader Monad (->) s.

 ipreviews :: IndexedGetter i s a     -> (i -> a -> r) -> s -> Maybe r
 ipreviews :: IndexedFold i s a       -> (i -> a -> r) -> s -> Maybe r
 ipreviews :: IndexedLens' i s a      -> (i -> a -> r) -> s -> Maybe r
 ipreviews :: IndexedTraversal' i s a -> (i -> a -> r) -> s -> Maybe r

However, it may be useful to think of its full generality when working with a Monad transformer stack:

 ipreviews :: MonadReader s m => IndexedGetter i s a     -> (i -> a -> r) -> m (Maybe r)
 ipreviews :: MonadReader s m => IndexedFold i s a       -> (i -> a -> r) -> m (Maybe r)
 ipreviews :: MonadReader s m => IndexedLens' i s a      -> (i -> a -> r) -> m (Maybe r)
 ipreviews :: MonadReader s m => IndexedTraversal' i s a -> (i -> a -> r) -> m (Maybe r)

preuse :: MonadState s m => Getting (First a) s a -> m (Maybe a)

Retrieve the first value targeted by a Fold or Traversal (or Just the result from a Getter or Lens) into the current state.

 preuse = use . pre

 preuse :: MonadState s m => Getter s a     -> m (Maybe a)
 preuse :: MonadState s m => Fold s a       -> m (Maybe a)
 preuse :: MonadState s m => Lens' s a      -> m (Maybe a)
 preuse :: MonadState s m => Iso' s a       -> m (Maybe a)
 preuse :: MonadState s m => Traversal' s a -> m (Maybe a)

preuses :: MonadState s m => Getting (First r) s a -> (a -> r) -> m (Maybe r)

Retrieve a function of the first value targeted by a Fold or Traversal (or Just the result from a Getter or Lens) into the current state.

 preuses = uses . pre

 preuses :: MonadState s m => Getter s a     -> (a -> r) -> m (Maybe r)
 preuses :: MonadState s m => Fold s a       -> (a -> r) -> m (Maybe r)
 preuses :: MonadState s m => Lens' s a      -> (a -> r) -> m (Maybe r)
 preuses :: MonadState s m => Iso' s a       -> (a -> r) -> m (Maybe r)
 preuses :: MonadState s m => Traversal' s a -> (a -> r) -> m (Maybe r)

ipreuse :: MonadState s m => IndexedGetting i (First (i, a)) s a -> m (Maybe (i, a))

Retrieve the first index and value targeted by an IndexedFold or IndexedTraversal (or Just the index and result from an IndexedGetter or IndexedLens) into the current state.

 ipreuse = use . ipre

 ipreuse :: MonadState s m => IndexedGetter i s a     -> m (Maybe (i, a))
 ipreuse :: MonadState s m => IndexedFold i s a       -> m (Maybe (i, a))
 ipreuse :: MonadState s m => IndexedLens' i s a      -> m (Maybe (i, a))
 ipreuse :: MonadState s m => IndexedTraversal' i s a -> m (Maybe (i, a))

ipreuses :: MonadState s m => IndexedGetting i (First r) s a -> (i -> a -> r) -> m (Maybe r)

Retrieve a function of the first index and value targeted by an IndexedFold or IndexedTraversal (or a function of Just the index and result from an IndexedGetter or IndexedLens) into the current state.

 ipreuses = uses . ipre

 ipreuses :: MonadState s m => IndexedGetter i s a     -> (i -> a -> r) -> m (Maybe r)
 ipreuses :: MonadState s m => IndexedFold i s a       -> (i -> a -> r) -> m (Maybe r)
 ipreuses :: MonadState s m => IndexedLens' i s a      -> (i -> a -> r) -> m (Maybe r)
 ipreuses :: MonadState s m => IndexedTraversal' i s a -> (i -> a -> r) -> m (Maybe r)

has :: Getting Any s a -> s -> Bool

Check to see if this Fold or Traversal matches 1 or more entries.

>>> has (element 0) []
False

>>> has _Left (Left 12)
True

>>> has _Right (Left 12)
False

This will always return True for a Lens or Getter.

>>> has _1 ("hello","world")
True

 has :: Getter s a     -> s -> Bool
 has :: Fold s a       -> s -> Bool
 has :: Iso' s a       -> s -> Bool
 has :: Lens' s a      -> s -> Bool
 has :: Traversal' s a -> s -> Bool

hasn't :: Getting All s a -> s -> Bool

Check to see if this Fold or Traversal has no matches.

>>> hasn't _Left (Right 12)
True

>>> hasn't _Left (Left 12)
False

Building Folds

folding :: (Foldable f, Contravariant g, Applicative g) => (s -> f a) -> LensLike g s t a b

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.

>>> [1,2,3,4]^..folding tail
[2,3,4]

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

Obtain a Fold from any Foldable.

>>> Just 3^..folded
[3]

>>> Nothing^..folded
[]

>>> [(1,2),(3,4)]^..folded.both
[1,2,3,4]

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

Build a Fold that unfolds its values from a seed.

 unfoldr ≡ toListOf . unfolded

>>> 10^..unfolded (\b -> if b == 0 then Nothing else Just (b, b-1))
[10,9,8,7,6,5,4,3,2,1]

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

x ^. iterated f returns an infinite Fold of repeated applications of f to x.

 toListOf (iterated f) a ≡ iterate f a

filtered :: (Choice p, Applicative f) => (a -> Bool) -> Overloaded' p f a a

Obtain a Fold that can be composed with to filter another Lens, Iso, Getter, Fold (or Traversal).

Note: This is not a legal Traversal, unless you are very careful not to invalidate the predicate on the target.

Note: This is also not a legal Prism, unless you are very careful not to inject a value that matches the predicate.

As a counter example, consider that given evens = filtered even the second Traversal law is violated:

 over evens succ . over evens succ /= over evens (succ . succ)

So, in order for this to qualify as a legal Traversal you can only use it for actions that preserve the result of the predicate!

>>> [1..10]^..folded.filtered even
[2,4,6,8,10]

This will preserve an index if it is present.

backwards :: (Profunctor p, Profunctor q) => Overloading p q (Backwards f) s t a b -> Overloading p q f s t a b

This allows you to traverse the elements of a pretty much any LensLike construction in the opposite order.

This will preserve indexes on Indexed types and will give you the elements of a (finite) Fold or Traversal in the opposite order.

This has no practical impact on a Getter, Setter, Lens or Iso.

NB: To write back through an Iso, you want to use from. Similarly, to write back through an Prism, you want to use re.

repeated :: Fold a a

Fold by repeating the input forever.

 repeat ≡ toListOf repeated

>>> timingOut $ 5^..taking 20 repeated
[5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5]

replicated :: Int -> Fold a a

A Fold that replicates its input n times.

 replicate n ≡ toListOf (replicated n)

>>> 5^..replicated 20
[5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5]

cycled :: (Contravariant f, Applicative f) => LensLike f s t a b -> LensLike f s t a b

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

>>> timingOut $ [1,2,3]^..taking 7 (cycled traverse)
[1,2,3,1,2,3,1]

takingWhile :: (Conjoined p, Applicative f) => (a -> Bool) -> Over p (TakingWhile p f a a) s t a a -> Over p f s t a a

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

 takeWhile p ≡ toListOf (takingWhile p folded)

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

 takingWhile :: (a -> Bool) -> Fold s a                         -> Fold s a
 takingWhile :: (a -> Bool) -> Getter s a                       -> Fold s a
 takingWhile :: (a -> Bool) -> Traversal' s a                   -> Fold s a -- * See note below
 takingWhile :: (a -> Bool) -> Lens' s a                        -> Fold s a -- * See note below
 takingWhile :: (a -> Bool) -> Prism' s a                       -> Fold s a -- * See note below
 takingWhile :: (a -> Bool) -> Iso' s a                         -> Fold s a -- * See note below
 takingWhile :: (a -> Bool) -> Action m s a                     -> MonadicFold m s a
 takingWhile :: (a -> Bool) -> MonadicFold m s a                -> MonadicFold m s a
 takingWhile :: (a -> Bool) -> IndexedTraversal' i s a          -> IndexedFold i s a -- * See note below
 takingWhile :: (a -> Bool) -> IndexedLens' i s a               -> IndexedFold i s a -- * See note below
 takingWhile :: (a -> Bool) -> IndexedFold i s a                -> IndexedFold i s a
 takingWhile :: (a -> Bool) -> IndexedGetter i s a              -> IndexedFold i s a
 takingWhile :: (a -> Bool) -> IndexedAction i m s a            -> IndexedMonadicFold i m s a
 takingWhile :: (a -> Bool) -> IndexedMonadicFold i m s a       -> IndexedMonadicFold i m s a

Note: When applied to a Traversal, takingWhile yields something that can be used as if it were a Traversal, but which is not a Traversal per the laws, unless you are careful to ensure that you do not invalidate the predicate when writing back through it.

droppingWhile :: (Conjoined p, Profunctor q, Applicative f) => (a -> Bool) -> Overloading p q (Compose (State Bool) f) s t a a -> Overloading p q f s t a a

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]

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

 droppingWhile :: (a -> Bool) -> Fold s a                         -> Fold s a
 droppingWhile :: (a -> Bool) -> Getter s a                       -> Fold s a
 droppingWhile :: (a -> Bool) -> Traversal' s a                   -> Fold s a                -- see notes
 droppingWhile :: (a -> Bool) -> Lens' s a                        -> Fold s a                -- see notes
 droppingWhile :: (a -> Bool) -> Prism' s a                       -> Fold s a                -- see notes
 droppingWhile :: (a -> Bool) -> Iso' s a                         -> Fold s a                -- see notes

 droppingWhile :: (a -> Bool) -> IndexPreservingTraversal' s a    -> IndexPreservingFold s a -- see notes
 droppingWhile :: (a -> Bool) -> IndexPreservingLens' s a         -> IndexPreservingFold s a -- see notes
 droppingWhile :: (a -> Bool) -> IndexPreservingGetter s a        -> IndexPreservingFold s a
 droppingWhile :: (a -> Bool) -> IndexPreservingFold s a          -> IndexPreservingFold s a
 droppingWhile :: (a -> Bool) -> IndexPreservingAction m s a      -> IndexPreservingFold s a

 droppingWhile :: (a -> Bool) -> IndexPreservingMonadicFold m s a -> IndexPreservingMonadicFold m s a

 droppingWhile :: (a -> Bool) -> IndexedTraversal' i s a          -> IndexedFold i s a       -- see notes
 droppingWhile :: (a -> Bool) -> IndexedLens' i s a               -> IndexedFold i s a       -- see notes
 droppingWhile :: (a -> Bool) -> IndexedGetter i s a              -> IndexedFold i s a
 droppingWhile :: (a -> Bool) -> IndexedFold i s a                -> IndexedFold i s a
 droppingWhile :: (a -> Bool) -> IndexedAction i m s a            -> IndexedFold i s a

 droppingWhile :: (a -> Bool) -> IndexedMonadicFold i m s a       -> IndexedMonadicFold i m s a

Note: Many uses of this combinator will yield something that meets the types, but not the laws of a valid Traversal or IndexedTraversal. The Traversal and IndexedTraversal laws are only satisfied if the new values you assign also pass the predicate! Otherwise subsequent traversals will visit fewer elements and Traversal fusion is not sound.

Folding

foldMapOf :: Profunctor p => Accessing p r s a -> p a r -> s -> r

 foldMap = foldMapOf folded

 foldMapOf ≡ views
 ifoldMapOf l = foldMapOf l . Indexed

 foldMapOf ::             Getter s a     -> (a -> r) -> s -> r
 foldMapOf :: Monoid r => Fold s a       -> (a -> r) -> s -> r
 foldMapOf ::             Lens' s a      -> (a -> r) -> s -> r
 foldMapOf ::             Iso' s a       -> (a -> r) -> s -> r
 foldMapOf :: Monoid r => Traversal' s a -> (a -> r) -> s -> r
 foldMapOf :: Monoid r => Prism' s a     -> (a -> r) -> s -> r

 foldMapOf :: Getting r s a -> (a -> r) -> s -> r

foldOf :: Getting a s a -> s -> a

 fold = foldOf folded

 foldOf ≡ view

 foldOf ::             Getter s m     -> s -> m
 foldOf :: Monoid m => Fold s m       -> s -> m
 foldOf ::             Lens' s m      -> s -> m
 foldOf ::             Iso' s m       -> s -> m
 foldOf :: Monoid m => Traversal' s m -> s -> m
 foldOf :: Monoid m => Prism' s m     -> s -> m

foldrOf :: Profunctor p => Accessing p (Endo r) s a -> p a (r -> r) -> r -> s -> r

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

 foldr ≡ foldrOf folded

 foldrOf :: Getter s a     -> (a -> r -> r) -> r -> s -> r
 foldrOf :: Fold s a       -> (a -> r -> r) -> r -> s -> r
 foldrOf :: Lens' s a      -> (a -> r -> r) -> r -> s -> r
 foldrOf :: Iso' s a       -> (a -> r -> r) -> r -> s -> r
 foldrOf :: Traversal' s a -> (a -> r -> r) -> r -> s -> r
 foldrOf :: Prism' s a     -> (a -> r -> r) -> r -> s -> r

 ifoldrOf l ≡ foldrOf l . Indexed

 foldrOf :: Getting (Endo r) s a -> (a -> r -> r) -> r -> s -> r

foldlOf :: Getting (Dual (Endo r)) s a -> (r -> a -> r) -> r -> s -> r

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

 foldl ≡ foldlOf folded

 foldlOf :: Getter s a     -> (r -> a -> r) -> r -> s -> r
 foldlOf :: Fold s a       -> (r -> a -> r) -> r -> s -> r
 foldlOf :: Lens' s a      -> (r -> a -> r) -> r -> s -> r
 foldlOf :: Iso' s a       -> (r -> a -> r) -> r -> s -> r
 foldlOf :: Traversal' s a -> (r -> a -> r) -> r -> s -> r
 foldlOf :: Prism' s a     -> (r -> a -> r) -> r -> s -> r

toListOf :: Getting (Endo [a]) s a -> s -> [a]

Extract a list of the targets of a Fold. See also (^..).

 toList ≡ toListOf folded
 (^..) ≡ flip toListOf

anyOf :: Profunctor p => Accessing p Any s a -> p a Bool -> s -> Bool

Returns True if any target of a Fold satisfies a predicate.

>>> anyOf both (=='x') ('x','y')
True
>>> import Data.Data.Lens
>>> anyOf biplate (== "world") (((),2::Int),"hello",("world",11))
True

 any ≡ anyOf folded

 ianyOf l ≡ allOf l . Indexed

 anyOf :: Getter s a     -> (a -> Bool) -> s -> Bool
 anyOf :: Fold s a       -> (a -> Bool) -> s -> Bool
 anyOf :: Lens' s a      -> (a -> Bool) -> s -> Bool
 anyOf :: Iso' s a       -> (a -> Bool) -> s -> Bool
 anyOf :: Traversal' s a -> (a -> Bool) -> s -> Bool
 anyOf :: Prism' s a     -> (a -> Bool) -> s -> Bool

allOf :: Profunctor p => Accessing p All s a -> p a Bool -> s -> Bool

Returns True if every target of a Fold satisfies a predicate.

>>> allOf both (>=3) (4,5)
True
>>> allOf folded (>=2) [1..10]
False

 all ≡ allOf folded

 iallOf l = allOf l . Indexed

 allOf :: Getter s a     -> (a -> Bool) -> s -> Bool
 allOf :: Fold s a       -> (a -> Bool) -> s -> Bool
 allOf :: Lens' s a      -> (a -> Bool) -> s -> Bool
 allOf :: Iso' s a       -> (a -> Bool) -> s -> Bool
 allOf :: Traversal' s a -> (a -> Bool) -> s -> Bool
 allOf :: Prism' s a     -> (a -> Bool) -> s -> Bool

andOf :: Getting All s Bool -> s -> Bool

Returns True if every target of a Fold is True.

>>> andOf both (True,False)
False
>>> andOf both (True,True)
True

 and ≡ andOf folded

 andOf :: Getter s Bool     -> s -> Bool
 andOf :: Fold s Bool       -> s -> Bool
 andOf :: Lens' s Bool      -> s -> Bool
 andOf :: Iso' s Bool       -> s -> Bool
 andOf :: Traversal' s Bool -> s -> Bool
 andOf :: Prism' s Bool     -> s -> Bool

orOf :: Getting Any s Bool -> s -> Bool

Returns True if any target of a Fold is True.

>>> orOf both (True,False)
True
>>> orOf both (False,False)
False

 or ≡ orOf folded

 orOf :: Getter s Bool     -> s -> Bool
 orOf :: Fold s Bool       -> s -> Bool
 orOf :: Lens' s Bool      -> s -> Bool
 orOf :: Iso' s Bool       -> s -> Bool
 orOf :: Traversal' s Bool -> s -> Bool
 orOf :: Prism' s Bool     -> s -> Bool

productOf :: Num a => Getting (Endo (Endo a)) s a -> s -> a

Calculate the Product of every number targeted by a Fold.

>>> productOf both (4,5)
20
>>> productOf folded [1,2,3,4,5]
120

 product ≡ productOf folded

sumOf :: Num a => Getting (Endo (Endo a)) s a -> s -> a

Calculate the Sum of every number targeted by a Fold.

>>> sumOf both (5,6)
11
>>> sumOf folded [1,2,3,4]
10
>>> sumOf (folded.both) [(1,2),(3,4)]
10
>>> import Data.Data.Lens
>>> sumOf biplate [(1::Int,[]),(2,[(3::Int,4::Int)])] :: Int
10

 sum ≡ sumOf folded

This operation may be more strict than you would expect. If you want a lazier version use ala Sum . foldMapOf

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

 sumOf :: Num a => Getter s a     -> s -> a
 sumOf :: Num a => Fold s a       -> s -> a
 sumOf :: Num a => Lens' s a      -> s -> a
 sumOf :: Num a => Iso' s a       -> s -> a
 sumOf :: Num a => Traversal' s a -> s -> a
 sumOf :: Num a => Prism' s a     -> s -> a

traverseOf_ :: (Profunctor p, Functor f) => Accessing p (Traversed r f) s a -> p a (f r) -> s -> f ()

Traverse over all of the targets of a Fold (or Getter), computing an Applicative (or Functor)-based answer, but unlike traverseOf do not construct a new structure. traverseOf_ generalizes traverse_ to work over any Fold.

When passed a Getter, traverseOf_ can work over any Functor, but when passed a Fold, traverseOf_ requires an Applicative.

>>> traverseOf_ both putStrLn ("hello","world")
hello
world

 traverse_ ≡ traverseOf_ folded

 traverseOf_ _2 :: Functor f => (c -> f r) -> (d, c) -> f ()
 traverseOf_ _Left :: Applicative f => (a -> f b) -> Either a c -> f ()

 itraverseOf_ l ≡ traverseOf_ l . Indexed

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

 traverseOf_ :: Functor f     => Getter s a     -> (a -> f r) -> s -> f ()
 traverseOf_ :: Applicative f => Fold s a       -> (a -> f r) -> s -> f ()
 traverseOf_ :: Functor f     => Lens' s a      -> (a -> f r) -> s -> f ()
 traverseOf_ :: Functor f     => Iso' s a       -> (a -> f r) -> s -> f ()
 traverseOf_ :: Applicative f => Traversal' s a -> (a -> f r) -> s -> f ()
 traverseOf_ :: Applicative f => Prism' s a     -> (a -> f r) -> s -> f ()

forOf_ :: (Profunctor p, Functor f) => Accessing p (Traversed r f) s a -> s -> p a (f r) -> f ()

Traverse over all of the targets of a Fold (or Getter), computing an Applicative (or Functor)-based answer, but unlike forOf do not construct a new structure. forOf_ generalizes for_ to work over any Fold.

When passed a Getter, forOf_ can work over any Functor, but when passed a Fold, forOf_ requires an Applicative.

 for_ ≡ forOf_ folded

>>> forOf_ both ("hello","world") putStrLn
hello
world

The rather specific signature of forOf_ allows it to be used as if the signature was any of:

 iforOf_ l s ≡ forOf_ l s . Indexed

 forOf_ :: Functor f     => Getter s a     -> s -> (a -> f r) -> f ()
 forOf_ :: Applicative f => Fold s a       -> s -> (a -> f r) -> f ()
 forOf_ :: Functor f     => Lens' s a      -> s -> (a -> f r) -> f ()
 forOf_ :: Functor f     => Iso' s a       -> s -> (a -> f r) -> f ()
 forOf_ :: Applicative f => Traversal' s a -> s -> (a -> f r) -> f ()
 forOf_ :: Applicative f => Prism' s a     -> s -> (a -> f r) -> f ()

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

Evaluate each action in observed by a Fold on a structure from left to right, ignoring the results.

 sequenceA_ ≡ sequenceAOf_ folded

>>> sequenceAOf_ both (putStrLn "hello",putStrLn "world")
hello
world

 sequenceAOf_ :: Functor f     => Getter s (f a)     -> s -> f ()
 sequenceAOf_ :: Applicative f => Fold s (f a)       -> s -> f ()
 sequenceAOf_ :: Functor f     => Lens' s (f a)      -> s -> f ()
 sequenceAOf_ :: Functor f     => Iso' s (f a)       -> s -> f ()
 sequenceAOf_ :: Applicative f => Traversal' s (f a) -> s -> f ()
 sequenceAOf_ :: Applicative f => Prism' s (f a)     -> s -> f ()

mapMOf_ :: (Profunctor p, Monad m) => Accessing p (Sequenced r m) s a -> p a (m r) -> s -> m ()

Map each target of a Fold on a structure to a monadic action, evaluate these actions from left to right, and ignore the results.

>>> mapMOf_ both putStrLn ("hello","world")
hello
world

 mapM_ ≡ mapMOf_ folded

 mapMOf_ :: Monad m => Getter s a     -> (a -> m r) -> s -> m ()
 mapMOf_ :: Monad m => Fold s a       -> (a -> m r) -> s -> m ()
 mapMOf_ :: Monad m => Lens' s a      -> (a -> m r) -> s -> m ()
 mapMOf_ :: Monad m => Iso' s a       -> (a -> m r) -> s -> m ()
 mapMOf_ :: Monad m => Traversal' s a -> (a -> m r) -> s -> m ()
 mapMOf_ :: Monad m => Prism' s a     -> (a -> m r) -> s -> m ()

forMOf_ :: (Profunctor p, Monad m) => Accessing p (Sequenced r m) s a -> s -> p a (m r) -> m ()

forMOf_ is mapMOf_ with two of its arguments flipped.

>>> forMOf_ both ("hello","world") putStrLn
hello
world

 forM_ ≡ forMOf_ folded

 forMOf_ :: Monad m => Getter s a     -> s -> (a -> m r) -> m ()
 forMOf_ :: Monad m => Fold s a       -> s -> (a -> m r) -> m ()
 forMOf_ :: Monad m => Lens' s a      -> s -> (a -> m r) -> m ()
 forMOf_ :: Monad m => Iso' s a       -> s -> (a -> m r) -> m ()
 forMOf_ :: Monad m => Traversal' s a -> s -> (a -> m r) -> m ()
 forMOf_ :: Monad m => Prism' s a     -> s -> (a -> m r) -> m ()

sequenceOf_ :: Monad m => Getting (Sequenced a m) s (m a) -> s -> m ()

Evaluate each monadic action referenced by a Fold on the structure from left to right, and ignore the results.

>>> sequenceOf_ both (putStrLn "hello",putStrLn "world")
hello
world

 sequence_ ≡ sequenceOf_ folded

 sequenceOf_ :: Monad m => Getter s (m a)     -> s -> m ()
 sequenceOf_ :: Monad m => Fold s (m a)       -> s -> m ()
 sequenceOf_ :: Monad m => Lens' s (m a)      -> s -> m ()
 sequenceOf_ :: Monad m => Iso' s (m a)       -> s -> m ()
 sequenceOf_ :: Monad m => Traversal' s (m a) -> s -> m ()
 sequenceOf_ :: Monad m => Prism' s (m a)     -> s -> m ()

asumOf :: Alternative f => Getting (Endo (f a)) s (f a) -> s -> f a

The sum of a collection of actions, generalizing concatOf.

>>> asumOf both ("hello","world")
"helloworld"

>>> asumOf each (Nothing, Just "hello", Nothing)
Just "hello"

 asum ≡ asumOf folded

 asumOf :: Alternative f => Getter s (f a)     -> s -> f a
 asumOf :: Alternative f => Fold s (f a)       -> s -> f a
 asumOf :: Alternative f => Lens' s (f a)      -> s -> f a
 asumOf :: Alternative f => Iso' s (f a)       -> s -> f a
 asumOf :: Alternative f => Traversal' s (f a) -> s -> f a
 asumOf :: Alternative f => Prism' s (f a)     -> s -> f a

msumOf :: MonadPlus m => Getting (Endo (m a)) s (m a) -> s -> m a

The sum of a collection of actions, generalizing concatOf.

>>> msumOf both ("hello","world")
"helloworld"

>>> msumOf each (Nothing, Just "hello", Nothing)
Just "hello"

 msum ≡ msumOf folded

 msumOf :: MonadPlus m => Getter s (m a)     -> s -> m a
 msumOf :: MonadPlus m => Fold s (m a)       -> s -> m a
 msumOf :: MonadPlus m => Lens' s (m a)      -> s -> m a
 msumOf :: MonadPlus m => Iso' s (m a)       -> s -> m a
 msumOf :: MonadPlus m => Traversal' s (m a) -> s -> m a
 msumOf :: MonadPlus m => Prism' s (m a)     -> s -> m a

concatMapOf :: Profunctor p => Accessing p [r] s a -> p a [r] -> s -> [r]

Map a function over all the targets of a Fold of a container and concatenate the resulting lists.

>>> concatMapOf both (\x -> [x, x + 1]) (1,3)
[1,2,3,4]

 concatMap ≡ concatMapOf folded

 concatMapOf :: Getter s a     -> (a -> [r]) -> s -> [r]
 concatMapOf :: Fold s a       -> (a -> [r]) -> s -> [r]
 concatMapOf :: Lens' s a      -> (a -> [r]) -> s -> [r]
 concatMapOf :: Iso' s a       -> (a -> [r]) -> s -> [r]
 concatMapOf :: Traversal' s a -> (a -> [r]) -> s -> [r]

concatOf :: Getting [r] s [r] -> s -> [r]

Concatenate all of the lists targeted by a Fold into a longer list.

>>> concatOf both ("pan","ama")
"panama"

 concat ≡ concatOf folded
 concatOf ≡ view

 concatOf :: Getter s [r]     -> s -> [r]
 concatOf :: Fold s [r]       -> s -> [r]
 concatOf :: Iso' s [r]       -> s -> [r]
 concatOf :: Lens' s [r]      -> s -> [r]
 concatOf :: Traversal' s [r] -> s -> [r]

elemOf :: Eq a => Getting Any s a -> a -> s -> Bool

Does the element occur anywhere within a given Fold of the structure?

>>> elemOf both "hello" ("hello","world")
True

 elem ≡ elemOf folded

 elemOf :: Eq a => Getter s a     -> a -> s -> Bool
 elemOf :: Eq a => Fold s a       -> a -> s -> Bool
 elemOf :: Eq a => Lens' s a      -> a -> s -> Bool
 elemOf :: Eq a => Iso' s a       -> a -> s -> Bool
 elemOf :: Eq a => Traversal' s a -> a -> s -> Bool
 elemOf :: Eq a => Prism' s a     -> a -> s -> Bool

notElemOf :: Eq a => Getting All s a -> a -> s -> Bool

Does the element not occur anywhere within a given Fold of the structure?

>>> notElemOf each 'd' ('a','b','c')
True

>>> notElemOf each 'a' ('a','b','c')
False

 notElem ≡ notElemOf folded

 notElemOf :: Eq a => Getter s a     -> a -> s -> Bool
 notElemOf :: Eq a => Fold s a       -> a -> s -> Bool
 notElemOf :: Eq a => Iso' s a       -> a -> s -> Bool
 notElemOf :: Eq a => Lens' s a      -> a -> s -> Bool
 notElemOf :: Eq a => Traversal' s a -> a -> s -> Bool
 notElemOf :: Eq a => Prism' s a     -> a -> s -> Bool

lengthOf :: Getting (Endo (Endo Int)) s a -> s -> Int

Calculate the number of targets there are for a Fold in a given container.

Note: This can be rather inefficient for large containers and just like length, this will not terminate for infinite folds.

 length ≡ lengthOf folded

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

>>> lengthOf traverse [1..10]
10

>>> lengthOf (traverse.traverse) [[1,2],[3,4],[5,6]]
6

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

 lengthOf :: Getter s a     -> s -> Int
 lengthOf :: Fold s a       -> s -> Int
 lengthOf :: Lens' s a      -> s -> Int
 lengthOf :: Iso' s a       -> s -> Int
 lengthOf :: Traversal' s a -> s -> Int

nullOf :: Getting All s a -> s -> 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 ignored ()
True

>>> nullOf traverse []
True

>>> nullOf (element 20) [1..10]
True

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

 nullOf :: Getter s a     -> s -> Bool
 nullOf :: Fold s a       -> s -> Bool
 nullOf :: Iso' s a       -> s -> Bool
 nullOf :: Lens' s a      -> s -> Bool
 nullOf :: Traversal' s a -> s -> Bool

notNullOf :: Getting Any s a -> s -> Bool

Returns True if this Fold or Traversal has any targets in the given container.

A more "conversational" alias for this combinator is has.

Note: notNullOf on a valid Iso, Lens or Getter should always return True.

 null ≡ notNullOf folded

This may be rather inefficient compared to the not . null check of many containers.

>>> notNullOf _1 (1,2)
True

>>> notNullOf traverse [1..10]
True

>>> notNullOf folded []
False

>>> notNullOf (element 20) [1..10]
False

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

 notNullOf :: Getter s a     -> s -> Bool
 notNullOf :: Fold s a       -> s -> Bool
 notNullOf :: Iso' s a       -> s -> Bool
 notNullOf :: Lens' s a      -> s -> Bool
 notNullOf :: Traversal' s a -> s -> Bool

firstOf :: Getting (Leftmost a) s a -> s -> Maybe a

Retrieve the First entry of a Fold or Traversal or retrieve Just the result from a Getter or Lens.

The answer is computed in a manner that leaks space less than ala First . foldMapOf and gives you back access to the outermost Just constructor more quickly, but may have worse constant factors.

>>> firstOf traverse [1..10]
Just 1

>>> firstOf both (1,2)
Just 1

>>> firstOf ignored ()
Nothing

 firstOf :: Getter s a     -> s -> Maybe a
 firstOf :: Fold s a       -> s -> Maybe a
 firstOf :: Lens' s a      -> s -> Maybe a
 firstOf :: Iso' s a       -> s -> Maybe a
 firstOf :: Traversal' s a -> s -> Maybe a

lastOf :: Getting (Rightmost a) s a -> s -> Maybe a

Retrieve the Last entry of a Fold or Traversal or retrieve Just the result from a Getter or Lens.

The answer is computed in a manner that leaks space less than ala Last . foldMapOf and gives you back access to the outermost Just constructor more quickly, but may have worse constant factors.

>>> lastOf traverse [1..10]
Just 10

>>> lastOf both (1,2)
Just 2

>>> lastOf ignored ()
Nothing

 lastOf :: Getter s a     -> s -> Maybe a
 lastOf :: Fold s a       -> s -> Maybe a
 lastOf :: Lens' s a      -> s -> Maybe a
 lastOf :: Iso' s a       -> s -> Maybe a
 lastOf :: Traversal' s a -> s -> Maybe a

maximumOf :: Ord a => Getting (Endo (Endo (Maybe a))) s a -> s -> Maybe a

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

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

>>> maximumOf traverse [1..10]
Just 10

>>> maximumOf traverse []
Nothing

>>> maximumOf (folded.filtered even) [1,4,3,6,7,9,2]
Just 6

 maximum ≡ fromMaybe (error "empty") . maximumOf folded

In the interest of efficiency, This operation has semantics more strict than strictly necessary. rmap getMax (foldMapOf l Max) has lazier semantics but could leak memory.

 maximumOf :: Ord a => Getter s a     -> s -> Maybe a
 maximumOf :: Ord a => Fold s a       -> s -> Maybe a
 maximumOf :: Ord a => Iso' s a       -> s -> Maybe a
 maximumOf :: Ord a => Lens' s a      -> s -> Maybe a
 maximumOf :: Ord a => Traversal' s a -> s -> Maybe a

minimumOf :: Ord a => Getting (Endo (Endo (Maybe a))) s a -> s -> Maybe a

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

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

>>> minimumOf traverse [1..10]
Just 1

>>> minimumOf traverse []
Nothing

>>> minimumOf (folded.filtered even) [1,4,3,6,7,9,2]
Just 2

 minimum ≡ fromMaybe (error "empty") . minimumOf folded

In the interest of efficiency, This operation has semantics more strict than strictly necessary. rmap getMin (foldMapOf l Min) has lazier semantics but could leak memory.

 minimumOf :: Ord a => Getter s a     -> s -> Maybe a
 minimumOf :: Ord a => Fold s a       -> s -> Maybe a
 minimumOf :: Ord a => Iso' s a       -> s -> Maybe a
 minimumOf :: Ord a => Lens' s a      -> s -> Maybe a
 minimumOf :: Ord a => Traversal' s a -> s -> Maybe a

maximumByOf :: Getting (Endo (Endo (Maybe a))) s a -> (a -> a -> Ordering) -> s -> Maybe a

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

>>> maximumByOf traverse (compare `on` length) ["mustard","relish","ham"]
Just "mustard"

In the interest of efficiency, This operation has semantics more strict than strictly necessary.

 maximumBy cmp ≡ fromMaybe (error "empty") . maximumByOf folded cmp

 maximumByOf :: Getter s a     -> (a -> a -> Ordering) -> s -> Maybe a
 maximumByOf :: Fold s a       -> (a -> a -> Ordering) -> s -> Maybe a
 maximumByOf :: Iso' s a       -> (a -> a -> Ordering) -> s -> Maybe a
 maximumByOf :: Lens' s a      -> (a -> a -> Ordering) -> s -> Maybe a
 maximumByOf :: Traversal' s a -> (a -> a -> Ordering) -> s -> Maybe a

minimumByOf :: Getting (Endo (Endo (Maybe a))) s a -> (a -> a -> Ordering) -> s -> Maybe a

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

In the interest of efficiency, This operation has semantics more strict than strictly necessary.

>>> minimumByOf traverse (compare `on` length) ["mustard","relish","ham"]
Just "ham"

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

 minimumByOf :: Getter s a     -> (a -> a -> Ordering) -> s -> Maybe a
 minimumByOf :: Fold s a       -> (a -> a -> Ordering) -> s -> Maybe a
 minimumByOf :: Iso' s a       -> (a -> a -> Ordering) -> s -> Maybe a
 minimumByOf :: Lens' s a      -> (a -> a -> Ordering) -> s -> Maybe a
 minimumByOf :: Traversal' s a -> (a -> a -> Ordering) -> s -> Maybe a

findOf :: Conjoined p => Accessing p (Endo (Maybe a)) s a -> p a Bool -> s -> Maybe a

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 each even (1,3,4,6)
Just 4

>>> findOf folded even [1,3,5,7]
Nothing

 findOf :: Getter s a     -> (a -> Bool) -> s -> Maybe a
 findOf :: Fold s a       -> (a -> Bool) -> s -> Maybe a
 findOf :: Iso' s a       -> (a -> Bool) -> s -> Maybe a
 findOf :: Lens' s a      -> (a -> Bool) -> s -> Maybe a
 findOf :: Traversal' s a -> (a -> Bool) -> s -> Maybe a

 find ≡ findOf folded
 ifindOf l ≡ findOf l . Indexed

A simpler version that didn't permit indexing, would be:

 findOf :: Getting (Endo (Maybe a)) s a -> (a -> Bool) -> s -> Maybe a
 findOf l p = foldrOf l (a y -> if p a then Just a else y) Nothing

foldrOf' :: Getting (Dual (Endo (Endo r))) s a -> (a -> r -> r) -> r -> s -> r

Strictly fold right over the elements of a structure.

 foldr' ≡ foldrOf' folded

 foldrOf' :: Getter s a     -> (a -> r -> r) -> r -> s -> r
 foldrOf' :: Fold s a       -> (a -> r -> r) -> r -> s -> r
 foldrOf' :: Iso' s a       -> (a -> r -> r) -> r -> s -> r
 foldrOf' :: Lens' s a      -> (a -> r -> r) -> r -> s -> r
 foldrOf' :: Traversal' s a -> (a -> r -> r) -> r -> s -> r

foldlOf' :: Getting (Endo (Endo r)) s a -> (r -> a -> r) -> r -> s -> r

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

 foldl' ≡ foldlOf' folded

 foldlOf' :: Getter s a     -> (r -> a -> r) -> r -> s -> r
 foldlOf' :: Fold s a       -> (r -> a -> r) -> r -> s -> r
 foldlOf' :: Iso' s a       -> (r -> a -> r) -> r -> s -> r
 foldlOf' :: Lens' s a      -> (r -> a -> r) -> r -> s -> r
 foldlOf' :: Traversal' s a -> (r -> a -> r) -> r -> s -> r

foldr1Of :: Getting (Endo (Maybe a)) s a -> (a -> a -> a) -> s -> a

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 each (+) (1,2,3,4)
10

 foldr1Of l f ≡ foldr1 f . toListOf l
 foldr1 ≡ foldr1Of folded

 foldr1Of :: Getter s a     -> (a -> a -> a) -> s -> a
 foldr1Of :: Fold s a       -> (a -> a -> a) -> s -> a
 foldr1Of :: Iso' s a       -> (a -> a -> a) -> s -> a
 foldr1Of :: Lens' s a      -> (a -> a -> a) -> s -> a
 foldr1Of :: Traversal' s a -> (a -> a -> a) -> s -> a

foldl1Of :: Getting (Dual (Endo (Maybe a))) s a -> (a -> a -> a) -> s -> a

A variant of foldlOf 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.

>>> foldl1Of each (+) (1,2,3,4)
10

 foldl1Of l f ≡ foldl1 f . toListOf l
 foldl1 ≡ foldl1Of folded

 foldl1Of :: Getter s a     -> (a -> a -> a) -> s -> a
 foldl1Of :: Fold s a       -> (a -> a -> a) -> s -> a
 foldl1Of :: Iso' s a       -> (a -> a -> a) -> s -> a
 foldl1Of :: Lens' s a      -> (a -> a -> a) -> s -> a
 foldl1Of :: Traversal' s a -> (a -> a -> a) -> s -> a

foldr1Of' :: Getting (Dual (Endo (Endo (Maybe a)))) s a -> (a -> a -> a) -> s -> a

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

 foldr1Of l f ≡ foldr1 f . toListOf l

 foldr1Of' :: Getter s a     -> (a -> a -> a) -> s -> a
 foldr1Of' :: Fold s a       -> (a -> a -> a) -> s -> a
 foldr1Of' :: Iso' s a       -> (a -> a -> a) -> s -> a
 foldr1Of' :: Lens' s a      -> (a -> a -> a) -> s -> a
 foldr1Of' :: Traversal' s a -> (a -> a -> a) -> s -> a

foldl1Of' :: Getting (Endo (Endo (Maybe a))) s a -> (a -> a -> a) -> s -> a

A variant of foldlOf' that has no base case and thus may only be applied to folds and structures such that the fold views at least one element of the structure.

 foldl1Of' l f ≡ foldl1' f . toListOf l

 foldl1Of' :: Getter s a     -> (a -> a -> a) -> s -> a
 foldl1Of' :: Fold s a       -> (a -> a -> a) -> s -> a
 foldl1Of' :: Iso' s a       -> (a -> a -> a) -> s -> a
 foldl1Of' :: Lens' s a      -> (a -> a -> a) -> s -> a
 foldl1Of' :: Traversal' s a -> (a -> a -> a) -> s -> a

foldrMOf :: Monad m => Getting (Dual (Endo (r -> m r))) s a -> (a -> r -> m r) -> r -> s -> m r

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 s a     -> (a -> r -> m r) -> r -> s -> m r
 foldrMOf :: Monad m => Fold s a       -> (a -> r -> m r) -> r -> s -> m r
 foldrMOf :: Monad m => Iso' s a       -> (a -> r -> m r) -> r -> s -> m r
 foldrMOf :: Monad m => Lens' s a      -> (a -> r -> m r) -> r -> s -> m r
 foldrMOf :: Monad m => Traversal' s a -> (a -> r -> m r) -> r -> s -> m r

foldlMOf :: Monad m => Getting (Endo (r -> m r)) s a -> (r -> a -> m r) -> r -> s -> m r

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 s a     -> (r -> a -> m r) -> r -> s -> m r
 foldlMOf :: Monad m => Fold s a       -> (r -> a -> m r) -> r -> s -> m r
 foldlMOf :: Monad m => Iso' s a       -> (r -> a -> m r) -> r -> s -> m r
 foldlMOf :: Monad m => Lens' s a      -> (r -> a -> m r) -> r -> s -> m r
 foldlMOf :: Monad m => Traversal' s a -> (r -> a -> m r) -> r -> s -> m r

Indexed Folds

(^@..) :: s -> IndexedGetting i (Endo [(i, a)]) s a -> [(i, a)]

An infix version of itoListOf.

(^@?) :: s -> IndexedGetting i (Endo (Maybe (i, a))) s a -> Maybe (i, a)

Perform a safe head (with index) of an IndexedFold or IndexedTraversal or retrieve Just the index and result from an IndexedGetter or IndexedLens.

When using a IndexedTraversal as a partial IndexedLens, or an IndexedFold as a partial IndexedGetter this can be a convenient way to extract the optional value.

 (^@?) :: s -> IndexedGetter i s a     -> Maybe (i, a)
 (^@?) :: s -> IndexedFold i s a       -> Maybe (i, a)
 (^@?) :: s -> IndexedLens' i s a      -> Maybe (i, a)
 (^@?) :: s -> IndexedTraversal' i s a -> Maybe (i, a)

(^@?!) :: s -> IndexedGetting i (Endo (i, a)) s a -> (i, a)

Perform an *UNSAFE* head (with index) of an IndexedFold or IndexedTraversal assuming that it is there.

 (^@?!) :: s -> IndexedGetter i s a     -> (i, a)
 (^@?!) :: s -> IndexedFold i s a       -> (i, a)
 (^@?!) :: s -> IndexedLens' i s a      -> (i, a)
 (^@?!) :: s -> IndexedTraversal' i s a -> (i, a)

Indexed Folding

ifoldMapOf :: IndexedGetting i m s a -> (i -> a -> m) -> s -> m

Fold an IndexedFold or IndexedTraversal by mapping indices and values to an arbitrary Monoid with access to the i.

When you don't need access to the index then foldMapOf is more flexible in what it accepts.

 foldMapOf l ≡ ifoldMapOf l . const

 ifoldMapOf ::             IndexedGetter i s a     -> (i -> a -> m) -> s -> m
 ifoldMapOf :: Monoid m => IndexedFold i s a       -> (i -> a -> m) -> s -> m
 ifoldMapOf ::             IndexedLens' i s a      -> (i -> a -> m) -> s -> m
 ifoldMapOf :: Monoid m => IndexedTraversal' i s a -> (i -> a -> m) -> s -> m

ifoldrOf :: IndexedGetting i (Endo r) s a -> (i -> a -> r -> r) -> r -> s -> r

Right-associative fold of parts of a structure that are viewed through an IndexedFold or IndexedTraversal with access to the i.

When you don't need access to the index then foldrOf is more flexible in what it accepts.

 foldrOf l ≡ ifoldrOf l . const

 ifoldrOf :: IndexedGetter i s a     -> (i -> a -> r -> r) -> r -> s -> r
 ifoldrOf :: IndexedFold i s a       -> (i -> a -> r -> r) -> r -> s -> r
 ifoldrOf :: IndexedLens' i s a      -> (i -> a -> r -> r) -> r -> s -> r
 ifoldrOf :: IndexedTraversal' i s a -> (i -> a -> r -> r) -> r -> s -> r

ifoldlOf :: IndexedGetting i (Dual (Endo r)) s a -> (i -> r -> a -> r) -> r -> s -> r

Left-associative fold of the parts of a structure that are viewed through an IndexedFold or IndexedTraversal with access to the i.

When you don't need access to the index then foldlOf is more flexible in what it accepts.

 foldlOf l ≡ ifoldlOf l . const

 ifoldlOf :: IndexedGetter i s a     -> (i -> r -> a -> r) -> r -> s -> r
 ifoldlOf :: IndexedFold i s a       -> (i -> r -> a -> r) -> r -> s -> r
 ifoldlOf :: IndexedLens' i s a      -> (i -> r -> a -> r) -> r -> s -> r
 ifoldlOf :: IndexedTraversal' i s a -> (i -> r -> a -> r) -> r -> s -> r

ianyOf :: IndexedGetting i Any s a -> (i -> a -> Bool) -> s -> Bool

Return whether or not any element viewed through an IndexedFold or IndexedTraversal satisfy a predicate, with access to the i.

When you don't need access to the index then anyOf is more flexible in what it accepts.

 anyOf l ≡ ianyOf l . const

 ianyOf :: IndexedGetter i s a     -> (i -> a -> Bool) -> s -> Bool
 ianyOf :: IndexedFold i s a       -> (i -> a -> Bool) -> s -> Bool
 ianyOf :: IndexedLens' i s a      -> (i -> a -> Bool) -> s -> Bool
 ianyOf :: IndexedTraversal' i s a -> (i -> a -> Bool) -> s -> Bool

iallOf :: IndexedGetting i All s a -> (i -> a -> Bool) -> s -> Bool

Return whether or not all elements viewed through an IndexedFold or IndexedTraversal satisfy a predicate, with access to the i.

When you don't need access to the index then allOf is more flexible in what it accepts.

 allOf l ≡ iallOf l . const

 iallOf :: IndexedGetter i s a     -> (i -> a -> Bool) -> s -> Bool
 iallOf :: IndexedFold i s a       -> (i -> a -> Bool) -> s -> Bool
 iallOf :: IndexedLens' i s a      -> (i -> a -> Bool) -> s -> Bool
 iallOf :: IndexedTraversal' i s a -> (i -> a -> Bool) -> s -> Bool

itraverseOf_ :: Functor f => IndexedGetting i (Traversed r f) s a -> (i -> a -> f r) -> s -> f ()

Traverse the targets of an IndexedFold or IndexedTraversal with access to the i, discarding the results.

When you don't need access to the index then traverseOf_ is more flexible in what it accepts.

 traverseOf_ l ≡ itraverseOf l . const

 itraverseOf_ :: Functor f     => IndexedGetter i s a     -> (i -> a -> f r) -> s -> f ()
 itraverseOf_ :: Applicative f => IndexedFold i s a       -> (i -> a -> f r) -> s -> f ()
 itraverseOf_ :: Functor f     => IndexedLens' i s a      -> (i -> a -> f r) -> s -> f ()
 itraverseOf_ :: Applicative f => IndexedTraversal' i s a -> (i -> a -> f r) -> s -> f ()

iforOf_ :: Functor f => IndexedGetting i (Traversed r f) s a -> s -> (i -> a -> f r) -> f ()

Traverse the targets of an IndexedFold or IndexedTraversal with access to the index, discarding the results (with the arguments flipped).

 iforOf_ ≡ flip . itraverseOf_

When you don't need access to the index then forOf_ is more flexible in what it accepts.

 forOf_ l a ≡ iforOf_ l a . const

 iforOf_ :: Functor f     => IndexedGetter i s a     -> s -> (i -> a -> f r) -> f ()
 iforOf_ :: Applicative f => IndexedFold i s a       -> s -> (i -> a -> f r) -> f ()
 iforOf_ :: Functor f     => IndexedLens' i s a      -> s -> (i -> a -> f r) -> f ()
 iforOf_ :: Applicative f => IndexedTraversal' i s a -> s -> (i -> a -> f r) -> f ()

imapMOf_ :: Monad m => IndexedGetting i (Sequenced r m) s a -> (i -> a -> m r) -> s -> m ()

Run monadic actions for each target of an IndexedFold or IndexedTraversal with access to the index, discarding the results.

When you don't need access to the index then mapMOf_ is more flexible in what it accepts.

 mapMOf_ l ≡ imapMOf l . const

 imapMOf_ :: Monad m => IndexedGetter i s a     -> (i -> a -> m r) -> s -> m ()
 imapMOf_ :: Monad m => IndexedFold i s a       -> (i -> a -> m r) -> s -> m ()
 imapMOf_ :: Monad m => IndexedLens' i s a      -> (i -> a -> m r) -> s -> m ()
 imapMOf_ :: Monad m => IndexedTraversal' i s a -> (i -> a -> m r) -> s -> m ()

iforMOf_ :: Monad m => IndexedGetting i (Sequenced r m) s a -> s -> (i -> a -> m r) -> m ()

Run monadic actions for each target of an IndexedFold or IndexedTraversal with access to the index, discarding the results (with the arguments flipped).

 iforMOf_ ≡ flip . imapMOf_

When you don't need access to the index then forMOf_ is more flexible in what it accepts.

 forMOf_ l a ≡ iforMOf l a . const

 iforMOf_ :: Monad m => IndexedGetter i s a     -> s -> (i -> a -> m r) -> m ()
 iforMOf_ :: Monad m => IndexedFold i s a       -> s -> (i -> a -> m r) -> m ()
 iforMOf_ :: Monad m => IndexedLens' i s a      -> s -> (i -> a -> m r) -> m ()
 iforMOf_ :: Monad m => IndexedTraversal' i s a -> s -> (i -> a -> m r) -> m ()

iconcatMapOf :: IndexedGetting i [r] s a -> (i -> a -> [r]) -> s -> [r]

Concatenate the results of a function of the elements of an IndexedFold or IndexedTraversal with access to the index.

When you don't need access to the index then concatMapOf is more flexible in what it accepts.

 concatMapOf l ≡ iconcatMapOf l . const
 iconcatMapOf ≡ ifoldMapOf

 iconcatMapOf :: IndexedGetter i s a     -> (i -> a -> [r]) -> s -> [r]
 iconcatMapOf :: IndexedFold i s a       -> (i -> a -> [r]) -> s -> [r]
 iconcatMapOf :: IndexedLens' i s a      -> (i -> a -> [r]) -> s -> [r]
 iconcatMapOf :: IndexedTraversal' i s a -> (i -> a -> [r]) -> s -> [r]

ifindOf :: IndexedGetting i (Endo (Maybe a)) s a -> (i -> a -> Bool) -> s -> Maybe a

The findOf function takes an IndexedFold or IndexedTraversal, a predicate that is also supplied the index, a structure and returns the left-most element of the structure matching the predicate, or Nothing if there is no such element.

When you don't need access to the index then findOf is more flexible in what it accepts.

 findOf l ≡ ifindOf l . const

 ifindOf :: IndexedGetter i s a     -> (i -> a -> Bool) -> s -> Maybe a
 ifindOf :: IndexedFold i s a       -> (i -> a -> Bool) -> s -> Maybe a
 ifindOf :: IndexedLens' i s a      -> (i -> a -> Bool) -> s -> Maybe a
 ifindOf :: IndexedTraversal' i s a -> (i -> a -> Bool) -> s -> Maybe a

ifoldrOf' :: IndexedGetting i (Dual (Endo (r -> r))) s a -> (i -> a -> r -> r) -> r -> s -> r

Strictly fold right over the elements of a structure with an index.

When you don't need access to the index then foldrOf' is more flexible in what it accepts.

 foldrOf' l ≡ ifoldrOf' l . const

 ifoldrOf' :: IndexedGetter i s a     -> (i -> a -> r -> r) -> r -> s -> r
 ifoldrOf' :: IndexedFold i s a       -> (i -> a -> r -> r) -> r -> s -> r
 ifoldrOf' :: IndexedLens' i s a      -> (i -> a -> r -> r) -> r -> s -> r
 ifoldrOf' :: IndexedTraversal' i s a -> (i -> a -> r -> r) -> r -> s -> r

ifoldlOf' :: IndexedGetting i (Endo (r -> r)) s a -> (i -> r -> a -> r) -> r -> s -> r

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

When you don't need access to the index then foldlOf' is more flexible in what it accepts.

 foldlOf' l ≡ ifoldlOf' l . const

 ifoldlOf' :: IndexedGetter i s a       -> (i -> r -> a -> r) -> r -> s -> r
 ifoldlOf' :: IndexedFold i s a         -> (i -> r -> a -> r) -> r -> s -> r
 ifoldlOf' :: IndexedLens' i s a        -> (i -> r -> a -> r) -> r -> s -> r
 ifoldlOf' :: IndexedTraversal' i s a   -> (i -> r -> a -> r) -> r -> s -> r

ifoldrMOf :: Monad m => IndexedGetting i (Dual (Endo (r -> m r))) s a -> (i -> a -> r -> m r) -> r -> s -> m r

Monadic fold right over the elements of a structure with an index.

When you don't need access to the index then foldrMOf is more flexible in what it accepts.

 foldrMOf l ≡ ifoldrMOf l . const

 ifoldrMOf :: Monad m => IndexedGetter i s a     -> (i -> a -> r -> m r) -> r -> s -> m r
 ifoldrMOf :: Monad m => IndexedFold i s a       -> (i -> a -> r -> m r) -> r -> s -> m r
 ifoldrMOf :: Monad m => IndexedLens' i s a      -> (i -> a -> r -> m r) -> r -> s -> m r
 ifoldrMOf :: Monad m => IndexedTraversal' i s a -> (i -> a -> r -> m r) -> r -> s -> m r

ifoldlMOf :: Monad m => IndexedGetting i (Endo (r -> m r)) s a -> (i -> r -> a -> m r) -> r -> s -> m r

Monadic fold over the elements of a structure with an index, associating to the left.

When you don't need access to the index then foldlMOf is more flexible in what it accepts.

 foldlMOf l ≡ ifoldlMOf l . const

 ifoldlMOf :: Monad m => IndexedGetter i s a     -> (i -> r -> a -> m r) -> r -> s -> m r
 ifoldlMOf :: Monad m => IndexedFold i s a       -> (i -> r -> a -> m r) -> r -> s -> m r
 ifoldlMOf :: Monad m => IndexedLens' i s a      -> (i -> r -> a -> m r) -> r -> s -> m r
 ifoldlMOf :: Monad m => IndexedTraversal' i s a -> (i -> r -> a -> m r) -> r -> s -> m r

itoListOf :: IndexedGetting i (Endo [(i, a)]) s a -> s -> [(i, a)]

Extract the key-value pairs from a structure.

When you don't need access to the indices in the result, then toListOf is more flexible in what it accepts.

 toListOf l ≡ map fst . itoListOf l

 itoListOf :: IndexedGetter i s a     -> s -> [(i,a)]
 itoListOf :: IndexedFold i s a       -> s -> [(i,a)]
 itoListOf :: IndexedLens' i s a      -> s -> [(i,a)]
 itoListOf :: IndexedTraversal' i s a -> s -> [(i,a)]

Building Indexed Folds

ifiltered :: (Indexable i p, Applicative f) => (i -> a -> Bool) -> Overloading' p (Indexed i) f a a

Filter an IndexedFold or IndexedGetter, obtaining an IndexedFold.

>>> [0,0,0,5,5,5]^..traversed.ifiltered (\i a -> i <= a)
[0,5,5,5]

Compose with filtered to filter another IndexedLens, IndexedIso, IndexedGetter, IndexedFold (or IndexedTraversal) with access to both the value and the index.

Note: As with filtered, this is not a legal IndexedTraversal, unless you are very careful not to invalidate the predicate on the target!

itakingWhile :: (Indexable i p, Profunctor q, Contravariant f, Applicative f) => (i -> a -> Bool) -> Overloading (Indexed i) q (Accessor (Endo (f s))) s s a a -> Overloading p q f s s a a

Obtain an IndexedFold by taking elements from another IndexedFold, IndexedLens, IndexedGetter or IndexedTraversal while a predicate holds.

 itakingWhile :: (i -> a -> Bool) -> IndexedFold i s a          -> IndexedFold i s a
 itakingWhile :: (i -> a -> Bool) -> IndexedTraversal' i s a    -> IndexedFold i s a
 itakingWhile :: (i -> a -> Bool) -> IndexedLens' i s a         -> IndexedFold i s a
 itakingWhile :: (i -> a -> Bool) -> IndexedGetter i s a        -> IndexedFold i s a
 itakingWhile :: (i -> a -> Bool) -> IndexedMonadicFold i m s a -> IndexedMonadicFold i m s a
 itakingWhile :: (i -> a -> Bool) -> IndexedAction i m s a      -> IndexedMonadicFold i m s a

idroppingWhile :: (Indexable i p, Profunctor q, Applicative f) => (i -> a -> Bool) -> Overloading (Indexed i) q (Compose (State Bool) f) s t a a -> Overloading p q f s t a a

Obtain an IndexedFold by dropping elements from another IndexedFold, IndexedLens, IndexedGetter or IndexedTraversal while a predicate holds.

 idroppingWhile :: (i -> a -> Bool) -> IndexedFold i s a          -> IndexedFold i s a
 idroppingWhile :: (i -> a -> Bool) -> IndexedTraversal' i s a    -> IndexedFold i s a -- see notes
 idroppingWhile :: (i -> a -> Bool) -> IndexedLens' i s a         -> IndexedFold i s a -- see notes
 idroppingWhile :: (i -> a -> Bool) -> IndexedGetter i s a        -> IndexedFold i s a
 idroppingWhile :: (i -> a -> Bool) -> IndexedMonadicFold i m s a -> IndexedMonadicFold i m s a
 idroppingWhile :: (i -> a -> Bool) -> IndexedAction i m s a      -> IndexedMonadicFold i m s a

Applying idroppingWhile to an IndexedLens or IndexedTraversal will still allow you to use it as a pseudo-IndexedTraversal, but if you change the value of the targets to ones where the predicate returns True, then you will break the Traversal laws and Traversal fusion will no longer be sound.

Deprecated

headOf :: Getting (First a) s a -> s -> Maybe a

Deprecated: headOf will be removed after GHC 7.8 is released. (Use preview or firstOf)

A deprecated alias for firstOf.

Internal types

data Leftmost a

Used for preview.

Instances

Monoid (Leftmost a)

Semigroup (Leftmost a)

data Rightmost a

Used for lastOf.

Instances

Monoid (Rightmost a)

Semigroup (Rightmost a)

data Traversed a f

Used internally by traverseOf_ and the like.

The argument a of the result should not be used!

Instances

Applicative f => Monoid (Traversed a f)

Apply f => Semigroup (Traversed a f)

data Sequenced a m

Used internally by mapM_ and the like.

The argument a of the result should not be used!

Instances

Monad m => Monoid (Sequenced a m)

Apply m => Semigroup (Sequenced a m)