This class permits overloading of function application for things that also admit a notion of a key or index.

This is a Profunctor that is both Corepresentable by f and Representable by g such that f is left adjoint to g. From this you can derive a lot of structure due to the preservation of limits and colimits.

A function with access to a index. This constructor may be useful when you need to store an Indexable in a container to avoid ImpredicativeTypes.

index :: Indexed i a b -> i -> a -> b

Compose an Indexed function with a non-indexed function.

Mnemonically, the < points to the indexing we want to preserve.

>>> let nestedMap = (fmap Map.fromList . Map.fromList) [(1, [(10, "one,ten"), (20, "one,twenty")]), (2, [(30, "two,thirty"), (40,"two,forty")])]
>>> nestedMap^..(itraversed<.itraversed).withIndex
[(1,"one,ten"),(1,"one,twenty"),(2,"two,thirty"),(2,"two,forty")]

Composition of Indexed functions.

Mnemonically, the < and > points to the fact that we want to preserve the indices.

>>> let nestedMap = (fmap Map.fromList . Map.fromList) [(1, [(10, "one,ten"), (20, "one,twenty")]), (2, [(30, "two,thirty"), (40,"two,forty")])]
>>> nestedMap^..(itraversed<.>itraversed).withIndex
[((1,10),"one,ten"),((1,20),"one,twenty"),((2,30),"two,thirty"),((2,40),"two,forty")]

Compose a non-indexed function with an Indexed function.

Mnemonically, the > points to the indexing we want to preserve.

This is the same as (.).

f . g (and f .> g) gives you the index of g unless g is index-preserving, like a Prism, Iso or Equality, in which case it'll pass through the index of f.

>>> let nestedMap = (fmap Map.fromList . Map.fromList) [(1, [(10, "one,ten"), (20, "one,twenty")]), (2, [(30, "two,thirty"), (40,"two,forty")])]
>>> nestedMap^..(itraversed.>itraversed).withIndex
[(10,"one,ten"),(20,"one,twenty"),(30,"two,thirty"),(40,"two,forty")]

Use a value itself as its own index. This is essentially an indexed version of id.

Note: When used to modify the value, this can break the index requirements assumed by indices and similar, so this is only properly an IndexedGetter, but it can be used as more.

 selfIndex ::IndexedGetter a a b

Remap the index.

Composition of Indexed functions with a user supplied function for combining indices.

Transform a Traversal into an IndexedTraversal or a Fold into an IndexedFold, etc.

 indexing :: Traversal s t a b -> IndexedTraversal Int s t a b
 indexing :: Prism s t a b     -> IndexedTraversal Int s t a b
 indexing :: Lens s t a b      -> IndexedLens Int  s t a b
 indexing :: Iso s t a b       -> IndexedLens Int s t a b
 indexing :: Fold s a          -> IndexedFold Int s a
 indexing :: Getter s a        -> IndexedGetter Int s a

indexing :: Indexable Int p => LensLike (Indexing f) s t a b -> Optical p (->) f s t a b

Transform a Traversal into an IndexedTraversal or a Fold into an IndexedFold, etc.

This combinator is like indexing except that it handles large traversals and folds gracefully.

 indexing64 :: Traversal s t a b -> IndexedTraversal Int64 s t a b
 indexing64 :: Prism s t a b     -> IndexedTraversal Int64 s t a b
 indexing64 :: Lens s t a b      -> IndexedLens Int64 s t a b
 indexing64 :: Iso s t a b       -> IndexedLens Int64 s t a b
 indexing64 :: Fold s a          -> IndexedFold Int64 s a
 indexing64 :: Getter s a        -> IndexedGetter Int64 s a

indexing64 :: Indexable Int64 p => LensLike (Indexing64 f) s t a b -> Over p f s t a b

A Functor with an additional index.

Instances must satisfy a modified form of the Functor laws:

 imap f . imap g ≡ imap (\i -> f i . g i)
 imap (\_ a -> a) ≡ id

A container that supports folding with an additional index.

Return whether or not any element in a container satisfies a predicate, with access to the index i.

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

 any ≡ iany . const

Return whether or not all elements in a container satisfy a predicate, with access to the index i.

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

 all ≡ iall . const

Return whether or not none of the elements in a container satisfy a predicate, with access to the index i.

When you don't need access to the index then none is more flexible in what it accepts.

 none ≡ inone . const
 inone f ≡ not . iany f

Determines whether no elements of the structure satisfy the predicate.

 none f ≡ not . any f

Traverse elements with access to the index i, discarding the results.

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

 traverse_ l = itraverse . const

Traverse elements with access to the index i, discarding the results (with the arguments flipped).

 ifor_ ≡ flip itraverse_

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

 for_ a ≡ ifor_ a . const

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.

 mapM_ ≡ imapM . const

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

 iforM_ ≡ flip imapM_

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

Concatenate the results of a function of the elements of an indexed container with access to the index.

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

 concatMap ≡ iconcatMap . const
 iconcatMap ≡ ifoldMap

Searches a container with a predicate that is also supplied the index, returning 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 find is more flexible in what it accepts.

 find ≡ ifind . const

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

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

 foldrM ≡ ifoldrM . const

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 foldlM is more flexible in what it accepts.

 foldlM ≡ ifoldlM . const

Extract the key-value pairs from a structure.

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

 toList ≡ map snd . itoList

Fold a container with indices returning both the indices and the values.

The result is only valid to compose in a Traversal, if you don't edit the index as edits to the index have no effect.

When composed with an IndexedFold or IndexedTraversal this yields an (Indexed) Fold of the indices.

This allows you to filter an IndexedFold, IndexedGetter, IndexedTraversal or IndexedLens based on a predicate on the indices.

>>> ["hello","the","world","!!!"]^..traversed.indices even
["hello","world"]

>>> over (traversed.indices (>0)) Prelude.reverse $ ["He","was","stressed","o_O"]
["He","saw","desserts","O_o"]

This allows you to filter an IndexedFold, IndexedGetter, IndexedTraversal or IndexedLens based on an index.

>>> ["hello","the","world","!!!"]^?traversed.index 2
Just "world"

A Traversable with an additional index.

An instance must satisfy a (modified) form of the Traversable laws:

 itraverse (const Identity) ≡ Identity
 fmap (itraverse f) . itraverse g ≡ getCompose . itraverse (\i -> Compose . fmap (f i) . g i)

Traverse with an index (and the arguments flipped).

 for a ≡ ifor a . const
 ifor ≡ flip itraverse

Map each element of a structure to a monadic action, evaluate these actions from left to right, and collect the results, with access the index.

When you don't need access to the index mapM is more liberal in what it can accept.

 mapM ≡ imapM . const

Map each element of a structure to a monadic action, evaluate these actions from left to right, and collect the results, with access its position (and the arguments flipped).

 forM a ≡ iforM a . const
 iforM ≡ flip imapM

Generalizes mapAccumR to add access to the index.

imapAccumROf accumulates state from right to left.

 mapAccumR ≡ imapAccumR . const

Generalizes mapAccumL to add access to the index.

imapAccumLOf accumulates state from left to right.

 mapAccumLOf ≡ imapAccumL . const