Portability  Rank2Types 

Stability  provisional 
Maintainer  Edward Kmett <ekmett@gmail.com> 
Safe Haskell  SafeInferred 
A
is a generalization of Traversal
a b c dtraverse
from
Traversable
. It allows you to traverse over a structure and change out
its contents with monadic or applicative sideeffects. Starting from
,
traverse
:: (Traversable
t, Applicative
f) => (c > f d) > t c > f (t d)
we monomorphize the contents and result to obtain
type Traversal a b c d = forall f. Applicative f => (c > f d) > a > f b
While a Traversal
isn't quite a Fold
, it _can_ be used for Getting
like a Fold
, because given a Monoid
m
, we have an Applicative
for (
. Everything you know how to do with a Const
m)Traversable
container, you can with with a Traversal
, and here we provide
combinators that generalize the usual Traversable
operations.
 type Traversal a b c d = forall f. Applicative f => (c > f d) > a > f b
 type :=> a b = forall f. Applicative f => (b > f b) > a > f a
 element :: Traversable t => Int > t a :> a
 elementOf :: Functor f => LensLike (ElementOf f) a b c c > Int > LensLike f a b c c
 traverseOf :: LensLike f a b c d > (c > f d) > a > f b
 forOf :: LensLike f a b c d > a > (c > f d) > f b
 sequenceAOf :: LensLike f a b (f c) c > a > f b
 mapMOf :: LensLike (WrappedMonad m) a b c d > (c > m d) > a > m b
 forMOf :: LensLike (WrappedMonad m) a b c d > a > (c > m d) > m b
 sequenceOf :: LensLike (WrappedMonad m) a b (m c) c > a > m b
 transposeOf :: LensLike ZipList a b [c] c > a > [b]
 mapAccumLOf :: LensLike (Backwards (State s)) a b c d > (s > c > (s, d)) > s > a > (s, b)
 mapAccumROf :: LensLike (State s) a b c d > (s > c > (s, d)) > s > a > (s, b)
 scanr1Of :: LensLike (State (Maybe c)) a b c c > (c > c > c) > a > b
 scanl1Of :: LensLike (Backwards (State (Maybe c))) a b c c > (c > c > c) > a > b
 class (Functor t, Foldable t) => Traversable t where
 traverse :: Applicative f => (a > f b) > t a > f (t b)
 ignored :: Traversal a a c d
 traverseLeft :: Traversal (Either a c) (Either b c) a b
 traverseRight :: Traversal (Either c a) (Either c b) a b
 both :: Traversal (a, a) (b, b) a b
 cloneTraversal :: Applicative f => ((c > Bazaar c d d) > a > Bazaar c d b) > (c > f d) > a > f b
 data ReifiedTraversal a b c d = ReifyTraversal {
 reflectTraversal :: Traversal a b c d
 type SimpleTraversal a b = Traversal a a b b
 type SimpleReifiedTraversal a b = ReifiedTraversal a a b b
Lenses
type Traversal a b c d = forall f. Applicative f => (c > f d) > a > f bSource
A Traversal
can be used directly as a Setter
or a Fold
(but not as a Lens
) and provides
the ability to both read and update multiple fields, subject to some relatively weak Traversal
laws.
These have also been known as multilenses, but they have the signature and spirit of
traverse
::Traversable
f =>Traversal
(f a) (f b) a b
and the more evocative name suggests their application.
Most of the time the Traversal
you will want to use is just traverse
, but you can also pass any
Lens
or Iso
as a Traversal
, and composition of a Traversal
(or Lens
or Iso
) with a Traversal
(or Lens
or Iso
)
using (.) forms a valid Traversal
.
The laws for a Traversal t
follow from the laws for Traversable as stated in "The Essence of the Iterator Pattern".
tpure
≡pure
fmap
(t f).
t g ≡getCompose
.
t (Compose
.
fmap
f.
g)
One consequence of this requirement is that a Traversal
needs to leave the same number of elements as a
candidate for subsequent Traversal
that it started with. Another testament to the strength of these laws
is that the caveat expressed in section 5.5 of the "Essence of the Iterator Pattern" about exotic
Traversable
instances that traverse
the same entry multiple times was actually already ruled out by the
second law in that same paper!
type :=> a b = forall f. Applicative f => (b > f b) > a > f aSource
Lensing Traversals
element :: Traversable t => Int > t a :> aSource
Access the nth element of a Traversable
container.
Attempts to access beyond the range of the Traversal
will cause an error.
element
≡elementOf
traverse
Traversing and Lensing
traverseOf :: LensLike f a b c d > (c > f d) > a > f bSource
Map each element of a structure targeted by a Lens or Traversal, evaluate these actions from left to right, and collect the results.
This function is only provided for consistency, id
is strictly more general.
traverseOf
≡id
This yields the obvious law:
traverse
≡traverseOf
traverse
traverseOf
::Iso
a b c d > (c > f d) > a > f btraverseOf
::Lens
a b c d > (c > f d) > a > f btraverseOf
::Traversal
a b c d > (c > f d) > a > f b
forOf :: LensLike f a b c d > a > (c > f d) > f bSource
A version of traverseOf
with the arguments flipped, such that:
forOf
l ≡flip
(traverseOf
l)
for
≡forOf
traverse
This function is only provided for consistency, flip
is strictly more general.
forOf
≡flip
forOf
::Iso
a b c d > a > (c > f d) > f bforOf
::Lens
a b c d > a > (c > f d) > f bforOf
::Traversal
a b c d > a > (c > f d) > f b
sequenceAOf :: LensLike f a b (f c) c > a > f bSource
Evaluate each action in the structure from left to right, and collect the results.
sequenceA
≡sequenceAOf
traverse
≡traverse
id
sequenceAOf
l ≡traverseOf
l id ≡ l id
sequenceAOf
::Iso
a b (f c) c > a > f bsequenceAOf
::Lens
a b (f c) c > a > f bsequenceAOf
::Applicative
f =>Traversal
a b (f c) c > a > f b
mapMOf :: LensLike (WrappedMonad m) a b c d > (c > m d) > a > m bSource
Map each element of a structure targeted by a lens to a monadic action, evaluate these actions from left to right, and collect the results.
mapM
≡mapMOf
traverse
mapMOf
::Iso
a b c d > (c > m d) > a > m bmapMOf
::Lens
a b c d > (c > m d) > a > m bmapMOf
::Monad
m =>Traversal
a b c d > (c > m d) > a > m b
forMOf :: LensLike (WrappedMonad m) a b c d > a > (c > m d) > m bSource
sequenceOf :: LensLike (WrappedMonad m) a b (m c) c > a > m bSource
Sequence the (monadic) effects targeted by a lens in a container from left to right.
sequence
≡sequenceOf
traverse
sequenceOf
l ≡mapMOf
l idsequenceOf
l ≡unwrapMonad
. lWrapMonad
sequenceOf
::Iso
a b (m c) c > a > m bsequenceOf
::Lens
a b (m c) c > a > m bsequenceOf
::Monad
m =>Traversal
a b (m c) c > a > m b
transposeOf :: LensLike ZipList a b [c] c > a > [b]Source
This generalizes transpose
to an arbitrary Traversal
.
Note: transpose
handles ragged inputs more intelligently, but for nonragged inputs:
transpose
≡transposeOf
traverse
>>>
transposeOf traverse [[1,2,3],[4,5,6]]
[[1,4],[2,5],[3,6]]
Since every Lens
is a Traversal
, we can use this as a form of
monadic strength as well:
transposeOf
_2
:: (b, [a]) > [(b, a)]
mapAccumLOf :: LensLike (Backwards (State s)) a b c d > (s > c > (s, d)) > s > a > (s, b)Source
This generalizes mapAccumL
to an arbitrary Traversal
.
mapAccumL
≡mapAccumLOf
traverse
mapAccumLOf
accumulates state from left to right.
mapAccumLOf
::Iso
a b c d > (s > c > (s, d)) > s > a > (s, b)mapAccumLOf
::Lens
a b c d > (s > c > (s, d)) > s > a > (s, b)mapAccumLOf
::Traversal
a b c d > (s > c > (s, d)) > s > a > (s, b)
mapAccumROf :: LensLike (State s) a b c d > (s > c > (s, d)) > s > a > (s, b)Source
This generalizes mapAccumR
to an arbitrary Traversal
.
mapAccumR
≡mapAccumROf
traverse
mapAccumROf
accumulates state from right to left.
mapAccumROf
::Iso
a b c d > (s > c > (s, d)) > s > a > (s, b)mapAccumROf
::Lens
a b c d > (s > c > (s, d)) > s > a > (s, b)mapAccumROf
::Traversal
a b c d > (s > c > (s, d)) > s > a > (s, b)
Common Traversals
class (Functor t, Foldable t) => Traversable t where
Functors representing data structures that can be traversed from left to right.
Minimal complete definition: traverse
or sequenceA
.
Instances are similar to Functor
, e.g. given a data type
data Tree a = Empty  Leaf a  Node (Tree a) a (Tree a)
a suitable instance would be
instance Traversable Tree where traverse f Empty = pure Empty traverse f (Leaf x) = Leaf <$> f x traverse f (Node l k r) = Node <$> traverse f l <*> f k <*> traverse f r
This is suitable even for abstract types, as the laws for <*>
imply a form of associativity.
The superclass instances should satisfy the following:
 In the
Functor
instance,fmap
should be equivalent to traversal with the identity applicative functor (fmapDefault
).  In the
Foldable
instance,foldMap
should be equivalent to traversal with a constant applicative functor (foldMapDefault
).
traverse :: Applicative f => (a > f b) > t a > f (t b)
Map each element of a structure to an action, evaluate these actions from left to right, and collect the results.
ignored :: Traversal a a c dSource
This is the trivial empty traversal.
ignored
::Applicative
f => (c > f d) > a > f a
ignored
≡const
pure
traverseLeft :: Traversal (Either a c) (Either b c) a bSource
A traversal for tweaking the lefthand value of an Either
:
>>>
over traverseLeft (+1) (Left 2)
Left 3>>>
over traverseLeft (+1) (Right 2)
Right 2>>>
Right 42 ^.traverseLeft :: String
"">>>
Left "hello" ^.traverseLeft
"hello"
traverseLeft ::Applicative
f => (a > f b) >Either
a c > f (Either
b c)
traverseRight :: Traversal (Either c a) (Either c b) a bSource
traverse the righthand value of an Either
:
traverseRight
≡traverse
Unfortunately the instance for
is still missing from base,
so this can't just be Traversable
(Either
c)traverse
>>>
over traverseRight (+1) (Left 2)
Left 2>>>
over traverseRight (+1) (Right 2)
Right 3>>>
Right "hello" ^.traverseRight
"hello">>>
Left "hello" ^.traverseRight :: [Double]
[]
traverseRight ::Applicative
f => (a > f b) >Either
c a > f (Either
c a)
both :: Traversal (a, a) (b, b) a bSource
Traverse both parts of a tuple with matching types.
>>>
both *~ 10 $ (1,2)
(10,20)>>>
over both length ("hello","world")
(5,5)>>>
("hello","world")^.both
"helloworld"
Cloning Traversals
cloneTraversal :: Applicative f => ((c > Bazaar c d d) > a > Bazaar c d b) > (c > f d) > a > f bSource
A Traversal
is completely characterized by its behavior on a Bazaar
.
Cloning a Traversal
is one way to make sure you aren't given
something weaker, such as a Fold
and can be
used as a way to pass around traversals that have to be monomorphic in f
.
Note: This only accepts a proper Traversal
(or Lens
). To clone a Lens
as such, use cloneLens
Note: It is usually better to ReifyTraversal
and use reflectTraversal
than to cloneTraversal
. The former can execute at full speed, while the
latter needs to round trip through the Bazaar
.
>>>
let foo l a = (view (cloneTraversal l) a, set (cloneTraversal l) 10 a)
>>>
foo both ("hello","world")
("helloworld",(10,10))
cloneTraversal
::LensLike
(Bazaar
c d) a b c d >Traversal
a b c d
data ReifiedTraversal a b c d Source
A form of Traversal
that can be stored monomorphically in a container.
ReifyTraversal  

Simple
type SimpleTraversal a b = Traversal a a b bSource
type SimpleReifiedTraversal a b = ReifiedTraversal a a b bSource
type SimpleReifiedTraversal =Simple
ReifiedTraversal