Copyright | (C) 2013-2016 Edward Kmett, 2015-2016 Artyom |
---|---|

License | BSD-style (see the file LICENSE) |

Safe Haskell | Trustworthy |

Language | Haskell2010 |

- Setter: modifies something in a structure
- Getter: extracts a value from a structure
- Fold: extracts multiple elements
- Lens: a combined getter-and-setter
- Iso: a lens that only changes the representation
- Traversal: a lens iterating over several elements
- Prism: a traversal iterating over at most 1 element
- Other types

- (&) :: a -> (a -> b) -> b
- type ASetter s t a b = (a -> Identity b) -> s -> Identity t
- type ASetter' s a = ASetter s s a a
- sets :: ((a -> b) -> s -> t) -> ASetter s t a b
- (%~) :: ASetter s t a b -> (a -> b) -> s -> t
- over :: ASetter s t a b -> (a -> b) -> s -> t
- (.~) :: ASetter s t a b -> b -> s -> t
- set :: ASetter s t a b -> b -> s -> t
- (?~) :: ASetter s t a (Maybe b) -> b -> s -> t
- (<%~) :: LensLike ((,) b) s t a b -> (a -> b) -> s -> (b, t)
- (<<%~) :: LensLike ((,) a) s t a b -> (a -> b) -> s -> (a, t)
- (<<.~) :: LensLike ((,) a) s t a b -> b -> s -> (a, t)
- mapped :: Functor f => ASetter (f a) (f b) a b
- type SimpleGetter s a = forall r. Getting r s a
- type Getting r s a = (a -> Const r a) -> s -> Const r s
- (^.) :: s -> Getting a s a -> a
- to :: (s -> a) -> SimpleGetter s a
- type SimpleFold s a = forall r. Monoid r => Getting r s a
- (^..) :: s -> Getting (Endo [a]) s a -> [a]
- toListOf :: Getting (Endo [a]) s a -> s -> [a]
- (^?) :: s -> Getting (First a) s a -> Maybe a
- (^?!) :: s -> Getting (Endo a) s a -> a
- has :: Getting Any s a -> s -> Bool
- folded :: Foldable f => SimpleFold (f a) a
- folding :: Foldable f => (s -> f a) -> SimpleFold s a
- type Lens s t a b = forall f. Functor f => (a -> f b) -> s -> f t
- type Lens' s a = Lens s s a a
- lens :: (s -> a) -> (s -> b -> t) -> Lens s t a b
- at :: At m => Index m -> Lens' m (Maybe (IxValue m))
- _1 :: Field1 s t a b => Lens s t a b
- _2 :: Field2 s t a b => Lens s t a b
- _3 :: Field3 s t a b => Lens s t a b
- _4 :: Field4 s t a b => Lens s t a b
- _5 :: Field5 s t a b => Lens s t a b
- strict :: Strict lazy strict => Lens' lazy strict
- lazy :: Strict lazy strict => Lens' strict lazy
- non :: Eq a => a -> Lens' (Maybe a) a
- type Traversal s t a b = forall f. Applicative f => (a -> f b) -> s -> f t
- type Traversal' s a = Traversal s s a a
- singular :: Traversal s t a a -> Lens s t a a
- failing :: Traversal s t a b -> Traversal s t a b -> Traversal s t a b
- filtered :: (a -> Bool) -> Traversal' a a
- both :: Traversal (a, a) (b, b) a b
- traversed :: Traversable f => Traversal (f a) (f b) a b
- each :: Each s t a b => Traversal s t a b
- ix :: Ixed m => Index m -> Traversal' m (IxValue m)
- _head :: Cons s s a a => Traversal' s a
- _tail :: Cons s s a a => Traversal' s s
- _init :: Snoc s s a a => Traversal' s s
- _last :: Snoc s s a a => Traversal' s a
- _Left :: Traversal (Either a b) (Either a' b) a a'
- _Right :: Traversal (Either a b) (Either a b') b b'
- _Just :: Traversal (Maybe a) (Maybe a') a a'
- _Nothing :: Traversal' (Maybe a) ()
- type LensLike f s t a b = (a -> f b) -> s -> f t
- type LensLike' f s a = LensLike f s s a a

# Documentation

(&) :: a -> (a -> b) -> b infixl 1

This operator is useful when you want to modify something several times. For instance, if you want to change 1st and 3rd elements of a tuple, you can write this:

(1,2,3)`&`

`_1`

`.~`

0`&`

`_3`

`%~`

`negate`

instead of e.g. this:

(`_1`

`.~`

0)`.`

(`_3`

`%~`

`negate`

)`$`

(1,2,3)

or this:

`set`

`_1`

0`.`

`over`

`_3`

`negate`

`$`

(1,2,3)

# Setter: modifies something in a structure

A setter is, broadly speaking, something that lets you modify a part of some value. Most likely you already know some setters:

`first`

:: (a -> b) -> (a, x) -> (b, x)(modifies 1st element of a pair; corresponds to

`_1`

)`left`

:: (a -> b) ->`Either`

a x ->`Either`

b x`map`

:: (a -> b) -> [a] -> [b](modifies every element in a list; corresponds to

`mapped`

)

As you see, a setter takes a function, a value, and applies the function to some part (or several parts) of the value. Moreover, setters can be pretty specific – for instance, a function that modifies the 3rd element of a list is a setter too:

-- Modify 3rd element in a list, if present. modify3rd :: (a -> a) -> [a] -> [a] modify3rd f (a:b:c:xs) = a : b : f c : xs modify3rd _ xs = xs

A nice thing about setters is that they compose easily – you can write

and it would be a function that takes a list of `map`

`.`

`left`

`Either`

s and modifies all of them that are `Left`

s.

This library provides its own type for setters – `ASetter`

; it's needed so that some functions in this library (like `_1`

) would be usable both as setters and as getters. You can turn an ordinary function like `map`

to a “lensy” setter with `sets`

.

To apply a setter to a value, use (`%~`

) or `over`

:

`>>>`

[2,3,4]`[1,2,3] & mapped %~ succ`

`>>>`

"Jane"`over _head toUpper "jane"`

To modify a value deeper inside the structure, use (`.`

):

`>>>`

["abc","deF","ghi"]`["abc","def","ghi"] & ix 1 . ix 2 %~ toUpper`

To set a value instead of modifying it, use `set`

or (`.~`

):

`>>>`

"xxx"`"abc" & mapped .~ 'x'`

`>>>`

('a','X','c')`set _2 'X' ('a','b','c')`

It's also possible to get both the old and the new value back – see (`<%~`

) and (`<<%~`

).

type ASetter s t a b = (a -> Identity b) -> s -> Identity t Source

`ASetter s t a b`

is something that turns a function modifying a value into a function modifying a *structure*. If you ignore `Identity`

(as `Identity a`

is the same thing as `a`

), the type is:

type ASetter s t a b = (a -> b) -> s -> t

The reason `Identity`

is used here is for `ASetter`

to be composable with other types, such as `Lens`

.

Technically, if you're writing a library, you shouldn't use this type for setters you are exporting from your library; the right type to use is `Setter`

, but it is not provided by this package (because then it'd have to depend on distributive). It's completely alright, however, to export functions which take an `ASetter`

as an argument.

(%~) :: ASetter s t a b -> (a -> b) -> s -> t infixr 4 Source

(`%~`

) applies a function to the target; an alternative explanation is that it is an inverse of `sets`

, which turns a setter into an ordinary function.

is the same thing as `mapped`

`%~`

`reverse`

.`fmap`

`reverse`

See `over`

if you want a non-operator synonym.

Negating the 1st element of a pair:

`>>>`

(-1,2)`(1,2) & _1 %~ negate`

Turning all `Left`

s in a list to upper case:

`>>>`

[Left "FOO",Right "bar"]`(mapped._Left.mapped %~ toUpper) [Left "foo", Right "bar"]`

over :: ASetter s t a b -> (a -> b) -> s -> t Source

Getting `fmap`

in a roundabout way:

`over`

`mapped`

::`Functor`

f => (a -> b) -> f a -> f b`over`

`mapped`

=`fmap`

Applying a function to both components of a pair:

`over`

`both`

:: (a -> b) -> (a, a) -> (b, b)`over`

`both`

= \f t -> (f (fst t), f (snd t))

Using

as a replacement for `over`

`_2`

`second`

:

`>>>`

(10,"20")`over _2 show (10,20)`

(<%~) :: LensLike ((,) b) s t a b -> (a -> b) -> s -> (b, t) Source

This is a version of (`%~`

) which modifies the structure and returns it along with the new value:

`>>>`

(-1, (-1, 2))`(1, 2) & _1 <%~ negate`

Simpler type signatures:

(`<%~`

) ::`Lens`

s t a b -> (a -> b) -> s -> (b, t) (`<%~`

) ::`Monoid`

b =>`Traversal`

s t a b -> (a -> b) -> s -> (b, t)

Since it does getting in addition to setting, you can't use it with `ASetter`

(but you can use it with lens and traversals).

mapped :: Functor f => ASetter (f a) (f b) a b Source

`mapped`

is a setter for everything contained in a functor. You can use it to map over lists, `Maybe`

, or even `IO`

(which is something you can't do with `traversed`

or `each`

).

Here `mapped`

is used to turn a value to all non-`Nothing`

values in a list:

`>>>`

[Just 0,Nothing,Just 0]`[Just 3,Nothing,Just 5] & mapped.mapped .~ 0`

Keep in mind that while `mapped`

is a more powerful setter than `each`

, it can't be used as a getter! This won't work (and will fail with a type error):

[(1,2),(3,4),(5,6)]`^..`

`mapped`

.`both`

# Getter: extracts a value from a structure

A getter extracts something from a value; in fact, any function is a getter. However, same as with setters, this library uses a special type for getters so that functions like `_1`

would be usable both as a setter and a getter. An ordinary function can be turned into a getter with `to`

.

Using a getter is done with (`^.`

) or `view`

from Lens.Micro.Extras:

`>>>`

'x'`('x','y') ^. _1`

`>>>`

2`view (ix 2) [0..5]`

Getters can be composed with (`.`

):

`>>>`

4`[(1,2),(3,4),(5,6)] ^. ix 1 . _2`

A getter always returns exactly 1 element (getters that can return more than one element are called folds and are present in this library as well).

type SimpleGetter s a = forall r. Getting r s a Source

A `SimpleGetter s a`

extracts `a`

from `s`

; so, it's the same thing as `(s -> a)`

, but you can use it in lens chains because its type looks like this:

type SimpleGetter s a = forall r. (a -> Const r a) -> s -> Const r s

Since `Const r`

is a functor, `SimpleGetter`

has the same shape as other lens types and can be composed with them. To get `(s -> a)`

out of a `SimpleGetter`

, choose `r ~ a`

and feed `Const :: a -> Const a a`

to the getter:

-- the actual signature is more permissive: --`view`

::`Getting`

a s a -> s -> a`view`

::`SimpleGetter`

s a -> s -> a`view`

getter =`getConst`

. getter`Const`

The actual `Getter`

from lens is more general:

type Getter s a = forall f. (Contravariant f, Functor f) => (a -> f a) -> s -> f s

I'm not currently aware of any functions that take lens's `Getter`

but won't accept `SimpleGetter`

, but you should try to avoid exporting `SimpleGetter`

s anyway to minimise confusion. Alternatively, look at microlens-contra, which provides a fully lens-compatible `Getter`

.

Lens users: you can convert a `SimpleGetter`

to `Getter`

by applying `to . view`

to it.

type Getting r s a = (a -> Const r a) -> s -> Const r s Source

Functions that operate on getters and folds – such as (`^.`

), (`^..`

), (`^?`

) – use `Getter r s a`

(with different values of `r`

) to describe what kind of result they need. For instance, (`^.`

) needs the getter to be able to return a single value, and so it accepts a getter of type `Getting a s a`

. (`^..`

) wants the getter to gather values together, so it uses `Getting (Endo [a]) s a`

(it could've used `Getting [a] s a`

instead, but it's faster with `Endo`

). The choice of `r`

depends on what you want to do with elements you're extracting from `s`

.

(^.) :: s -> Getting a s a -> a infixl 8 Source

(`^.`

) applies a getter to a value; in other words, it gets a value out of a structure using a getter (which can be a lens, traversal, fold, etc.).

Getting 1st field of a tuple:

(`^.`

`_1`

) :: (a, b) -> a (`^.`

`_1`

) =`fst`

When (`^.`

) is used with a traversal, it combines all results using the `Monoid`

instance for the resulting type. For instance, for lists it would be simple concatenation:

`>>>`

"string"`("str","ing") ^. each`

The reason for this is that traversals use `Applicative`

, and the `Applicative`

instance for `Const`

uses monoid concatenation to combine “effects” of `Const`

.

A non-operator version of (`^.`

) is called `view`

, and it's a bit more general than (`^.`

) (it works in `MonadReader`

). If you need the general version, you can get it from microlens-mtl; otherwise there's `view`

available in Lens.Micro.Extras.

to :: (s -> a) -> SimpleGetter s a Source

`to`

creates a getter from any function:

a`^.`

`to`

f = f a

It's most useful in chains, because it lets you mix lenses and ordinary functions. Suppose you have a record which comes from some third-party library and doesn't have any lens accessors. You want to do something like this:

value ^. _1 . field . at 2

However, `field`

isn't a getter, and you have to do this instead:

field (value ^. _1) ^. at 2

but now `value`

is in the middle and it's hard to read the resulting code. A variant with `to`

is prettier and more readable:

value ^. _1 . to field . at 2

# Fold: extracts multiple elements

Folds are getters that can return more than one element (or no elements at all). Except for some rare cases, a fold is the same thing as `(s -> [a])`

; you can use `folding`

to turn any function of type `(s -> f a)`

(where `f`

is `Foldable`

) into a fold.

Folds can be applied to values by using operators like (`^..`

), (`^?`

), etc:

`>>>`

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

A nice thing about folds is that you can combine them with (`<>`

) to concatenate their outputs:

`>>>`

[2,1]`(1,2,3) ^.. (_2 <> _1)`

When you need to get all elements of the same type in a complicated structure, (`<>`

) can be more helpful than `each`

:

`>>>`

[1,2,3,4]`([1,2], 3, [Nothing, Just 4]) ^.. (_1.each <> _2 <> _3.each._Just)`

(Just like setters and getters before, folds can be composed with (`.`

).)

The (`<>`

) trick works nicely with (`^?`

), too. For instance, if you want to get the 9th element of the list, but would be fine with 5th too if the list is too short, you could combine `ix 9`

and `ix 5`

:

`>>>`

Just 9`[0..9] ^? (ix 9 <> ix 5)`

`>>>`

Just 5`[0..8] ^? (ix 9 <> ix 5)`

`>>>`

Nothing`[0..3] ^? (ix 9 <> ix 5)`

(Unfortunately, this trick won't help you with setting or modifying.)

type SimpleFold s a = forall r. Monoid r => Getting r s a Source

A `SimpleFold s a`

extracts several `a`

s from `s`

; so, it's pretty much the same thing as `(s -> [a])`

, but you can use it with lens operators.

The actual `Fold`

from lens is more general:

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

There are several functions in lens that accept lens's `Fold`

but won't accept `SimpleFold`

; I'm aware of
`takingWhile`

,
`droppingWhile`

,
`backwards`

,
`foldByOf`

,
`foldMapByOf`

.
For this reason, try not to export `SimpleFold`

s if at all possible. microlens-contra provides a fully lens-compatible `Fold`

.

Lens users: you can convert a `SimpleFold`

to `Fold`

by applying `folded . toListOf`

to it.

(^..) :: s -> Getting (Endo [a]) s a -> [a] infixl 8 Source

`s ^.. t`

returns the list of all values that `t`

gets from `s`

.

A `Maybe`

contains either 0 or 1 values:

`>>>`

[3]`Just 3 ^.. _Just`

Gathering all values in a list of tuples:

`>>>`

[1,2,3,4]`[(1,2),(3,4)] ^.. each.each`

(^?) :: s -> Getting (First a) s a -> Maybe a infixl 8 Source

`s ^? t`

returns the 1st element `t`

returns, or `Nothing`

if `t`

doesn't return anything. It's trivially implemented by passing the `First`

monoid to the getter.

Safe `head`

:

`>>>`

Nothing`[] ^? each`

`>>>`

Just 1`[1..3] ^? each`

`>>>`

Nothing`Left 1 ^? _Right`

`>>>`

Just 1`Right 1 ^? _Right`

A non-operator version of (`^?`

) is called `preview`

, and – like `view`

– it's a bit more general than (`^?`

) (it works in `MonadReader`

). If you need the general version, you can get it from microlens-mtl; otherwise there's `preview`

available in Lens.Micro.Extras.

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

`has`

checks whether a getter (any getter, including lenses, traversals, and folds) returns at least 1 value.

Checking whether a list is non-empty:

`>>>`

False`has each []`

You can also use it with e.g. `_Left`

(and other 0-or-1 traversals) as a replacement for `isNothing`

, `isJust`

and other `isConstructorName`

functions:

`>>>`

True`has _Left (Left 1)`

folded :: Foldable f => SimpleFold (f a) a Source

folding :: Foldable f => (s -> f a) -> SimpleFold s a Source

# Lens: a combined getter-and-setter

type Lens s t a b = forall f. Functor f => (a -> f b) -> s -> f t Source

Lenses in a nutshell: use (`^.`

) to get, (`.~`

) to set, (`%~`

) to modify. (`.`

) composes lenses (i.e. if a `B`

is a part of `A`

, and a `C`

is a part of in `B`

, then `b.c`

lets you operate on `C`

inside `A`

). You can create lenses with `lens`

, or you can write them by hand (see below).

`Lens s t a b`

is the lowest common denominator of a setter and a getter, something that has the power of both; it has a `Functor`

constraint, and since both `Const`

and `Identity`

are functors, it can be used whenever a getter or a setter is needed.

`a`

is the type of the value inside of structure`b`

is the type of the replaced value`s`

is the type of the whole structure`t`

is the type of the structure after replacing`a`

in it with`b`

A `Lens`

can only point at a single value inside a structure (unlike a `Traversal`

).

It is easy to write lenses manually. The generic template is:

```
somelens :: Lens s t a b
-- “f” is the “a -> f b” function, “s” is the structure.
somelens f s =
let
a = ... -- Extract the value from “s”.
rebuildWith b = ... -- Write a function which would
-- combine “s” and modified value
-- to produce new structure.
in
rebuildWith
````<$>`

f a -- Apply the structure-producing
-- function to the modified value.

Here's the `_1`

lens:

`_1`

::`Lens`

(a, x) (b, x) a b`_1`

f (a, x) = (\b -> (b, x))`<$>`

f a

Here's a more complicated lens, which extracts *several* values from a structure (in a tuple):

type Age = Int type City = String type Country = String data Person = Person Age City Country -- This lens lets you access all location-related information about a person. location ::`Lens'`

Person (City, Country) location f (Person age city country) = (\(city', country') -> Person age city' country')`<$>`

f (city, country)

You even can choose to use a lens to present *all* information contained in the structure (in a different way). Such lenses are called `Iso`

in lens's terminology. For instance (assuming you don't mind functions that can error out), here's a lens which lets you act on the string representation of a value:

string :: (Read a, Show a) =>`Lens'`

a String string f s = read`<$>`

f (show s)

Using it to reverse a number:

>>> 123`&`

string`%~`

reverse 321

type Lens' s a = Lens s s a a Source

This is a type alias for monomorphic lenses which don't change the type of the container (or of the value inside).

lens :: (s -> a) -> (s -> b -> t) -> Lens s t a b Source

`lens`

creates a `Lens`

from a getter and a setter. The resulting lens isn't the most effective one (because of having to traverse the structure twice when modifying), but it shouldn't matter much.

A (partial) lens for list indexing:

ix :: Int ->`Lens'`

[a] a ix i =`lens`

(`!!`

i) -- getter (\s b -> take i s ++ b : drop (i+1) s) -- setter

Usage:

>>> [1..9]`^.`

ix 3 4 >>> [1..9] & ix 3`%~`

negate [1,2,3,-4,5,6,7,8,9]

When getting, the setter is completely unused; when setting, the getter is unused. Both are used only when the value is being modified. For instance, here we define a lens for the 1st element of a list, but instead of a legitimate getter we use `undefined`

. Then we use the resulting lens for *setting* and it works, which proves that the getter wasn't used:

`>>>`

[10,2,3]`[1,2,3] & lens undefined (\s b -> b : tail s) .~ 10`

at :: At m => Index m -> Lens' m (Maybe (IxValue m)) Source

This lens lets you read, write, or delete elements in `Map`

-like structures. It returns `Nothing`

when the value isn't found, just like `lookup`

:

`Data.Map.lookup k m = m ``^.`

at k

However, it also lets you insert and delete values by setting the value to

or `Just`

value`Nothing`

:

Data.Map.insert k a m = m`&`

at k`.~`

Just a Data.Map.delete k m = m`&`

at k`.~`

Nothing

Or you could use (`?~`

) instead of (`.~`

):

Data.Map.insert k a m = m`&`

at k`?~`

a

Note that `at`

doesn't work for arrays or lists. You can't delete an arbitrary element from an array (what would be left in its place?), and you can't set an arbitrary element in a list because if the index is out of list's bounds, you'd have to somehow fill the stretch between the last element and the element you just inserted (i.e. `[1,2,3] & at 10 .~ 5`

is undefined). If you want to modify an already existing value in an array or list, you should use `ix`

instead.

`at`

is often used with `non`

. See the documentation of `non`

for examples.

Note that `at`

isn't strict for `Map`

, even if you're using `Data.Map.Strict`

:

`>>>`

1`Data.Map.Strict.size (Data.Map.Strict.empty & at 1 .~ Just undefined)`

The reason for such behavior is that there's actually no “strict `Map`

” type; `Data.Map.Strict`

just provides some strict functions for ordinary `Map`

s.

This package doesn't actually provide any instances for `at`

, but there are instances for `Map`

and `IntMap`

in microlens-ghc and an instance for `HashMap`

in microlens-platform.

_1 :: Field1 s t a b => Lens s t a b Source

Gives access to the 1st field of a tuple (up to 5-tuples).

Getting the 1st component:

`>>>`

1`(1,2,3,4,5) ^. _1`

Setting the 1st component:

`>>>`

(10,2,3)`(1,2,3) & _1 .~ 10`

Note that this lens is lazy, and can set fields even of `undefined`

:

`>>>`

(10,*** Exception: Prelude.undefined`set _1 10 undefined :: (Int, Int)`

This is done to avoid violating a lens law stating that you can get back what you put:

`>>>`

10`view _1 . set _1 10 $ (undefined :: (Int, Int))`

The implementation (for 2-tuples) is:

`_1`

f t = (,)`<$>`

f (`fst`

t)`<*>`

`pure`

(`snd`

t)

or, alternatively,

`_1`

f ~(a,b) = (\a' -> (a',b))`<$>`

f a

(where `~`

means a lazy pattern).

# Iso: a lens that only changes the representation

Isos (or isomorphisms) are lenses that convert a value instead of targeting a part of it; in other words, inside of every list lives a reversed list, inside of every strict `Text`

lives a lazy `Text`

, and inside of every `(a, b)`

lives a `(b, a)`

. Since an isomorphism doesn't lose any information, it's possible to *reverse* it and use it in the opposite direction by using `from`

from the lens library:

from :: Iso' s a -> Iso' a s

However, it's not possible for microlens to export isomorphisms, because their type depends on `Profunctor`

, which resides in the profunctors library, which is a somewhat huge dependency. So, all isomorphisms included here are lenses instead (and thus you can't use them in the opposite direction).

strict :: Strict lazy strict => Lens' lazy strict Source

`strict`

lets you convert between strict and lazy versions of a datatype:

`>>>`

`let someText = "hello" :: Lazy.Text`

`>>>`

"hello" :: Strict.Text`someText ^. strict`

It can also be useful if you have a function that works on a strict type but your type is lazy:

stripDiacritics :: Strict.Text -> Strict.Text stripDiacritics = ...

`>>>`

`let someText = "Paul Erdős" :: Lazy.Text`

`>>>`

"Paul Erdos" :: Lazy.Text`someText & strict %~ stripDiacritics`

`strict`

works on `ByteString`

and `StateT`

/`WriterT`

/`RWST`

if you use microlens-ghc, and additionally on `Text`

if you use microlens-platform.

non :: Eq a => a -> Lens' (Maybe a) a Source

`non`

lets you “relabel” a `Maybe`

by equating `Nothing`

to an arbitrary value (which you can choose):

`>>>`

1`Just 1 ^. non 0`

`>>>`

0`Nothing ^. non 0`

The most useful thing about `non`

is that relabeling also works in other direction. If you try to `set`

the “forbidden” value, it'll be turned to `Nothing`

:

`>>>`

Nothing`Just 1 & non 0 .~ 0`

Setting anything else works just fine:

`>>>`

Just 5`Just 1 & non 0 .~ 5`

Same happens if you try to modify a value:

`>>>`

Nothing`Just 1 & non 0 %~ subtract 1`

`>>>`

Just 2`Just 1 & non 0 .~ (+ 1)`

`non`

is often useful when combined with `at`

. For instance, if you have a map of songs and their playcounts, it makes sense not to store songs with 0 plays in the map; `non`

can act as a filter that wouldn't pass such entries.

Decrease playcount of a song to 0, and it'll be gone:

`>>>`

fromList [("Yesterday",3)]`fromList [("Soon",1),("Yesterday",3)] & at "Soon" . non 0 %~ subtract 1`

Try to add a song with 0 plays, and it won't be added:

`>>>`

fromList [("Yesterday",3)]`fromList [("Yesterday",3)] & at "Soon" . non 0 .~ 0`

But it will be added if you set any other number:

`>>>`

fromList [("Soon",1),("Yesterday",3)]`fromList [("Yesterday",3)] & at "Soon" . non 0 .~ 1`

`non`

is also useful when working with nested maps. Here a nested map is created when it's missing:

`>>>`

fromList [("Dez Mona",fromList [("Soon",1)])]`Map.empty & at "Dez Mona" . non Map.empty . at "Soon" .~ Just 1`

and here it is deleted when its last entry is deleted (notice that `non`

is used twice here):

`>>>`

fromList []`fromList [("Dez Mona",fromList [("Soon",1)])] & at "Dez Mona" . non Map.empty . at "Soon" . non 0 %~ subtract 1`

To understand the last example better, observe the flow of values in it:

- the map goes into
`at "Dez Mona"`

- the nested map (wrapped into
`Just`

) goes into`non Map.empty`

`Just`

is unwrapped and the nested map goes into`at "Soon"`

`Just 1`

is unwrapped by`non 0`

Then the final value – i.e. 1 – is modified by `subtract 1`

and the result (which is 0) starts flowing backwards:

`non 0`

sees the 0 and produces a`Nothing`

`at "Soon"`

sees`Nothing`

and deletes the corresponding value from the map- the resulting empty map is passed to
`non Map.empty`

, which sees that it's empty and thus produces`Nothing`

`at "Dez Mona"`

sees`Nothing`

and removes the key from the map

# Traversal: a lens iterating over several elements

type Traversal s t a b = forall f. Applicative f => (a -> f b) -> s -> f t Source

Traversals in a nutshell: they're like lenses but they can point at multiple values. Use (`^..`

) to get all values, (`^?`

) to get the 1st value, (`.~`

) to set values, (`%~`

) to modify them. (`.`

) composes traversals just as it composes lenses. (`^.`

) can be used with traversals as well, but don't confuse it with (`^..`

).

`Traversal s t a b`

is a generalisation of `Lens`

which allows many targets (possibly 0). It's achieved by changing the constraint to `Applicative`

instead of `Functor`

– indeed, the point of `Applicative`

is that you can combine effects, which is just what we need to have many targets.

Traversals don't differ from lenses when it comes to setting – you can use usual (`%~`

) and (`.~`

) to modify and set values. Getting is a bit different, because you have to decide what to do in the case of multiple values. In particular, you can use these combinators (as well as everything else in the “Folds” section):

- (
`^..`

) gets a list of values - (
`^?`

) gets the 1st value (or`Nothing`

if there are no values) - (
`^?!`

) gets the 1st value and throws an exception if there are no values

In addition, (`^.`

) works for traversals as well – it combines traversed values using the (`<>`

) operation (if the values are instances of `Monoid`

).

Traversing any value twice is a violation of traversal laws. You can, however, traverse values in any order.

Ultimately, traversals should follow 2 laws:

t pure ≡ pure fmap (t f) . t g ≡ getCompose . t (Compose . fmap f . g)

The 1st law states that you can't change the shape of the structure or do anything funny with elements (traverse elements which aren't in the structure, create new elements out of thin air, etc.). The 2nd law states that you should be able to fuse 2 identical traversals into one. For a more detailed explanation of the laws, see this blog post (if you prefer rambling blog posts), or The Essence Of The Iterator Pattern (if you prefer papers).

type Traversal' s a = Traversal s s a a Source

This is a type alias for monomorphic traversals which don't change the type of the container (or of the values inside).

singular :: Traversal s t a a -> Lens s t a a Source

`singular`

turns a traversal into a lens that behaves like a single-element traversal:

`>>>`

1`[1,2,3] ^. signular each`

`>>>`

[-1,2,3]`[1,2,3] & singular each %~ negate`

If there is nothing to return, it'll throw an error:

`>>>`

*** Exception: Lens.Micro.singular: empty traversal`[] ^. singular each`

However, it won't fail if you are merely setting the value:

`>>>`

`[] & singular each %~ negate`

failing :: Traversal s t a b -> Traversal s t a b -> Traversal s t a b infixl 5 Source

`failing`

lets you chain traversals together; if the 1st traversal fails, the 2nd traversal will be used.

`>>>`

([0,0],[3])`([1,2],[3]) & failing (_1.each) (_2.each) .~ 0`

`>>>`

([],[0])`([],[3]) & failing (_1.each) (_2.each) .~ 0`

Note that the resulting traversal won't be valid unless either both traversals don't touch each others' elements, or both traversals return exactly the same results. To see an example of how `failing`

can generate invalid traversals, see this Stackoverflow question.

filtered :: (a -> Bool) -> Traversal' a a Source

`filtered`

is a traversal that filters elements “passing” thru it:

`>>>`

[1,2,3,4]`(1,2,3,4) ^.. each`

`>>>`

[2,4]`(1,2,3,4) ^.. each . filtered even`

It also can be used to modify elements selectively:

`>>>`

(1,200,3,400)`(1,2,3,4) & each . filtered even %~ (*100)`

The implementation of `filtered`

is very simple. Consider this traversal, which always “traverses” just the value it's given:

`id :: ``Traversal'`

a a
id f s = f s

And this traversal, which traverses nothing (in other words, *doesn't* traverse the value it's given):

ignored ::`Traversal'`

a a ignored f s =`pure`

s

And now combine them into a traversal that conditionally traverses the value it's given, and you get `filtered`

:

filtered :: (a -> Bool) ->`Traversal'`

a a filtered p s = if p s then f s else`pure`

s

By the way, note that `filtered`

can generate illegal traversals – sometimes this can bite you. For instance, take `evens`

:

evens =`filtered`

`even`

If `evens`

was a legal traversal, you'd be able to fuse several applications of `evens`

like this:

`over`

evens f`.`

`over`

evens g =`over`

evens (f`.`

g)

Unfortunately, in case of `evens`

this isn't a correct optimisation:

- the left-side variant applies
`g`

to all even numbers, and then applies`f`

to all even numbers that are left after`f`

(because`f`

might've turned some even numbers into odd ones) - the right-side variant applies
`f`

and`g`

to all even numbers

Of course, when you are careful and know what you're doing, you won't try to make such an optimisation. However, if you export an illegal traversal created with `filtered`

and someone tries to use it, ne might mistakenly assume that it's legal, do the optimisation, and silently get an incorrect result.

If you are using `filtered`

with some another traversal that doesn't overlap with -whatever the predicate checks-, the resulting traversal will be legal. For instance, here the predicate looks at the 1st element of a tuple, but the resulting traversal only gives you access to the 2nd:

pairedWithEvens ::`Traversal`

[(Int, a)] [(Int, b)] a b pairedWithEvens =`each`

`.`

`filtered`

(`even`

`.`

`fst`

)`.`

`_2`

Since you can't do anything with the 1st components thru this traversal, the following holds for any `f`

and `g`

:

`over`

pairedWithEvens f`.`

`over`

pairedWithEvens g =`over`

pairedWithEvens (f`.`

g)

traversed :: Traversable f => Traversal (f a) (f b) a b Source

`traversed`

traverses any `Traversable`

container (list, vector, `Map`

, `Maybe`

, you name it):

`>>>`

[1]`Just 1 ^.. traversed`

`traversed`

is the same as `traverse`

, but can be faster thanks to magic rewrite rules.

each :: Each s t a b => Traversal s t a b Source

`each`

tries to be a universal `Traversal`

– it behaves like `traversed`

in most situations, but also adds support for e.g. tuples with same-typed values:

`>>>`

(2,3)`(1,2) & each %~ succ`

`>>>`

"xyz"`["x", "y", "z"] ^. each`

However, note that `each`

doesn't work on *every* instance of `Traversable`

. If you have a `Traversable`

which isn't supported by `each`

, you can use `traversed`

instead. Personally, I like using `each`

instead of `traversed`

whenever possible – it's shorter and more descriptive.

You can use `each`

with these things:

`each`

::`Traversal`

[a] [b] a b`each`

::`Traversal`

(`Maybe`

a) (`Maybe`

b) a b`each`

::`Traversal`

(a,a) (b,b) a b`each`

::`Traversal`

(a,a,a) (b,b,b) a b`each`

::`Traversal`

(a,a,a,a) (b,b,b,b) a b`each`

::`Traversal`

(a,a,a,a,a) (b,b,b,b,b) a b`each`

:: (`RealFloat`

a,`RealFloat`

b) =>`Traversal`

(`Complex`

a) (`Complex`

b) a b

You can also use `each`

with types from array, bytestring, and containers by using microlens-ghc, or additionally with types from vector, text, and unordered-containers by using microlens-platform.

ix :: Ixed m => Index m -> Traversal' m (IxValue m) Source

This traversal lets you access (and update) an arbitrary element in a list, array, `Map`

, etc. (If you want to insert or delete elements as well, look at `at`

.)

An example for lists:

`>>>`

[0,1,2,10,4,5]`[0..5] & ix 3 .~ 10`

You can use it for getting, too:

`>>>`

Just 3`[0..5] ^? ix 3`

Of course, the element may not be present (which means that you can use `ix`

as a safe variant of (`!!`

)):

`>>>`

Nothing`[0..5] ^? ix 10`

Another useful instance is the one for functions – it lets you modify their outputs for specific inputs. For instance, here's `maximum`

that returns 0 when the list is empty (instead of throwing an exception):

maximum0 =`maximum`

`&`

`ix`

[]`.~`

0

The following instances are provided in this package:

`ix`

::`Int`

->`Traversal'`

[a] a`ix`

:: (`Eq`

e) => e ->`Traversal'`

(e -> a) a

You can also use `ix`

with types from array, bytestring, and containers by using microlens-ghc, or additionally with types from vector, text, and unordered-containers by using microlens-platform.

_head :: Cons s s a a => Traversal' s a Source

`_head`

traverses the 1st element of something (usually a list, but can also be a `Seq`

, etc):

`>>>`

Just 1`[1..5] ^? _head`

It can be used to modify too, as in this example where the 1st letter of a sentence is capitalised:

`>>>`

"Mary had a little lamb."`"mary had a little lamb." & _head %~ toTitle`

The reason it's a traversal and not a lens is that there's nothing to traverse when the list is empty:

`>>>`

Nothing`[] ^? _head`

This package only lets you use `_head`

on lists, but if you use microlens-ghc you get instances for `ByteString`

and `Seq`

, and if you use microlens-platform you additionally get instances for `Text`

and `Vector`

.

_tail :: Cons s s a a => Traversal' s s Source

`_tail`

gives you access to the tail of a list (or `Seq`

, etc):

`>>>`

Just [2,3,4,5]`[1..5] ^? _tail`

You can modify the tail as well:

`>>>`

[4,3,2,1]`[4,1,2,3] & _tail %~ reverse`

Since lists are monoids, you can use `_tail`

with plain (`^.`

) (and then it'll return an empty list if you give it an empty list):

`>>>`

[2,3,4,5]`[1..5] ^. _tail`

`>>>`

[]`[] ^. _tail`

If you want to traverse each *element* of the tail, use `_tail`

with `each`

:

`>>>`

"I hate caps."`"I HATE CAPS." & _tail.each %~ toLower`

This package only lets you use `_tail`

on lists, but if you use microlens-ghc you get instances for `ByteString`

and `Seq`

, and if you use microlens-platform you additionally get instances for `Text`

and `Vector`

.

_init :: Snoc s s a a => Traversal' s s Source

_last :: Snoc s s a a => Traversal' s a Source

# Prism: a traversal iterating over at most 1 element

Prisms are traversals that always target 0 or 1 values. Moreover, it's possible to *reverse* a prism, using it to construct a structure instead of peeking into it. Here's an example from the lens library:

>>> over _Left (+1) (Left 2) Left 3 >>> _Left # 5 Left 5

However, it's not possible for microlens to export prisms, because their type depends on `Choice`

from profunctors. So, all prisms included here are traversals instead (and you can't reverse them).

_Left :: Traversal (Either a b) (Either a' b) a a' Source

`_Left`

targets the value contained in an `Either`

, provided it's a `Left`

.

Gathering all `Left`

s in a structure (like the `lefts`

function, but not necessarily just for lists):

`>>>`

[1,3]`[Left 1, Right 'c', Left 3] ^.. each._Just`

Checking whether an `Either`

is a `Left`

(like `isLeft`

):

`>>>`

True`has _Left (Left 1)`

`>>>`

False`has _Left (Right 1)`

Extracting a value (if you're sure it's a `Left`

):

`>>>`

1`Left 1 ^?! _Left`

Mapping over all `Left`

s:

`>>>`

[Left "FOO",Right "bar"]`(each._Left %~ map toUpper) [Left "foo", Right "bar"]`

Implementation:

`_Left`

f (Left a) =`Left`

`<$>`

f a`_Left`

_ (Right b) =`pure`

(`Right`

b)

_Just :: Traversal (Maybe a) (Maybe a') a a' Source

`_Just`

targets the value contained in a `Maybe`

, provided it's a `Just`

.

See documentation for `_Left`

(as these 2 are pretty similar). In particular, it can be used to write these:

- Unsafely extracting a value from a
`Just`

:

`fromJust`

= (`^?!`

`_Just`

)

- Checking whether a value is a
`Just`

:

`isJust`

=`has`

`_Just`

- Converting a
`Maybe`

to a list (empty or consisting of a single element):

`maybeToList`

= (`^..`

`_Just`

)

- Gathering all
`Just`

s in a list:

`catMaybes`

= (`^..`

`each`

`.`

`_Just`

)

_Nothing :: Traversal' (Maybe a) () Source

`_Nothing`

targets a `()`

if the `Maybe`

is a `Nothing`

, and doesn't target anything otherwise:

`>>>`

[]`Just 1 ^.. _Nothing`

`>>>`

[()]`Nothing ^.. _Nothing`

It's not particularly useful (unless you want to use

as a replacement for `has`

`_Nothing`

`isNothing`

), and provided mainly for consistency.

Implementation:

`_Nothing`

f Nothing =`const`

`Nothing`

`<$>`

f ()`_Nothing`

_ j =`pure`

j

# Other types

type LensLike f s t a b = (a -> f b) -> s -> f t Source

`LensLike`

is a type that is often used to make combinators as general as possible. For instance, take (`<<%~`

), which only requires the passed lens to be able to work with the `(,) a`

functor (lenses and traversals can do that). The fully expanded type is as follows:

`(``<<%~`

) :: ((a -> (a, b)) -> s -> (a, t)) -> (a -> b) -> s -> (a, t)

With `LensLike`

, the intent to use the `(,) a`

functor can be made a bit clearer:

`(``<<%~`

) :: LensLike ((,) a) s t a b -> (a -> b) -> s -> (a, t)