Safe Haskell | Safe |
---|---|

Language | Haskell2010 |

- Setting (applying a function to values)
- Getting (retrieving a value)
- Folds (getters which return multiple elements)
- Lenses (things which are both setters and getters)
- Traversals (lenses which iterate over several elements)
- Prisms (traversals which iterate over at most 1 element)
- Each (an universal traversal for various structures)
- Tuples

- (&) :: 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
- mapped :: Functor f => ASetter (f a) (f b) a b
- type Getting r s a = (a -> Const r a) -> s -> Const r s
- (^.) :: s -> Getting a s a -> 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
- folded :: (Foldable f, Applicative (Const r)) => Getting r (f a) a
- has :: Getting Any s a -> s -> Bool
- 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
- 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
- both :: Traversal (a, a) (b, b) a b
- _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) ()
- class Each s t a b | s -> a, t -> b, s b -> t, t a -> s where
- class Field1 s t a b | s -> a, t -> b, s b -> t, t a -> s where
- class Field2 s t a b | s -> a, t -> b, s b -> t, t a -> s where
- class Field3 s t a b | s -> a, t -> b, s b -> t, t a -> s where
- class Field4 s t a b | s -> a, t -> b, s b -> t, t a -> s where
- class Field5 s t a b | s -> a, t -> b, s b -> t, t a -> s where

# 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)

# Setting (applying a function to values)

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

This means that examples of setters you might've already seen are:

`map`

:: (a -> b) -> [a] -> [b](which corresponds to

`mapped`

)`fmap`

::`Functor`

f => (a -> b) -> f a -> f b(which corresponds to

`mapped`

as well)`first`

:: (a -> b) -> (a, x) -> (b, x)(which corresponds to

`_1`

)`left`

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

a x ->`Either`

b x(which corresponds to

`_Left`

)

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 we'd have to depend on distributive). It's completely alright, however, to export functions which take an `ASetter`

as an argument.

type ASetter' s a = ASetter s s a a Source

This is a type alias for monomorphic setters which don't change the type of the container (or of the value inside). It's useful more often than the same type in lens, because we can't provide real setters and so it does the job of both `ASetter'`

and `Setter'`

.

(%~) :: 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)`

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 `traverse`

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`

# Getting (retrieving a value)

Getters are a not-entirely-obvious way to use lenses to *carry out* information from a structure (instead of changing something inside the structure). Any lens or traversal is a getter.

For details, see the documentation for `Getting`

.

Including `Getter`

is impossible, as then this package would have to depend on contravariant and it's a big dependency.

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

If you take a lens or a traversal and choose

as your functor, you will get `Const`

r`Getting r s a`

. This can be used to get something out of the structure instead of modifying it:

s`^.`

l =`getConst`

(l`Const`

s)

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

), (`^..`

), (`^?`

) – use `Getter r s a`

(with different values of `r`

) to describe what kind of getter 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`

.

# Folds (getters which return multiple elements)

(^..) :: 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`

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)`

# Lenses (things which are both setters and getters)

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`

# Traversals (lenses which iterate 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).

# Prisms (traversals which iterate over at most 1 element)

Prisms are traversals which 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`

, which resides in the profunctors library, which is a somewhat huge dependency. So, all prisms included here are traversals instead.

_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):

`toListOf`

(`each`

.`_Left`

) :: [`Either`

a b] -> [a]`toListOf`

(`each`

.`_Left`

) =`lefts`

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)

_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

# Each (an universal traversal for various structures)

class Each s t a b | s -> a, t -> b, s b -> t, t a -> s where Source

A class to support `each`

. If you're writing a library, don't write instances of this class which would be exported – other users won't be able to use them if they use lens.

Nothing

each :: Traversal s t a b Source

`each`

tries to be a universal `Traversal`

– it behaves like `traverse`

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 `traverse`

instead. Personally, I like using `each`

instead of `traverse`

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

Each [a] [b] a b Source | |

Each (Complex a) (Complex b) a b Source | |

Each (Maybe a) (Maybe b) a b Source | |

((~) * a b, (~) * q r) => Each (a, b) (q, r) a q Source | |

((~) * a b, (~) * a c, (~) * q r, (~) * q s) => Each (a, b, c) (q, r, s) a q Source | |

((~) * a b, (~) * a c, (~) * a d, (~) * q r, (~) * q s, (~) * q t) => Each (a, b, c, d) (q, r, s, t) a q Source | |

((~) * a b, (~) * a c, (~) * a d, (~) * a e, (~) * q r, (~) * q s, (~) * q t, (~) * q u) => Each (a, b, c, d, e) (q, r, s, t, u) a q Source |

# Tuples

class Field1 s t a b | s -> a, t -> b, s b -> t, t a -> s where Source

Nothing

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).

class Field2 s t a b | s -> a, t -> b, s b -> t, t a -> s where Source

Nothing

Gives access to the 2nd field of a tuple (up to 5-tuples).

See documentation for `_1`

.

class Field3 s t a b | s -> a, t -> b, s b -> t, t a -> s where Source

Nothing

Gives access to the 3rd field of a tuple (up to 5-tuples).

See documentation for `_1`

.