Copyright | (C) 2012-15 Edward Kmett |
---|---|

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

Maintainer | Edward Kmett <ekmett@gmail.com> |

Stability | provisional |

Portability | Rank2Types |

Safe Haskell | None |

Language | Haskell98 |

- type Iso s t a b = forall p f. (Profunctor p, Functor f) => p a (f b) -> p s (f t)
- type Iso' s a = Iso s s a a
- type AnIso s t a b = Exchange a b a (Identity b) -> Exchange a b s (Identity t)
- type AnIso' s a = AnIso s s a a
- iso :: (s -> a) -> (b -> t) -> Iso s t a b
- from :: AnIso s t a b -> Iso b a t s
- cloneIso :: AnIso s t a b -> Iso s t a b
- withIso :: AnIso s t a b -> ((s -> a) -> (b -> t) -> r) -> r
- au :: Functor f => AnIso s t a b -> ((b -> t) -> f s) -> f a
- auf :: (Functor f, Functor g) => Optic (Costar f) g s t a b -> (f a -> g b) -> f s -> g t
- under :: AnIso s t a b -> (t -> s) -> b -> a
- mapping :: (Functor f, Functor g) => AnIso s t a b -> Iso (f s) (g t) (f a) (g b)
- simple :: Equality' a a
- non :: Eq a => a -> Iso' (Maybe a) a
- non' :: APrism' a () -> Iso' (Maybe a) a
- anon :: a -> (a -> Bool) -> Iso' (Maybe a) a
- enum :: Enum a => Iso' Int a
- curried :: Iso ((a, b) -> c) ((d, e) -> f) (a -> b -> c) (d -> e -> f)
- uncurried :: Iso (a -> b -> c) (d -> e -> f) ((a, b) -> c) ((d, e) -> f)
- flipped :: Iso (a -> b -> c) (a' -> b' -> c') (b -> a -> c) (b' -> a' -> c')
- class Bifunctor p => Swapped p where
- class Strict lazy strict | lazy -> strict, strict -> lazy where
- lazy :: Strict lazy strict => Iso' strict lazy
- class Reversing t where
- reversing :: t -> t

- reversed :: Reversing a => Iso' a a
- involuted :: (a -> a) -> Iso' a a
- magma :: LensLike (Mafic a b) s t a b -> Iso s u (Magma Int t b a) (Magma j u c c)
- imagma :: Over (Indexed i) (Molten i a b) s t a b -> Iso s t' (Magma i t b a) (Magma j t' c c)
- data Magma i t b a
- contramapping :: Contravariant f => AnIso s t a b -> Iso (f a) (f b) (f s) (f t)
- class Profunctor p where
- dimapping :: (Profunctor p, Profunctor q) => AnIso s t a b -> AnIso s' t' a' b' -> Iso (p a s') (q b t') (p s a') (q t b')
- lmapping :: (Profunctor p, Profunctor q) => AnIso s t a b -> Iso (p a x) (q b y) (p s x) (q t y)
- rmapping :: (Profunctor p, Profunctor q) => AnIso s t a b -> Iso (p x s) (q y t) (p x a) (q y b)
- bimapping :: (Bifunctor f, Bifunctor g) => AnIso s t a b -> AnIso s' t' a' b' -> Iso (f s s') (g t t') (f a a') (g b b')

# Isomorphism Lenses

type Iso s t a b = forall p f. (Profunctor p, Functor f) => p a (f b) -> p s (f t) Source

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

When you see this as an argument to a function, it expects an `Iso`

.

# Isomorphism Construction

# Consuming Isomorphisms

cloneIso :: AnIso s t a b -> Iso s t a b Source

Convert from `AnIso`

back to any `Iso`

.

This is useful when you need to store an isomorphism as a data type inside a container and later reconstitute it as an overloaded function.

See `cloneLens`

or `cloneTraversal`

for more information on why you might want to do this.

withIso :: AnIso s t a b -> ((s -> a) -> (b -> t) -> r) -> r Source

Extract the two functions, one from `s -> a`

and
one from `b -> t`

that characterize an `Iso`

.

# Working with isomorphisms

auf :: (Functor f, Functor g) => Optic (Costar f) g s t a b -> (f a -> g b) -> f s -> g t Source

Based on `ala'`

from Conor McBride's work on Epigram.

This version is generalized to accept any `Iso`

, not just a `newtype`

.

For a version you pass the name of the `newtype`

constructor to, see `alaf`

.

`>>>`

10`auf (_Unwrapping Sum) (foldMapOf both) Prelude.length ("hello","world")`

Mnemonically, the German *auf* plays a similar role to *à la*, and the combinator
is `au`

with an extra function argument:

`auf`

::`Iso`

s t a b -> ((r -> a) -> e -> b) -> (r -> s) -> e -> t

but the signature is general.

## Common Isomorphisms

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

If `v`

is an element of a type `a`

, and `a'`

is `a`

sans the element `v`

, then

is an isomorphism from
`non`

v

to `Maybe`

a'`a`

.

`non`

≡`non'`

`.`

`only`

Keep in mind this is only a real isomorphism if you treat the domain as being

.`Maybe`

(a sans v)

This is practically quite useful when you want to have a `Map`

where all the entries should have non-zero values.

`>>>`

fromList [("hello",3)]`Map.fromList [("hello",1)] & at "hello" . non 0 +~ 2`

`>>>`

fromList []`Map.fromList [("hello",1)] & at "hello" . non 0 -~ 1`

`>>>`

1`Map.fromList [("hello",1)] ^. at "hello" . non 0`

`>>>`

0`Map.fromList [] ^. at "hello" . non 0`

This combinator is also particularly useful when working with nested maps.

*e.g.* When you want to create the nested `Map`

when it is missing:

`>>>`

fromList [("hello",fromList [("world","!!!")])]`Map.empty & at "hello" . non Map.empty . at "world" ?~ "!!!"`

and when have deleting the last entry from the nested `Map`

mean that we
should delete its entry from the surrounding one:

`>>>`

fromList []`fromList [("hello",fromList [("world","!!!")])] & at "hello" . non Map.empty . at "world" .~ Nothing`

It can also be used in reverse to exclude a given value:

`>>>`

Just 2`non 0 # rem 10 4`

`>>>`

Nothing`non 0 # rem 10 5`

non' :: APrism' a () -> Iso' (Maybe a) a Source

generalizes `non'`

p

to take any unit `non`

(p # ())`Prism`

This function generates an isomorphism between

and `Maybe`

(a | `isn't`

p a)`a`

.

`>>>`

fromList [("hello",fromList [("world","!!!")])]`Map.singleton "hello" Map.empty & at "hello" . non' _Empty . at "world" ?~ "!!!"`

`>>>`

fromList []`fromList [("hello",fromList [("world","!!!")])] & at "hello" . non' _Empty . at "world" .~ Nothing`

anon :: a -> (a -> Bool) -> Iso' (Maybe a) a Source

generalizes `anon`

a p

to take any value and a predicate.`non`

a

This function assumes that `p a`

holds

and generates an isomorphism between `True`

and `Maybe`

(a | `not`

(p a))`a`

.

`>>>`

fromList [("hello",fromList [("world","!!!")])]`Map.empty & at "hello" . anon Map.empty Map.null . at "world" ?~ "!!!"`

`>>>`

fromList []`fromList [("hello",fromList [("world","!!!")])] & at "hello" . anon Map.empty Map.null . at "world" .~ Nothing`

enum :: Enum a => Iso' Int a Source

This isomorphism can be used to convert to or from an instance of `Enum`

.

`>>>`

0`LT^.from enum`

`>>>`

'a'`97^.enum :: Char`

Note: this is only an isomorphism from the numeric range actually used
and it is a bit of a pleasant fiction, since there are questionable
`Enum`

instances for `Double`

, and `Float`

that exist solely for
`[1.0 .. 4.0]`

sugar and the instances for those and `Integer`

don't
cover all values in their range.

flipped :: Iso (a -> b -> c) (a' -> b' -> c') (b -> a -> c) (b' -> a' -> c') Source

The isomorphism for flipping a function.

`>>>`

(2,1)`((,)^.flipped) 1 2`

class Strict lazy strict | lazy -> strict, strict -> lazy where Source

Ad hoc conversion between "strict" and "lazy" versions of a structure,
such as `Text`

or `ByteString`

.

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

An `Iso`

between the strict variant of a structure and its lazy
counterpart.

`lazy`

=`from`

`strict`

See http://hackage.haskell.org/package/strict-base-types for an example use.

class Reversing t where Source

This class provides a generalized notion of list reversal extended to other containers.

reversed :: Reversing a => Iso' a a Source

An `Iso`

between a list, `ByteString`

, `Text`

fragment, etc. and its reversal.

`>>>`

"evil"`"live" ^. reversed`

`>>>`

"lived"`"live" & reversed %~ ('d':)`

## Uncommon Isomorphisms

imagma :: Over (Indexed i) (Molten i a b) s t a b -> Iso s t' (Magma i t b a) (Magma j t' c c) Source

This isomorphism can be used to inspect an `IndexedTraversal`

to see how it associates
the structure and it can also be used to bake the `IndexedTraversal`

into a `Magma`

so
that you can traverse over it multiple times with access to the original indices.

This provides a way to peek at the internal structure of a
`Traversal`

or `IndexedTraversal`

TraversableWithIndex i (Magma i t b) | |

FoldableWithIndex i (Magma i t b) | |

FunctorWithIndex i (Magma i t b) | |

Functor (Magma i t b) | |

Foldable (Magma i t b) | |

Traversable (Magma i t b) | |

(Show i, Show a) => Show (Magma i t b a) |

## Contravariant functors

contramapping :: Contravariant f => AnIso s t a b -> Iso (f a) (f b) (f s) (f t) Source

Lift an `Iso`

into a `Contravariant`

functor.

contramapping ::`Contravariant`

f =>`Iso`

s t a b ->`Iso`

(f a) (f b) (f s) (f t) contramapping ::`Contravariant`

f =>`Iso'`

s a ->`Iso'`

(f a) (f s)

# Profunctors

class Profunctor p where

Formally, the class `Profunctor`

represents a profunctor
from `Hask`

-> `Hask`

.

Intuitively it is a bifunctor where the first argument is contravariant and the second argument is covariant.

You can define a `Profunctor`

by either defining `dimap`

or by defining both
`lmap`

and `rmap`

.

If you supply `dimap`

, you should ensure that:

`dimap`

`id`

`id`

≡`id`

If you supply `lmap`

and `rmap`

, ensure:

`lmap`

`id`

≡`id`

`rmap`

`id`

≡`id`

If you supply both, you should also ensure:

`dimap`

f g ≡`lmap`

f`.`

`rmap`

g

These ensure by parametricity:

`dimap`

(f`.`

g) (h`.`

i) ≡`dimap`

g h`.`

`dimap`

f i`lmap`

(f`.`

g) ≡`lmap`

g`.`

`lmap`

f`rmap`

(f`.`

g) ≡`rmap`

f`.`

`rmap`

g

dimap :: (a -> b) -> (c -> d) -> p b c -> p a d

lmap :: (a -> b) -> p b c -> p a c

rmap :: (b -> c) -> p a b -> p a c

Profunctor (->) | |

Profunctor ReifiedFold | |

Profunctor ReifiedGetter | |

Monad m => Profunctor (Kleisli m) | |

Functor w => Profunctor (Cokleisli w) | |

Functor f => Profunctor (Star f) | |

Functor f => Profunctor (Costar f) | |

Arrow p => Profunctor (WrappedArrow p) | |

Profunctor (Forget r) | |

Profunctor (Tagged *) | |

Profunctor (Indexed i) | |

Profunctor (ReifiedIndexedFold i) | |

Profunctor (ReifiedIndexedGetter i) | |

Profunctor (Market a b) | |

Profunctor (Exchange a b) |

dimapping :: (Profunctor p, Profunctor q) => AnIso s t a b -> AnIso s' t' a' b' -> Iso (p a s') (q b t') (p s a') (q t b') Source

Lift two `Iso`

s into both arguments of a `Profunctor`

simultaneously.

dimapping ::`Profunctor`

p =>`Iso`

s t a b ->`Iso`

s' t' a' b' ->`Iso`

(p a s') (p b t') (p s a') (p t b') dimapping ::`Profunctor`

p =>`Iso'`

s a ->`Iso'`

s' a' ->`Iso'`

(p a s') (p s a')

lmapping :: (Profunctor p, Profunctor q) => AnIso s t a b -> Iso (p a x) (q b y) (p s x) (q t y) Source

Lift an `Iso`

contravariantly into the left argument of a `Profunctor`

.

lmapping ::`Profunctor`

p =>`Iso`

s t a b ->`Iso`

(p a x) (p b y) (p s x) (p t y) lmapping ::`Profunctor`

p =>`Iso'`

s a ->`Iso'`

(p a x) (p s x)

rmapping :: (Profunctor p, Profunctor q) => AnIso s t a b -> Iso (p x s) (q y t) (p x a) (q y b) Source

Lift an `Iso`

covariantly into the right argument of a `Profunctor`

.

rmapping ::`Profunctor`

p =>`Iso`

s t a b ->`Iso`

(p x s) (p y t) (p x a) (p y b) rmapping ::`Profunctor`

p =>`Iso'`

s a ->`Iso'`

(p x s) (p x a)