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

Language | Haskell2010 |

Bidirectional arrows. Taken directly from

- Artem Alimarine, et al.
*There and Back Again: Arrows for Invertible Programming*. Haskell '05. http://citeseer.ist.psu.edu/alimarine05there.html

## Synopsis

- class (Groupoid a, Category a) => BiArrow a where
- class (BiArrow a, Arrow a) => BiArrow' a
- biarr :: BiArrow a => (b <-> c) -> a b c
- involve :: BiArrow a => (b -> b) -> a b b
- (^^>>) :: BiArrow a => (b <-> c) -> a c d -> a b d
- (>>^^) :: BiArrow a => a b c -> (c <-> d) -> a b d
- (<<^^) :: BiArrow a => a c d -> (b <-> c) -> a b d
- (^^<<) :: BiArrow a => (c <-> d) -> a b c -> a b d
- type BiKleisli m = Bijection (Kleisli m)

# Documentation

class (Groupoid a, Category a) => BiArrow a where Source #

The bidirectional arrow class.

Instances should satisfy the following laws:

f1 <-> g2 >>> g1 <-> f2 = (f1 >>> g1) <-> (f2 >>> g2)

invert (invert f) = f

invert (f <-> g) = g <-> f

first (f <-> g) = f *** id <-> g *** id

first h >>> id *** f <-> id *** g = id *** f <-> id *** g >>> first h

first (first f) >>> assoc = assoc >>> first f

where `assoc = [`

`biCase`

|((x,y),z) <-> (x,(y,z))|]

(<->) :: (b -> c) -> (c -> b) -> a b c infix 2 Source #

Take two functions and lift them into a bidirectional arrow. The intention is that these functions are each other's inverse.

invert :: a b c -> a c b Source #

Inverse: reverse the direction of a bidirectional arrow.

## Instances

BiArrow Iso Source # | |

(Semigroupoid a, Arrow a) => BiArrow (Bijection a) Source # | |

(Semigroupoid a, Arrow a) => BiArrow (Iso a) Source # | |

(BiArrow a, Monad m) => BiArrow (MonadArrow a m) Source # | |

Defined in Control.Invertible.MonadArrow (<->) :: (b -> c) -> (c -> b) -> MonadArrow a m b c Source # invert :: MonadArrow a m b c -> MonadArrow a m c b Source # |

class (BiArrow a, Arrow a) => BiArrow' a Source #

Bidirectional arrows under `Arrow`

.

Although `BiArrow`

should not, strictly speaking, be a subclass of `Arrow`

(as it is often impossible to define `arr`

), this is done because (as the paper says) "conceptually bi-arrows form an extension of the arrow class. Moreover, it allows us to use bi-arrows as normal arrows." This class exists to register this confound.

## Instances

(Semigroupoid a, Arrow a) => BiArrow' (Bijection a) Source # | |

Defined in Control.Invertible.BiArrow | |

Monad m => BiArrow' (MonadArrow (<->) m) Source # | |

Defined in Control.Invertible.MonadArrow |

biarr :: BiArrow a => (b <-> c) -> a b c Source #

Lift a bidirectional function to an arbitrary arrow using `BiArrow`

.

involve :: BiArrow a => (b -> b) -> a b b Source #

Construct an involution (a biarrow where the function and inverse are the same).

(^^>>) :: BiArrow a => (b <-> c) -> a c d -> a b d infixr 1 Source #

Precomposition with a pure bijection.

(>>^^) :: BiArrow a => a b c -> (c <-> d) -> a b d infixr 1 Source #

Postcomposition with a pure bijection.

(<<^^) :: BiArrow a => a c d -> (b <-> c) -> a b d infixr 1 Source #

Precomposition with a pure bijection (right-to-left variant).

(^^<<) :: BiArrow a => (c <-> d) -> a b c -> a b d infixr 1 Source #

Postcomposition with a pure bijection (right-to-left variant).