Safe Haskell	Trustworthy
Language	Haskell2010

Control.Invertible.BiArrow

Contents

Orphan instances

Description

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

Documentation

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

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))|]

Minimal complete definition

(<->)

Methods

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

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 #

Inverse: reverse the direction of a bidirectional arrow.

Instances

BiArrow Iso #
Methods (<->) :: (b -> c) -> (c -> b) -> Iso b c # invert :: Iso b c -> Iso c b #
(Semigroupoid * a, Arrow a) => BiArrow (Bijection a) #
Methods (<->) :: (b -> c) -> (c -> b) -> Bijection a b c # invert :: Bijection a b c -> Bijection a c b #
(Semigroupoid * a, Arrow a) => BiArrow (Bijection a) #
Methods (<->) :: (b -> c) -> (c -> b) -> Bijection a b c # invert :: Bijection a b c -> Bijection a c b #
(Semigroupoid * a, Arrow a) => BiArrow (Iso * a) #
Methods (<->) :: (b -> c) -> (c -> b) -> Iso * a b c # invert :: Iso * a b c -> Iso * a c b #
(BiArrow a, Monad m) => BiArrow (MonadArrow a m) #
Methods (<->) :: (b -> c) -> (c -> b) -> MonadArrow a m b c # invert :: MonadArrow a m b c -> MonadArrow a m c b #

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

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) #

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

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

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

Lift a bidirectional function to an arbitrary arrow using <->.

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

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 #

Precomposition with a pure bijection.

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

Postcomposition with a pure bijection.

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

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

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

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

type BiKleisli m = Bijection (Kleisli m) #

Bidirectional Kleisli monad arrow transformer.

Orphan instances

Groupoid * Iso #	Poor orphans. Please will someone adopt us?
Methods inv :: k1 a b -> k1 b a #
Semigroupoid * Iso #	Poor orphans. Please will someone adopt us?
Methods o :: c j k1 -> c i j -> c i k1 #
Semigroupoid * a => Groupoid * (Bijection a) #	Poor orphans. Please will someone adopt us?
Methods inv :: k1 a b -> k1 b a #
Semigroupoid * a => Semigroupoid * (Bijection a) #	Poor orphans. Please will someone adopt us?
Methods o :: c j k1 -> c i j -> c i k1 #