Control.Invertible.BiArrow

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

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

(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

BiArrow a => BiArrow (StreamArrow a)

Methods

(<->) :: (b -> c) -> (c -> b) -> StreamArrow a b c

invert :: StreamArrow a b c -> StreamArrow 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

(Arrow a, BiArrow a) => BiArrow (StateArrow s a)

Methods

(<->) :: (b -> c) -> (c -> b) -> StateArrow s a b c

invert :: StateArrow s a b c -> StateArrow s a c b

BiArrow a => BiArrow (CoStateArrow s a)

Methods

(<->) :: (b -> c) -> (c -> b) -> CoStateArrow s a b c

invert :: CoStateArrow s a b c -> CoStateArrow s 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)

BiArrow' a => BiArrow' (StreamArrow a)

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

BiArrow' a => BiArrow' (StateArrow s a)

BiArrow' a => BiArrow' (CoStateArrow s 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).

Orphan instances

Semigroupoid * a => Groupoid * (Bijection a)

Poor orphans. Please will someone adopt us?

Methods

inv :: k1 a b -> k1 b a

Groupoid * a => Groupoid * (StreamArrow 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

Semigroupoid * a => Semigroupoid * (StreamArrow a)

Poor orphans. Please will someone adopt us?

Methods

o :: c j k1 -> c i j -> c i k1

Groupoid * a => Groupoid * (StateArrow s a)

Poor orphans. Please will someone adopt us?

Methods

inv :: k1 a b -> k1 b a

Groupoid * a => Groupoid * (CoStateArrow s a)

Poor orphans. Please will someone adopt us?

Methods

inv :: k1 a b -> k1 b a

Semigroupoid * a => Semigroupoid * (StateArrow s a)

Poor orphans. Please will someone adopt us?

Methods

o :: c j k1 -> c i j -> c i k1

Semigroupoid * a => Semigroupoid * (CoStateArrow s a)

Poor orphans. Please will someone adopt us?

Methods

o :: c j k1 -> c i j -> c i k1