| Safe Haskell | Trustworthy |
|---|---|
| Language | Haskell2010 |
Control.Invertible.BiArrow
Contents
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
- 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))|]
Minimal complete definition
Methods
(<->) :: (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 # | |
| (BiArrow a, Monad m) => BiArrow (MonadArrow a m) Source # | |
Defined in Control.Invertible.MonadArrow Methods (<->) :: (b -> c) -> (c -> b) -> MonadArrow a m b c Source # invert :: MonadArrow a m b c -> MonadArrow a m c b Source # | |
| (Semigroupoid a, Arrow a) => BiArrow (Iso a) 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).