Safe Haskell	Trustworthy
Language	Haskell2010

Data.Invertible.Bijection

Description

The base representation for bidirectional arrows (bijections).

Synopsis

Documentation

data Bijection a b c Source #

A representation of a bidirectional arrow (embedding-projection pair of arrows transformer): an arrow and its inverse. Most uses will prefer the specialized <-> type for function arrows.

To constitute a valid bijection, biTo and biFrom should be inverses:

```
biTo . biFrom = id
```
```
biFrom . biTo = id
```

It may be argued that the arguments should be in the opposite order due to the arrow syntax, but it makes more sense to me to have the forward function come first.

Constructors

(:<->:) infix 2
Fields biTo :: a b c biFrom :: a c b

Instances

Category * a => Category * (Bijection a) Source #
Methods id :: cat a a # (.) :: cat b c -> cat a b -> cat a c #
Semigroupoid * a => Groupoid * (Bijection a) Source #
Methods inv :: k1 a b -> k1 b a #
Semigroupoid * a => Semigroupoid * (Bijection a) Source #
Methods o :: c j k1 -> c i j -> c i k1 #
Arrow a => Arrow (Bijection a) Source #	In order to use all the `Arrow` functions, we make a partially broken instance, where `arr` creates a bijection with a broken `biFrom`. See note on `BiArrow'`. `&&&` is first-biased, and uses only the left argument's `biFrom`.
Methods arr :: (b -> c) -> Bijection a b c # first :: Bijection a b c -> Bijection a (b, d) (c, d) # second :: Bijection a b c -> Bijection a (d, b) (d, c) # (***) :: Bijection a b c -> Bijection a b' c' -> Bijection a (b, b') (c, c') # (&&&) :: Bijection a b c -> Bijection a b c' -> Bijection a b (c, c') #
ArrowChoice a => ArrowChoice (Bijection a) Source #	`\|\|\|` is Left-biased, and uses only the left argument's `biFrom`.
Methods left :: Bijection a b c -> Bijection a (Either b d) (Either c d) # right :: Bijection a b c -> Bijection a (Either d b) (Either d c) # (+++) :: Bijection a b c -> Bijection a b' c' -> Bijection a (Either b b') (Either c c') # (\|\|\|) :: Bijection a b d -> Bijection a c d -> Bijection a (Either b c) d #
ArrowZero a => ArrowZero (Bijection a) Source #
Methods zeroArrow :: Bijection a b c #
Invariant2 (Bijection (->)) Source #
Methods invmap2 :: (a -> c) -> (c -> a) -> (b -> d) -> (d -> b) -> Bijection (->) a b -> Bijection (->) c d #
(Semigroupoid * a, Arrow a) => BiArrow' (Bijection a) Source #

(Semigroupoid * a, Arrow a) => BiArrow (Bijection a) Source #
Methods (<->) :: (b -> c) -> (c -> b) -> Bijection a b c Source # invert :: Bijection a b c -> Bijection a c b Source #
Monad m => Arrow (MonadArrow (<->) m) #
Methods arr :: (b -> c) -> MonadArrow (<->) m b c # first :: MonadArrow (<->) m b c -> MonadArrow (<->) m (b, d) (c, d) # second :: MonadArrow (<->) m b c -> MonadArrow (<->) m (d, b) (d, c) # (***) :: MonadArrow (<->) m b c -> MonadArrow (<->) m b' c' -> MonadArrow (<->) m (b, b') (c, c') # (&&&) :: MonadArrow (<->) m b c -> MonadArrow (<->) m b c' -> MonadArrow (<->) m b (c, c') #
Monad m => ArrowChoice (MonadArrow (<->) m) #
Methods left :: MonadArrow (<->) m b c -> MonadArrow (<->) m (Either b d) (Either c d) # right :: MonadArrow (<->) m b c -> MonadArrow (<->) m (Either d b) (Either d c) # (+++) :: MonadArrow (<->) m b c -> MonadArrow (<->) m b' c' -> MonadArrow (<->) m (Either b b') (Either c c') # (\|\|\|) :: MonadArrow (<->) m b d -> MonadArrow (<->) m c d -> MonadArrow (<->) m (Either b c) d #
MonadPlus m => ArrowZero (MonadArrow (<->) m) #
Methods zeroArrow :: MonadArrow (<->) m b c #
MonadPlus m => ArrowPlus (MonadArrow (<->) m) #
Methods (<+>) :: MonadArrow (<->) m b c -> MonadArrow (<->) m b c -> MonadArrow (<->) m b c #
Invariant (Bijection (->) b) Source #
Methods invmap :: (a -> b) -> (b -> a) -> Bijection (->) b a -> Bijection (->) b b #
Monad m => BiArrow' (MonadArrow (<->) m) Source #

(Semigroupoid * a, Arrow a) => Functor (Bijection a b) Source #
Methods fmap :: (a <-> b) -> Bijection a b a -> Bijection a b b Source #
Monoidal (Bijection (->) ()) Source #
Methods unit :: Bijection (->) () () Source # (>*<) :: Bijection (->) () a -> Bijection (->) () b -> Bijection (->) () (a, b) Source #

type (<->) = Bijection (->) infix 2 Source #

Specialization of Bijection to function arrows. Represents both a function, f, and its (presumed) inverse, g, represented as f :<->: g.