Safe Haskell	Trustworthy
Language	Haskell2010

Data.Invertible.Bijection

Description

The base representation for bidirectional arrows (bijections).

Synopsis

data Bijection (a :: * -> * -> *) b c = (:<->:) {
- biTo :: a b c
- biFrom :: a c b
}
type (<->) = Bijection (->)

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

Instances details

Category a => Category (Bijection a :: Type -> Type -> Type) Source #
Instance details Defined in Data.Invertible.Bijection Methods id :: forall (a0 :: k). Bijection a a0 a0 # (.) :: forall (b :: k) (c :: k) (a0 :: k). Bijection a b c -> Bijection a a0 b -> Bijection a a0 c #
Semigroupoid a => Groupoid (Bijection a :: Type -> Type -> Type) Source #
Instance details Defined in Data.Invertible.Bijection Methods inv :: forall (a0 :: k) (b :: k). Bijection a a0 b -> Bijection a b a0 Source #
Semigroupoid a => Semigroupoid (Bijection a :: Type -> Type -> Type) Source #
Instance details Defined in Data.Invertible.Bijection Methods o :: forall (j :: k) (k1 :: k) (i :: k). Bijection a j k1 -> Bijection a i j -> Bijection a i k1 Source #
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`.
Instance details Defined in Data.Invertible.Bijection 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`.
Instance details Defined in Data.Invertible.Bijection 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 #
Instance details Defined in Data.Invertible.Bijection Methods zeroArrow :: Bijection a b c #
Invariant2 (Bijection (->)) Source #
Instance details Defined in Data.Invertible.Bijection Methods invmap2 :: (a -> c) -> (c -> a) -> (b -> d) -> (d -> b) -> Bijection (->) a b -> Bijection (->) c d Source #
(Semigroupoid a, Arrow a) => BiArrow (Bijection a) Source #
Instance details Defined in Control.Invertible.BiArrow Methods (<->) :: (b -> c) -> (c -> b) -> Bijection a b c Source # invert :: Bijection a b c -> Bijection a c b Source #
(Semigroupoid a, Arrow a) => BiArrow' (Bijection a) Source #
Instance details Defined in Control.Invertible.BiArrow
Monad m => Arrow (MonadArrow (<->) m) Source #
Instance details Defined in Control.Invertible.MonadArrow 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) Source #
Instance details Defined in Control.Invertible.MonadArrow 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 => ArrowPlus (MonadArrow (<->) m) Source #
Instance details Defined in Control.Invertible.MonadArrow Methods (<+>) :: MonadArrow (<->) m b c -> MonadArrow (<->) m b c -> MonadArrow (<->) m b c #
MonadPlus m => ArrowZero (MonadArrow (<->) m) Source #
Instance details Defined in Control.Invertible.MonadArrow Methods zeroArrow :: MonadArrow (<->) m b c #
Invariant (Bijection (->) b) Source #
Instance details Defined in Data.Invertible.Bijection Methods invmap :: (a -> b0) -> (b0 -> a) -> Bijection (->) b a -> Bijection (->) b b0 Source #
Monad m => BiArrow' (MonadArrow (<->) m) Source #
Instance details Defined in Control.Invertible.MonadArrow
(Semigroupoid a, Arrow a) => Functor (Bijection a b) Source #
Instance details Defined in Control.Invertible.Functor Methods fmap :: (a0 <-> b0) -> Bijection a b a0 -> Bijection a b b0 Source #
Monoidal (Bijection (->) ()) Source #
Instance details Defined in Control.Invertible.Monoidal 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.

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