Data.Invertible.Bijection

The base representation for bidirectional arrows (bijections).

Documentation

data Bijection (a :: * -> * -> *) b c

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

Arrow a => Arrow (Bijection a)

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.

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')

ArrowZero a => ArrowZero (Bijection a)

Defined in Data.Invertible.Bijection

Methods

zeroArrow :: Bijection a b c

ArrowChoice a => ArrowChoice (Bijection a)

||| is Left-biased, and uses only the left argument's biFrom.

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

Invariant2 (Bijection ((->) :: Type -> Type -> Type))

Defined in Data.Invertible.Bijection

Methods

invmap2 :: (a -> c) -> (c -> a) -> (b -> d) -> (d -> b) -> Bijection (->) a b -> Bijection (->) c d

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

Defined in Control.Invertible.BiArrow

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

Defined in Control.Invertible.BiArrow

Methods

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

invert :: Bijection a b c -> Bijection a c b

Category a => Category (Bijection a :: Type -> Type -> Type)

Defined in Data.Invertible.Bijection

Methods

id :: Bijection a a0 a0

(.) :: Bijection a b c -> Bijection a a0 b -> Bijection a a0 c

Semigroupoid a => Groupoid (Bijection a :: Type -> Type -> Type)

Defined in Data.Invertible.Bijection

Methods

inv :: Bijection a a0 b -> Bijection a b a0

Semigroupoid a => Semigroupoid (Bijection a :: Type -> Type -> Type)

Defined in Data.Invertible.Bijection

Methods

o :: Bijection a j k1 -> Bijection a i j -> Bijection a i k1

Monad m => Arrow (MonadArrow (<->) m)

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')

MonadPlus m => ArrowZero (MonadArrow (<->) m)

Defined in Control.Invertible.MonadArrow

Methods

zeroArrow :: MonadArrow (<->) m b c

MonadPlus m => ArrowPlus (MonadArrow (<->) m)

Defined in Control.Invertible.MonadArrow

Methods

(<+>) :: MonadArrow (<->) m b c -> MonadArrow (<->) m b c -> MonadArrow (<->) m b c

Monad m => ArrowChoice (MonadArrow (<->) m)

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

Invariant (Bijection ((->) :: Type -> Type -> Type) b)

Defined in Data.Invertible.Bijection

Methods

invmap :: (a -> b0) -> (b0 -> a) -> Bijection (->) b a -> Bijection (->) b b0

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

Defined in Control.Invertible.MonadArrow

(Semigroupoid a, Arrow a) => Functor (Bijection a b)

Defined in Control.Invertible.Functor

Methods

fmap :: (a0 <-> b0) -> Bijection a b a0 -> Bijection a b b0

Monoidal (Bijection ((->) :: Type -> Type -> Type) ())

Defined in Control.Invertible.Monoidal

Methods

unit :: Bijection (->) () ()

(>*<) :: Bijection (->) () a -> Bijection (->) () b -> Bijection (->) () (a, b)

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

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