class Bifunctor p where

A bifunctor is a type constructor that takes two type arguments and is a functor in both arguments. That is, unlike with Functor, a type constructor such as Either does not need to be partially applied for a Bifunctor instance, and the methods in this class permit mapping functions over the Left value or the Right value, or both at the same time.

It is a bifunctor where both the first and second arguments are covariant.

You can define a Bifunctor by either defining bimap or by defining both first and second.

If you supply bimap, you should ensure that:

`bimap identity identity`  `identity`

If you supply first and second, ensure:

first identity  identity
second identity  identity

If you supply both, you should also ensure:

bimap f g  first f . second g

By parametricity, these will ensure that:

bimap  (f . g) (h . i)  bimap f h . bimap g i
first  (f . g)  first  f . first  g
second (f . g)  second f . second g

: (a -> b) -> (c -> d) -> p a c -> p b d

Map over both arguments at the same time.

bimap f g  first f . second g


>>> bimap not (+1) (True, 3)

>>> bimap not (+1) (Left True)
Left False

>>> bimap not (+1) (Right 3)
Right 4

: (a -> b) -> p a c -> p b c

Map covariantly over the first argument.

first f  bimap f identity


>>> first not (True, 3)

>>> first not (Left True : Either Bool Int)
Left False

: (b -> c) -> p a b -> p a c

Map covariantly over the second argument.

second  bimap identity


>>> second (+1) (True, 3)

>>> second (+1) (Right 3 : Either Bool Int)
Right 4

instance Bifunctor Either

instance Bifunctor ()

instance Bifunctor x1

instance Bifunctor (x1, x2)

instance Bifunctor (x1, x2, x3)

instance Bifunctor (x1, x2, x3, x4)

instance Bifunctor (x1, x2, x3, x4, x5)