class Bifunctor p

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

Methods:

• bimap : (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

  Examples:

  >>> bimap not (+1) (True, 3)
  (False,4)
  
  >>> bimap not (+1) (Left True)
  Left False
  
  >>> bimap not (+1) (Right 3)
  Right 4

• first : (a -> b) -> p a c -> p b c Map covariantly over the first argument.

  first f ≡ bimap f identity

  Examples:

  >>> first not (True, 3)
  (False,3)
  
  >>> first not (Left True : Either Bool Int)
  Left False

• second : (b -> c) -> p a b -> p a c Map covariantly over the second argument.

  second ≡ bimap identity

  Examples:

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

Instances:

• 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)

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