Control.Applicative.Cancellative

This module contains definitions for Cancellative functors along with the relevant combinators.

Cancellative

class Alternative f => Cancellative f where

A group on Applicative functors.

Cancellative functors have the following laws in addition to those of Alternative:

Left Cancellation

cancel a <|> a = empty

Rigth Cancellation

a <|> cancel a = empty

This is analogous to a group operation on applicative functors, in the sense that Alternative forms a monoid. A straight- forward implementation exists whenever f a forms a Group for all a, in which case, cancel == invert.

Minimal complete definition

Nothing

Methods

cancel :: f a -> f a

Invert (or cancel) a Cancellative functor, such that, if the functor is also a GroupFoldable, then gold . cancel amounts to evaluating the inverse of a word in the functor.

Examples:

>>> let x = FreeGroup [Left (Sum (2 :: Word8)), Right (Sum 3)]
>>> cancel x
FreeGroup {runFreeGroup = [Left (Sum {getSum = 3}),Right (Sum {getSum = 2})]}

cancel :: Group (f a) => f a -> f a

Invert (or cancel) a Cancellative functor, such that, if the functor is also a GroupFoldable, then gold . cancel amounts to evaluating the inverse of a word in the functor.

Examples:

>>> let x = FreeGroup [Left (Sum (2 :: Word8)), Right (Sum 3)]
>>> cancel x
FreeGroup {runFreeGroup = [Left (Sum {getSum = 3}),Right (Sum {getSum = 2})]}

Instances

Cancellative FreeGroup

Defined in Control.Applicative.Cancellative

Methods

cancel :: FreeGroup a -> FreeGroup a

Cancellative FA

Defined in Control.Applicative.Cancellative

Methods

cancel :: FA a -> FA a

Cancellative FG

Defined in Control.Applicative.Cancellative

Methods

cancel :: FG a -> FG a

Cancellative (Proxy :: Type -> Type)

Defined in Control.Applicative.Cancellative

Methods

cancel :: Proxy a -> Proxy a

Cancellative combinators

cancel1 :: (Group a, Cancellative f) => a -> f a -> f a

Cancel a single element in a Cancellative functor.

Examples:

>>> let x = FreeGroup [Left (Sum (2 :: Word8)), Right (Sum 3)]
>>> gold x
Sum {getSum = 1}
>>> gold $ cancel1 (Sum 1) x
Sum {getSum = 0}

annihilate :: (Cancellative f, Traversable t) => (a -> f a) -> t a -> f (t a)

Annihilate a 'Traversable'\'s worth of elements in a Cancellative functor.