contravariant-1.3.1.1: Contravariant functors

Copyright(C) 2007-2015 Edward Kmett
LicenseBSD-style (see the file LICENSE)
MaintainerEdward Kmett <ekmett@gmail.com>
Stabilityprovisional
Portabilityportable
Safe HaskellSafe
LanguageHaskell98

Data.Functor.Contravariant

Contents

Description

Contravariant functors, sometimes referred to colloquially as Cofunctor, even though the dual of a Functor is just a Functor. As with Functor the definition of Contravariant for a given ADT is unambiguous.

Synopsis

Contravariant Functors

class Contravariant f where

Any instance should be subject to the following laws:

contramap id = id
contramap f . contramap g = contramap (g . f)

Note, that the second law follows from the free theorem of the type of contramap and the first law, so you need only check that the former condition holds.

Minimal complete definition

contramap

Methods

contramap :: (a -> b) -> f b -> f a

(>$) :: b -> f b -> f a infixl 4

Replace all locations in the output with the same value. The default definition is contramap . const, but this may be overridden with a more efficient version.

Instances

Contravariant V1 
Contravariant U1 
Contravariant SettableStateVar 
Contravariant Equivalence

Equivalence relations are Contravariant, because you can apply the contramapped function to each input to the equivalence relation.

Contravariant Comparison

A Comparison is a Contravariant Functor, because contramap can apply its function argument to each input of the comparison function.

Contravariant Predicate

A Predicate is a Contravariant Functor, because contramap can apply its function argument to the input of the predicate.

Contravariant f => Contravariant (Rec1 f) 
Contravariant (Const a) 
Contravariant (Proxy *) 
Contravariant f => Contravariant (Reverse f) 
Contravariant f => Contravariant (Backwards f) 
Contravariant (Constant a) 
Contravariant (Op a) 
Contravariant (K1 i c) 
(Contravariant f, Contravariant g) => Contravariant ((:+:) f g) 
(Contravariant f, Contravariant g) => Contravariant ((:*:) f g) 
(Functor f, Contravariant g) => Contravariant ((:.:) f g) 
(Functor f, Contravariant g) => Contravariant (Compose f g) 
(Contravariant f, Contravariant g) => Contravariant (Product f g) 
(Contravariant f, Contravariant g) => Contravariant (Sum f g) 
(Contravariant f, Functor g) => Contravariant (ComposeCF f g) 
(Functor f, Contravariant g) => Contravariant (ComposeFC f g) 
Contravariant f => Contravariant (M1 i c f) 

phantom :: (Functor f, Contravariant f) => f a -> f b

If f is both Functor and Contravariant then by the time you factor in the laws of each of those classes, it can't actually use it's argument in any meaningful capacity.

This method is surprisingly useful. Where both instances exist and are lawful we have the following laws:

fmap f ≡ phantom
contramap f ≡ phantom

Operators

(>$<) :: Contravariant f => (a -> b) -> f b -> f a infixl 4

This is an infix alias for contramap

(>$$<) :: Contravariant f => f b -> (a -> b) -> f a infixl 4

This is an infix version of contramap with the arguments flipped.

Predicates

newtype Predicate a

Constructors

Predicate 

Fields

getPredicate :: a -> Bool
 

Instances

Contravariant Predicate

A Predicate is a Contravariant Functor, because contramap can apply its function argument to the input of the predicate.

Decidable Predicate 
Divisible Predicate 
Typeable (* -> *) Predicate 

Comparisons

newtype Comparison a

Defines a total ordering on a type as per compare

This condition is not checked by the types. You must ensure that the supplied values are valid total orderings yourself.

Constructors

Comparison 

Fields

getComparison :: a -> a -> Ordering
 

Instances

Contravariant Comparison

A Comparison is a Contravariant Functor, because contramap can apply its function argument to each input of the comparison function.

Decidable Comparison 
Divisible Comparison 
Monoid (Comparison a) 
Semigroup (Comparison a) 
Typeable (* -> *) Comparison 

defaultComparison :: Ord a => Comparison a

Compare using compare

Equivalence Relations

newtype Equivalence a

This data type represents an equivalence relation.

Equivalence relations are expected to satisfy three laws:

Reflexivity:

getEquivalence f a a = True

Symmetry:

getEquivalence f a b = getEquivalence f b a

Transitivity:

If getEquivalence f a b and getEquivalence f b c are both True then so is getEquivalence f a c

The types alone do not enforce these laws, so you'll have to check them yourself.

Constructors

Equivalence 

Fields

getEquivalence :: a -> a -> Bool
 

Instances

Contravariant Equivalence

Equivalence relations are Contravariant, because you can apply the contramapped function to each input to the equivalence relation.

Decidable Equivalence 
Divisible Equivalence 
Monoid (Equivalence a) 
Semigroup (Equivalence a) 
Typeable (* -> *) Equivalence 

defaultEquivalence :: Eq a => Equivalence a

Check for equivalence with ==

Note: The instances for Double and Float violate reflexivity for NaN.

Dual arrows

newtype Op a b

Dual function arrows.

Constructors

Op 

Fields

getOp :: b -> a
 

Instances

Category * Op 
Contravariant (Op a) 
Monoid r => Decidable (Op r) 
Monoid r => Divisible (Op r) 
Floating a => Floating (Op a b) 
Fractional a => Fractional (Op a b) 
Num a => Num (Op a b) 
Monoid a => Monoid (Op a b) 
Semigroup a => Semigroup (Op a b) 
Typeable (* -> * -> *) Op