| Copyright | (C) 2007-2015 Edward Kmett |
|---|---|
| License | BSD-style (see the file LICENSE) |
| Maintainer | Edward Kmett <ekmett@gmail.com> |
| Stability | provisional |
| Portability | portable |
| Safe Haskell | Safe |
| Language | Haskell98 |
Data.Functor.Contravariant
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.
- class Contravariant f where
- phantom :: (Functor f, Contravariant f) => f a -> f b
- (>$<) :: Contravariant f => (a -> b) -> f b -> f a
- (>$$<) :: Contravariant f => f b -> (a -> b) -> f a
- ($<) :: Contravariant f => f b -> b -> f a
- newtype Predicate a = Predicate {
- getPredicate :: a -> Bool
- newtype Comparison a = Comparison {
- getComparison :: a -> a -> Ordering
- defaultComparison :: Ord a => Comparison a
- newtype Equivalence a = Equivalence {
- getEquivalence :: a -> a -> Bool
- defaultEquivalence :: Eq a => Equivalence a
- comparisonEquivalence :: Comparison a -> Equivalence a
- newtype Op a b = Op {
- getOp :: b -> a
Contravariant Functors
class Contravariant f where #
Any instance should be subject to the following laws:
contramap id = id contramap f . contramap g = contramap (g . f)
Note, that the second law follows from the free theorem of the type of
contramap and the first law, so you need only check that the former
condition holds.
Minimal complete definition
Instances
phantom :: (Functor f, Contravariant f) => f a -> f b #
If f is both Functor and Contravariant then by the time you factor in the laws
of each of those classes, it can't actually use it's argument in any meaningful capacity.
This method is surprisingly useful. Where both instances exist and are lawful we have the following laws:
fmapf ≡phantomcontramapf ≡phantom
Operators
(>$<) :: Contravariant f => (a -> b) -> f b -> f a infixl 4 #
This is an infix alias for contramap
(>$$<) :: Contravariant f => f b -> (a -> b) -> f a infixl 4 #
This is an infix version of contramap with the arguments flipped.
($<) :: Contravariant f => f b -> b -> f a infixl 4 #
This is >$ with its arguments flipped.
Predicates
Constructors
| Predicate | |
Fields
| |
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
| |
Instances
| Contravariant Comparison # | A |
| Decidable Comparison # | |
| Divisible Comparison # | |
| Semigroup (Comparison a) # | |
| Monoid (Comparison a) # | |
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:
getEquivalencef a b =getEquivalencef b a
Transitivity:
If getEquivalence f a bgetEquivalence f b cTrue 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
| |
Instances
| Contravariant Equivalence # | Equivalence relations are |
| Decidable Equivalence # | |
| Divisible Equivalence # | |
| Semigroup (Equivalence a) # | |
| Monoid (Equivalence a) # | |
defaultEquivalence :: Eq a => Equivalence a #
comparisonEquivalence :: Comparison a -> Equivalence a #
Dual arrows
Dual function arrows.