Copyright  (C) 20072015 Edward Kmett 

License  BSDstyle (see the file LICENSE) 
Maintainer  Edward Kmett <ekmett@gmail.com> 
Stability  provisional 
Portability  portable 
Safe Haskell  Safe 
Language  Haskell98 
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
 (>$$<) :: Contravariant f => f b > (a > b) > f a
 ($<) :: Contravariant f => f b > b > f a
 newtype Predicate a = Predicate {
 getPredicate :: a > Bool
 newtype Comparison a = Comparison {
 getComparison :: a > a > Ordering
 defaultComparison :: Ord a => Comparison a
 newtype Equivalence a = Equivalence {
 getEquivalence :: a > a > Bool
 defaultEquivalence :: Eq a => Equivalence a
 comparisonEquivalence :: Comparison a > Equivalence a
 newtype Op a b = Op {
 getOp :: b > a
Contravariant Functors
class Contravariant f where Source
Any instance should be subject to the following laws:
contramap id = id contramap f . contramap g = contramap (g . f)
Note, that the second law follows from the free theorem of the type of
contramap
and the first law, so you need only check that the former
condition holds.
phantom :: (Functor f, Contravariant f) => f a > f b Source
If f
is both Functor
and Contravariant
then by the time you factor in the laws
of each of those classes, it can't actually use it's argument in any meaningful capacity.
This method is surprisingly useful. Where both instances exist and are lawful we have the following laws:
fmap
f ≡phantom
contramap
f ≡phantom
Operators
(>$<) :: Contravariant f => (a > b) > f b > f a infixl 4 Source
This is an infix alias for contramap
(>$$<) :: Contravariant f => f b > (a > b) > f a infixl 4 Source
This is an infix version of contramap
with the arguments flipped.
($<) :: Contravariant f => f b > b > f a infixl 4 Source
This is >$
with its arguments flipped.
Predicates
Predicate  

Comparisons
newtype Comparison a Source
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.
Comparison  

Contravariant Comparison  A 
Decidable Comparison  
Divisible Comparison  
Monoid (Comparison a)  
Semigroup (Comparison a)  
Typeable (* > *) Comparison 
defaultComparison :: Ord a => Comparison a Source
Compare using compare
Equivalence Relations
newtype Equivalence a Source
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
and getEquivalence
f a b
are both getEquivalence
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.
Equivalence  

Contravariant Equivalence  Equivalence relations are 
Decidable Equivalence  
Divisible Equivalence  
Monoid (Equivalence a)  
Semigroup (Equivalence a)  
Typeable (* > *) Equivalence 
defaultEquivalence :: Eq a => Equivalence a Source
comparisonEquivalence :: Comparison a > Equivalence a Source
Dual arrows
Dual function arrows.