Data.Functor.Contravariant

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.

Since: base-4.12.0.0

Contravariant Functors

class Contravariant f where

The class of contravariant functors.

Whereas in Haskell, one can think of a Functor as containing or producing values, a contravariant functor is a functor that can be thought of as consuming values.

As an example, consider the type of predicate functions a -> Bool. One such predicate might be negative x = x < 0, which classifies integers as to whether they are negative. However, given this predicate, we can re-use it in other situations, providing we have a way to map values to integers. For instance, we can use the negative predicate on a person's bank balance to work out if they are currently overdrawn:

newtype Predicate a = Predicate { getPredicate :: a -> Bool }

instance Contravariant Predicate where
  contramap :: (a' -> a) -> (Predicate a -> Predicate a')
  contramap f (Predicate p) = Predicate (p . f)
                                         |   `- First, map the input...
                                         `----- then apply the predicate.

overdrawn :: Predicate Person
overdrawn = contramap personBankBalance negative

Any instance should be subject to the following laws:

Identity

contramap id = id

Composition

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

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' -> a) -> f a -> 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 Comparison

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

Defined in Data.Functor.Contravariant

Methods

contramap :: (a' -> a) -> Comparison a -> Comparison a'

(>$) :: b -> Comparison b -> Comparison a

Contravariant Equivalence

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

Defined in Data.Functor.Contravariant

Methods

contramap :: (a' -> a) -> Equivalence a -> Equivalence a'

(>$) :: b -> Equivalence b -> Equivalence a

Contravariant Predicate

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

Without newtypes contramap f equals precomposing with f (= (. f)).

contramap :: (a' -> a) -> (Predicate a -> Predicate a')
contramap f (Predicate g) = Predicate (g . f)

Defined in Data.Functor.Contravariant

Methods

contramap :: (a' -> a) -> Predicate a -> Predicate a'

(>$) :: b -> Predicate b -> Predicate a

Contravariant (Op a)

Defined in Data.Functor.Contravariant

Methods

contramap :: (a' -> a0) -> Op a a0 -> Op a a'

(>$) :: b -> Op a b -> Op a a0

Contravariant (Proxy :: Type -> Type)

Defined in Data.Functor.Contravariant

Methods

contramap :: (a' -> a) -> Proxy a -> Proxy a'

(>$) :: b -> Proxy b -> Proxy a

Contravariant (U1 :: Type -> Type)

Defined in Data.Functor.Contravariant

Methods

contramap :: (a' -> a) -> U1 a -> U1 a'

(>$) :: b -> U1 b -> U1 a

Contravariant (V1 :: Type -> Type)

Defined in Data.Functor.Contravariant

Methods

contramap :: (a' -> a) -> V1 a -> V1 a'

(>$) :: b -> V1 b -> V1 a

Contravariant (Const a :: Type -> Type)

Defined in Data.Functor.Contravariant

Methods

contramap :: (a' -> a0) -> Const a a0 -> Const a a'

(>$) :: b -> Const a b -> Const a a0

Contravariant f => Contravariant (Alt f)

Defined in Data.Functor.Contravariant

Methods

contramap :: (a' -> a) -> Alt f a -> Alt f a'

(>$) :: b -> Alt f b -> Alt f a

Contravariant f => Contravariant (Rec1 f)

Defined in Data.Functor.Contravariant

Methods

contramap :: (a' -> a) -> Rec1 f a -> Rec1 f a'

(>$) :: b -> Rec1 f b -> Rec1 f a

(Contravariant f, Contravariant g) => Contravariant (Product f g)

Defined in Data.Functor.Contravariant

Methods

contramap :: (a' -> a) -> Product f g a -> Product f g a'

(>$) :: b -> Product f g b -> Product f g a

(Contravariant f, Contravariant g) => Contravariant (Sum f g)

Defined in Data.Functor.Contravariant

Methods

contramap :: (a' -> a) -> Sum f g a -> Sum f g a'

(>$) :: b -> Sum f g b -> Sum f g a

(Contravariant f, Contravariant g) => Contravariant (f :*: g)

Defined in Data.Functor.Contravariant

Methods

contramap :: (a' -> a) -> (f :*: g) a -> (f :*: g) a'

(>$) :: b -> (f :*: g) b -> (f :*: g) a

(Contravariant f, Contravariant g) => Contravariant (f :+: g)

Defined in Data.Functor.Contravariant

Methods

contramap :: (a' -> a) -> (f :+: g) a -> (f :+: g) a'

(>$) :: b -> (f :+: g) b -> (f :+: g) a

Contravariant (K1 i c :: Type -> Type)

Defined in Data.Functor.Contravariant

Methods

contramap :: (a' -> a) -> K1 i c a -> K1 i c a'

(>$) :: b -> K1 i c b -> K1 i c a

(Functor f, Contravariant g) => Contravariant (Compose f g)

Defined in Data.Functor.Contravariant

Methods

contramap :: (a' -> a) -> Compose f g a -> Compose f g a'

(>$) :: b -> Compose f g b -> Compose f g a

(Functor f, Contravariant g) => Contravariant (f :.: g)

Defined in Data.Functor.Contravariant

Methods

contramap :: (a' -> a) -> (f :.: g) a -> (f :.: g) a'

(>$) :: b -> (f :.: g) b -> (f :.: g) a

Contravariant f => Contravariant (M1 i c f)

Defined in Data.Functor.Contravariant

Methods

contramap :: (a' -> a) -> M1 i c f a -> M1 i c f a'

(>$) :: b -> M1 i c f b -> M1 i c f a

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 its 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.

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

This is >$ with its 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.

Without newtypes contramap f equals precomposing with f (= (. f)).

contramap :: (a' -> a) -> (Predicate a -> Predicate a')
contramap f (Predicate g) = Predicate (g . f)

Defined in Data.Functor.Contravariant

Methods

contramap :: (a' -> a) -> Predicate a -> Predicate a'

(>$) :: b -> Predicate b -> Predicate a

Monoid (Predicate a)

mempty on predicates always returns True. Without newtypes this equals pure True.

mempty :: Predicate a
mempty = _ -> True

Defined in Data.Functor.Contravariant

Methods

mempty :: Predicate a

mappend :: Predicate a -> Predicate a -> Predicate a

mconcat :: [Predicate a] -> Predicate a

Semigroup (Predicate a)

(<>) on predicates uses logical conjunction (&&) on the results. Without newtypes this equals liftA2 (&&).

(<>) :: Predicate a -> Predicate a -> Predicate a
Predicate pred <> Predicate pred' = Predicate a ->
  pred a && pred' a

Defined in Data.Functor.Contravariant

Methods

(<>) :: Predicate a -> Predicate a -> Predicate a

sconcat :: NonEmpty (Predicate a) -> Predicate a

stimes :: Integral b => b -> Predicate a -> Predicate a

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.

Defined in Data.Functor.Contravariant

Methods

contramap :: (a' -> a) -> Comparison a -> Comparison a'

(>$) :: b -> Comparison b -> Comparison a

Monoid (Comparison a)

mempty on comparisons always returns EQ. Without newtypes this equals pure (pure EQ).

mempty :: Comparison a
mempty = Comparison _ _ -> EQ

Defined in Data.Functor.Contravariant

Methods

mempty :: Comparison a

mappend :: Comparison a -> Comparison a -> Comparison a

mconcat :: [Comparison a] -> Comparison a

Semigroup (Comparison a)

(<>) on comparisons combines results with (<>) @Ordering. Without newtypes this equals liftA2 (liftA2 (<>)).

(<>) :: Comparison a -> Comparison a -> Comparison a
Comparison cmp <> Comparison cmp' = Comparison a a' ->
  cmp a a' <> cmp a a'

Defined in Data.Functor.Contravariant

Methods

(<>) :: Comparison a -> Comparison a -> Comparison a

sconcat :: NonEmpty (Comparison a) -> Comparison a

stimes :: Integral b => b -> Comparison a -> 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

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.

Defined in Data.Functor.Contravariant

Methods

contramap :: (a' -> a) -> Equivalence a -> Equivalence a'

(>$) :: b -> Equivalence b -> Equivalence a

Monoid (Equivalence a)

mempty on equivalences always returns True. Without newtypes this equals pure (pure True).

mempty :: Equivalence a
mempty = Equivalence _ _ -> True

Defined in Data.Functor.Contravariant

Methods

mempty :: Equivalence a

mappend :: Equivalence a -> Equivalence a -> Equivalence a

mconcat :: [Equivalence a] -> Equivalence a

Semigroup (Equivalence a)

(<>) on equivalences uses logical conjunction (&&) on the results. Without newtypes this equals liftA2 (liftA2 (&&)).

(<>) :: Equivalence a -> Equivalence a -> Equivalence a
Equivalence equiv <> Equivalence equiv' = Equivalence a b ->
  equiv a b && equiv a b

Defined in Data.Functor.Contravariant

Methods

(<>) :: Equivalence a -> Equivalence a -> Equivalence a

sconcat :: NonEmpty (Equivalence a) -> Equivalence a

stimes :: Integral b => b -> Equivalence a -> Equivalence a

defaultEquivalence :: Eq a => Equivalence a

Check for equivalence with ==.

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

comparisonEquivalence :: Comparison a -> Equivalence a

Dual arrows

newtype Op a b

Dual function arrows.

Constructors

Op

Fields

• getOp :: b -> a

Instances

Category Op

Defined in Data.Functor.Contravariant

Methods

id :: forall (a :: k). Op a a

(.) :: forall (b :: k) (c :: k) (a :: k). Op b c -> Op a b -> Op a c

Contravariant (Op a)

Defined in Data.Functor.Contravariant

Methods

contramap :: (a' -> a0) -> Op a a0 -> Op a a'

(>$) :: b -> Op a b -> Op a a0

Monoid a => Monoid (Op a b)

mempty @(Op a b) without newtypes is mempty @(b->a) = _ -> mempty.

mempty :: Op a b
mempty = Op _ -> mempty

Defined in Data.Functor.Contravariant

Methods

mempty :: Op a b

mappend :: Op a b -> Op a b -> Op a b

mconcat :: [Op a b] -> Op a b

Semigroup a => Semigroup (Op a b)

(<>) @(Op a b) without newtypes is (<>) @(b->a) = liftA2 (<>). This lifts the Semigroup operation (<>) over the output of a.

(<>) :: Op a b -> Op a b -> Op a b
Op f <> Op g = Op a -> f a <> g a

Defined in Data.Functor.Contravariant

Methods

(<>) :: Op a b -> Op a b -> Op a b

sconcat :: NonEmpty (Op a b) -> Op a b

stimes :: Integral b0 => b0 -> Op a b -> Op a b

Floating a => Floating (Op a b)

Defined in Data.Functor.Contravariant

Methods

pi :: Op a b

exp :: Op a b -> Op a b

log :: Op a b -> Op a b

sqrt :: Op a b -> Op a b

(**) :: Op a b -> Op a b -> Op a b

logBase :: Op a b -> Op a b -> Op a b

sin :: Op a b -> Op a b

cos :: Op a b -> Op a b

tan :: Op a b -> Op a b

asin :: Op a b -> Op a b

acos :: Op a b -> Op a b

atan :: Op a b -> Op a b

sinh :: Op a b -> Op a b

cosh :: Op a b -> Op a b

tanh :: Op a b -> Op a b

asinh :: Op a b -> Op a b

acosh :: Op a b -> Op a b

atanh :: Op a b -> Op a b

log1p :: Op a b -> Op a b

expm1 :: Op a b -> Op a b

log1pexp :: Op a b -> Op a b

log1mexp :: Op a b -> Op a b

Num a => Num (Op a b)

Defined in Data.Functor.Contravariant

Methods

(+) :: Op a b -> Op a b -> Op a b

(-) :: Op a b -> Op a b -> Op a b

(*) :: Op a b -> Op a b -> Op a b

negate :: Op a b -> Op a b

abs :: Op a b -> Op a b

signum :: Op a b -> Op a b

fromInteger :: Integer -> Op a b

Fractional a => Fractional (Op a b)

Defined in Data.Functor.Contravariant

Methods

(/) :: Op a b -> Op a b -> Op a b

recip :: Op a b -> Op a b

fromRational :: Rational -> Op a b