Data.Functor.Contravariant

base-4.12.0.0: Basic libraries

Copyright | (C) 2007-2015 Edward Kmett
License | BSD-style (see the file LICENSE)
Maintainer | libraries@haskell.org
Stability | provisional
Portability | portable
Safe Haskell | Trustworthy
Language | Haskell2010

Data.Functor.Contravariant

Contents

- Contravariant Functors
- Operators
- Predicates
- Comparisons
- Equivalence Relations
- Dual arrows

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.

Since: base-4.12.0.0

Synopsis

- class Contravariant f where
  - contramap :: (a -> b) -> f b -> f a
  - (>$) :: b -> f b -> f a
- phantom :: (Functor f, Contravariant f) => f a -> f b
- (>$<) :: 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 #

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 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:

    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 Equivalence # | Equivalence relations are `Contravariant`, because you can apply the contramapped function to each input to the equivalence relation.
Instance details Defined in Data.Functor.Contravariant Methods contramap :: (a -> b) -> Equivalence b -> Equivalence a # (>$) :: b -> Equivalence b -> Equivalence a #
Contravariant Comparison # | A `Comparison` is a `Contravariant` `Functor`, because `contramap` can apply its function argument to each input of the comparison function.
Instance details Defined in Data.Functor.Contravariant Methods contramap :: (a -> b) -> Comparison b -> Comparison a # (>$) :: b -> Comparison b -> Comparison a #
Contravariant Predicate # | A `Predicate` is a `Contravariant` `Functor`, because `contramap` can apply its function argument to the input of the predicate.
Instance details Defined in Data.Functor.Contravariant Methods contramap :: (a -> b) -> Predicate b -> Predicate a # (>$) :: b -> Predicate b -> Predicate a #
Contravariant (V1 :: Type -> Type) # |
Instance details Defined in Data.Functor.Contravariant Methods contramap :: (a -> b) -> V1 b -> V1 a # (>$) :: b -> V1 b -> V1 a #
Contravariant (U1 :: Type -> Type) # |
Instance details Defined in Data.Functor.Contravariant Methods contramap :: (a -> b) -> U1 b -> U1 a # (>$) :: b -> U1 b -> U1 a #
Contravariant (Proxy :: Type -> Type) # |
Instance details Defined in Data.Functor.Contravariant Methods contramap :: (a -> b) -> Proxy b -> Proxy a # (>$) :: b -> Proxy b -> Proxy a #
Contravariant (Op a) # |
Instance details Defined in Data.Functor.Contravariant Methods contramap :: (a0 -> b) -> Op a b -> Op a a0 # (>$) :: b -> Op a b -> Op a a0 #
Contravariant f => Contravariant (Rec1 f) # |
Instance details Defined in Data.Functor.Contravariant Methods contramap :: (a -> b) -> Rec1 f b -> Rec1 f a # (>$) :: b -> Rec1 f b -> Rec1 f a #
Contravariant f => Contravariant (Alt f) # |
Instance details Defined in Data.Functor.Contravariant Methods contramap :: (a -> b) -> Alt f b -> Alt f a # (>$) :: b -> Alt f b -> Alt f a #
Contravariant (Const a :: Type -> Type) # |
Instance details Defined in Data.Functor.Contravariant Methods contramap :: (a0 -> b) -> Const a b -> Const a a0 # (>$) :: b -> Const a b -> Const a a0 #
Contravariant (K1 i c :: Type -> Type) # |
Instance details Defined in Data.Functor.Contravariant Methods contramap :: (a -> b) -> K1 i c b -> K1 i c a # (>$) :: b -> K1 i c b -> K1 i c a #
(Contravariant f, Contravariant g) => Contravariant (f :+: g) # |
Instance details Defined in Data.Functor.Contravariant Methods contramap :: (a -> b) -> (f :+: g) b -> (f :+: g) a # (>$) :: b -> (f :+: g) b -> (f :+: g) a #
(Contravariant f, Contravariant g) => Contravariant (f :*: g) # |
Instance details Defined in Data.Functor.Contravariant Methods contramap :: (a -> b) -> (f :*: g) b -> (f :*: g) a # (>$) :: b -> (f :*: g) b -> (f :*: g) a #
(Contravariant f, Contravariant g) => Contravariant (Sum f g) # |
Instance details Defined in Data.Functor.Contravariant Methods contramap :: (a -> b) -> Sum f g b -> Sum f g a # (>$) :: b -> Sum f g b -> Sum f g a #
(Contravariant f, Contravariant g) => Contravariant (Product f g) # |
Instance details Defined in Data.Functor.Contravariant Methods contramap :: (a -> b) -> Product f g b -> Product f g a # (>$) :: b -> Product f g b -> Product f g a #
Contravariant f => Contravariant (M1 i c f) # |
Instance details Defined in Data.Functor.Contravariant Methods contramap :: (a -> b) -> M1 i c f b -> M1 i c f a # (>$) :: b -> M1 i c f b -> M1 i c f a #
(Functor f, Contravariant g) => Contravariant (f :.: g) # |
Instance details Defined in Data.Functor.Contravariant Methods contramap :: (a -> b) -> (f :.: g) b -> (f :.: g) a # (>$) :: b -> (f :.: g) b -> (f :.: g) a #
(Functor f, Contravariant g) => Contravariant (Compose f g) # |
Instance details Defined in Data.Functor.Contravariant Methods contramap :: (a -> b) -> Compose f g b -> Compose f g a # (>$) :: b -> Compose f g b -> Compose f g 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.
Instance details Defined in Data.Functor.Contravariant Methods contramap :: (a -> b) -> Predicate b -> Predicate a # (>$) :: b -> Predicate b -> Predicate a #
Semigroup (Predicate a) # |
Instance details 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 #
Monoid (Predicate a) # |
Instance details Defined in Data.Functor.Contravariant Methods mempty :: Predicate a # mappend :: Predicate a -> Predicate a -> Predicate a # mconcat :: [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.
Instance details Defined in Data.Functor.Contravariant Methods contramap :: (a -> b) -> Comparison b -> Comparison a # (>$) :: b -> Comparison b -> Comparison a #
Semigroup (Comparison a) # |
Instance details 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 #
Monoid (Comparison a) # |
Instance details Defined in Data.Functor.Contravariant Methods mempty :: Comparison a # mappend :: Comparison a -> Comparison a -> Comparison a # mconcat :: [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.
Instance details Defined in Data.Functor.Contravariant Methods contramap :: (a -> b) -> Equivalence b -> Equivalence a # (>$) :: b -> Equivalence b -> Equivalence a #
Semigroup (Equivalence a) # |
Instance details 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 #
Monoid (Equivalence a) # |
Instance details Defined in Data.Functor.Contravariant Methods mempty :: Equivalence a # mappend :: Equivalence a -> Equivalence a -> Equivalence a # mconcat :: [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

Contravariant (Op a) # |
Instance details Defined in Data.Functor.Contravariant Methods contramap :: (a0 -> b) -> Op a b -> Op a a0 # (>$) :: b -> Op a b -> Op a a0 #
Category Op # |
Instance details Defined in Data.Functor.Contravariant Methods id :: Op a a # (.) :: Op b c -> Op a b -> Op a c #
Floating a => Floating (Op a b) # |
Instance details 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 #
Fractional a => Fractional (Op a b) # |
Instance details 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 #
Num a => Num (Op a b) # |
Instance details 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 #
Semigroup a => Semigroup (Op a b) # |
Instance details 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 #
Monoid a => Monoid (Op a b) # |
Instance details 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 #