{-# LANGUAGE EmptyCase #-}{-# LANGUAGE GeneralizedNewtypeDeriving #-}{-# LANGUAGE StandaloneDeriving #-}{-# LANGUAGE Trustworthy #-}{-# LANGUAGE TypeOperators #-}------------------------------------------------------------------------------- |-- Module : Data.Functor.Contravariant-- Copyright : (C) 2007-2015 Edward Kmett-- License : BSD-style (see the file LICENSE)---- Maintainer : libraries@haskell.org-- Stability : provisional-- Portability : portable---- '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 4.12.0.0----------------------------------------------------------------------------moduleData.Functor.Contravariant(-- * Contravariant FunctorsContravariant (..),phantom -- * Operators,(>$<) ,(>$$<) ,($<) -- * Predicates,Predicate (..)-- * Comparisons,Comparison (..),defaultComparison -- * Equivalence Relations,Equivalence (..),defaultEquivalence ,comparisonEquivalence -- * Dual arrows,Op (..))whereimportControl.Applicative importControl.Category importData.Function (on )importData.Functor.Product importData.Functor.Sum importData.Functor.Compose importData.Monoid (Alt (..))importData.Proxy importGHC.Generics importPrelude hiding((.) ,id )-- | 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:---- [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.classContravariant f wherecontramap ::(a ->b )->f b ->f a -- | 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.(>$) ::b ->f b ->f a (>$) =(a -> b) -> f b -> f a
forall (f :: * -> *) a b. Contravariant f => (a -> b) -> f b -> f a
contramap ((a -> b) -> f b -> f a) -> (b -> a -> b) -> b -> f b -> f a
forall k (cat :: k -> k -> *) (b :: k) (c :: k) (a :: k).
Category cat =>
cat b c -> cat a b -> cat a c
. b -> a -> b
forall a b. a -> b -> a
const -- | 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'-- @phantom ::(Functor f ,Contravariant f )=>f a ->f b phantom :: f a -> f b
phantom f a
x =()() -> f a -> f ()
forall (f :: * -> *) a b. Functor f => a -> f b -> f a
<$ f a
x f () -> () -> f b
forall (f :: * -> *) b a. Contravariant f => f b -> b -> f a
$< ()infixl4>$ ,$< ,>$< ,>$$< -- | This is '>$' with its arguments flipped.($<) ::Contravariant f =>f b ->b ->f a $< :: f b -> b -> f a
($<) =(b -> f b -> f a) -> f b -> b -> f a
forall a b c. (a -> b -> c) -> b -> a -> c
flip b -> f b -> f a
forall (f :: * -> *) b a. Contravariant f => b -> f b -> f a
(>$) -- | This is an infix alias for 'contramap'.(>$<) ::Contravariant f =>(a ->b )->f b ->f a >$< :: (a -> b) -> f b -> f a
(>$<) =(a -> b) -> f b -> f a
forall (f :: * -> *) a b. Contravariant f => (a -> b) -> f b -> f a
contramap -- | This is an infix version of 'contramap' with the arguments flipped.(>$$<) ::Contravariant f =>f b ->(a ->b )->f a >$$< :: f b -> (a -> b) -> f a
(>$$<) =((a -> b) -> f b -> f a) -> f b -> (a -> b) -> f a
forall a b c. (a -> b -> c) -> b -> a -> c
flip (a -> b) -> f b -> f a
forall (f :: * -> *) a b. Contravariant f => (a -> b) -> f b -> f a
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forall (f :: * -> *) a b. Contravariant f => (a -> b) -> f b -> f a
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forall (f :: * -> *) a b. Contravariant f => (a -> b) -> f b -> f a
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f (g p) -> (:.:) f g p
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forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
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forall (f :: * -> *) a b. Contravariant f => (a -> b) -> f b -> f a
contramap a -> b
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contramap a -> b
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forall (f :: * -> *) a b. Contravariant f => (a -> b) -> f b -> f a
contramap a -> b
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contramap a -> b
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forall k (t :: k). Proxy t
Proxy newtypePredicate a =Predicate {Predicate a -> a -> Bool
getPredicate ::a ->Bool}-- | A 'Predicate' is a 'Contravariant' 'Functor', because 'contramap' can-- apply its function argument to the input of the predicate.instanceContravariant Predicate wherecontramap :: (a -> b) -> Predicate b -> Predicate a
contramap a -> b
f Predicate b
g =(a -> Bool) -> Predicate a
forall a. (a -> Bool) -> Predicate a
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forall a b. (a -> b) -> a -> b
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p <> :: Predicate a -> Predicate a -> Predicate a
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q =(a -> Bool) -> Predicate a
forall a. (a -> Bool) -> Predicate a
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forall a b. (a -> b) -> a -> b
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a instanceMonoid (Predicate a )wheremempty :: Predicate a
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forall a. (a -> Bool) -> Predicate a
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forall a b. (a -> b) -> a -> b
$ Bool -> a -> Bool
forall a b. a -> b -> a
const Bool
True-- | 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.newtypeComparison a =Comparison {Comparison a -> a -> a -> Ordering
getComparison ::a ->a ->Ordering}derivinginstanceSemigroup (Comparison a )derivinginstanceMonoid (Comparison a )-- | A 'Comparison' is a 'Contravariant' 'Functor', because 'contramap' can-- apply its function argument to each input of the comparison function.instanceContravariant Comparison wherecontramap :: (a -> b) -> Comparison b -> Comparison a
contramap a -> b
f Comparison b
g =(a -> a -> Ordering) -> Comparison a
forall a. (a -> a -> Ordering) -> Comparison a
Comparison ((a -> a -> Ordering) -> Comparison a)
-> (a -> a -> Ordering) -> Comparison a
forall a b. (a -> b) -> a -> b
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forall b c a. (b -> b -> c) -> (a -> b) -> a -> a -> c
on (Comparison b -> b -> b -> Ordering
forall a. Comparison a -> a -> a -> Ordering
getComparison Comparison b
g )a -> b
f -- | Compare using 'compare'.defaultComparison ::Orda =>Comparison a defaultComparison :: Comparison a
defaultComparison =(a -> a -> Ordering) -> Comparison a
forall a. (a -> a -> Ordering) -> Comparison a
Comparison a -> a -> Ordering
forall a. Ord a => a -> a -> Ordering
compare-- | 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.newtypeEquivalence a =Equivalence {Equivalence a -> a -> a -> Bool
getEquivalence ::a ->a ->Bool}-- | Equivalence relations are 'Contravariant', because you can-- apply the contramapped function to each input to the equivalence-- relation.instanceContravariant Equivalence wherecontramap :: (a -> b) -> Equivalence b -> Equivalence a
contramap a -> b
f Equivalence b
g =(a -> a -> Bool) -> Equivalence a
forall a. (a -> a -> Bool) -> Equivalence a
Equivalence ((a -> a -> Bool) -> Equivalence a)
-> (a -> a -> Bool) -> Equivalence a
forall a b. (a -> b) -> a -> b
$ (b -> b -> Bool) -> (a -> b) -> a -> a -> Bool
forall b c a. (b -> b -> c) -> (a -> b) -> a -> a -> c
on (Equivalence b -> b -> b -> Bool
forall a. Equivalence a -> a -> a -> Bool
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f instanceSemigroup (Equivalence a )whereEquivalence a -> a -> Bool
p <> :: Equivalence a -> Equivalence a -> Equivalence a
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forall a. (a -> a -> Bool) -> Equivalence a
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forall a b. (a -> b) -> a -> b
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b instanceMonoid (Equivalence a )wheremempty :: Equivalence a
mempty =(a -> a -> Bool) -> Equivalence a
forall a. (a -> a -> Bool) -> Equivalence a
Equivalence (\a
_a
_->Bool
True)-- | Check for equivalence with '=='.---- Note: The instances for 'Double' and 'Float' violate reflexivity for @NaN@.defaultEquivalence ::Eqa =>Equivalence a defaultEquivalence :: Equivalence a
defaultEquivalence =(a -> a -> Bool) -> Equivalence a
forall a. (a -> a -> Bool) -> Equivalence a
Equivalence a -> a -> Bool
forall a. Eq a => a -> a -> Bool
(==)comparisonEquivalence ::Comparison a ->Equivalence a comparisonEquivalence :: Comparison a -> Equivalence a
comparisonEquivalence (Comparison a -> a -> Ordering
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EQ-- | Dual function arrows.newtypeOp a b =Op {Op a b -> b -> a
getOp ::b ->a }derivinginstanceSemigroup a =>Semigroup (Op a b )derivinginstanceMonoid a =>Monoid (Op a b )instanceCategory Op whereid :: Op a a
id =(a -> a) -> Op a a
forall a b. (b -> a) -> Op a b
Op a -> a
forall k (cat :: k -> k -> *) (a :: k). Category cat => cat a a
id Op c -> b
f . :: Op b c -> Op a b -> Op a c
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g =(c -> a) -> Op a c
forall a b. (b -> a) -> Op a b
Op (b -> a
g (b -> a) -> (c -> b) -> c -> a
forall k (cat :: k -> k -> *) (b :: k) (c :: k) (a :: k).
Category cat =>
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f )instanceContravariant (Op a )wherecontramap :: (a -> b) -> Op a b -> Op a a
contramap a -> b
f Op a b
g =(a -> a) -> Op a a
forall a b. (b -> a) -> Op a b
Op (Op a b -> b -> a
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f )instanceNum a =>Num (Op a b )whereOp b -> a
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f fromInteger :: Integer -> Op a b
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forall a b. (b -> a) -> Op a b
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f fromRational :: Rational -> Op a b
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f cos :: Op a b -> Op a b
cos (Op b -> a
f )=(b -> a) -> Op a b
forall a b. (b -> a) -> Op a b
Op ((b -> a) -> Op a b) -> (b -> a) -> Op a b
forall a b. (a -> b) -> a -> b
$ a -> a
forall a. Floating a => a -> a
cos (a -> a) -> (b -> a) -> b -> a
forall k (cat :: k -> k -> *) (b :: k) (c :: k) (a :: k).
Category cat =>
cat b c -> cat a b -> cat a c
. b -> a
f asin :: Op a b -> Op a b
asin (Op b -> a
f )=(b -> a) -> Op a b
forall a b. (b -> a) -> Op a b
Op ((b -> a) -> Op a b) -> (b -> a) -> Op a b
forall a b. (a -> b) -> a -> b
$ a -> a
forall a. Floating a => a -> a
asin (a -> a) -> (b -> a) -> b -> a
forall k (cat :: k -> k -> *) (b :: k) (c :: k) (a :: k).
Category cat =>
cat b c -> cat a b -> cat a c
. b -> a
f atan :: Op a b -> Op a b
atan (Op b -> a
f )=(b -> a) -> Op a b
forall a b. (b -> a) -> Op a b
Op ((b -> a) -> Op a b) -> (b -> a) -> Op a b
forall a b. (a -> b) -> a -> b
$ a -> a
forall a. Floating a => a -> a
atan (a -> a) -> (b -> a) -> b -> a
forall k (cat :: k -> k -> *) (b :: k) (c :: k) (a :: k).
Category cat =>
cat b c -> cat a b -> cat a c
. b -> a
f acos :: Op a b -> Op a b
acos (Op b -> a
f )=(b -> a) -> Op a b
forall a b. (b -> a) -> Op a b
Op ((b -> a) -> Op a b) -> (b -> a) -> Op a b
forall a b. (a -> b) -> a -> b
$ a -> a
forall a. Floating a => a -> a
acos (a -> a) -> (b -> a) -> b -> a
forall k (cat :: k -> k -> *) (b :: k) (c :: k) (a :: k).
Category cat =>
cat b c -> cat a b -> cat a c
. b -> a
f sinh :: Op a b -> Op a b
sinh (Op b -> a
f )=(b -> a) -> Op a b
forall a b. (b -> a) -> Op a b
Op ((b -> a) -> Op a b) -> (b -> a) -> Op a b
forall a b. (a -> b) -> a -> b
$ a -> a
forall a. Floating a => a -> a
sinh (a -> a) -> (b -> a) -> b -> a
forall k (cat :: k -> k -> *) (b :: k) (c :: k) (a :: k).
Category cat =>
cat b c -> cat a b -> cat a c
. b -> a
f tanh :: Op a b -> Op a b
tanh (Op b -> a
f )=(b -> a) -> Op a b
forall a b. (b -> a) -> Op a b
Op ((b -> a) -> Op a b) -> (b -> a) -> Op a b
forall a b. (a -> b) -> a -> b
$ a -> a
forall a. Floating a => a -> a
tanh (a -> a) -> (b -> a) -> b -> a
forall k (cat :: k -> k -> *) (b :: k) (c :: k) (a :: k).
Category cat =>
cat b c -> cat a b -> cat a c
. b -> a
f cosh :: Op a b -> Op a b
cosh (Op b -> a
f )=(b -> a) -> Op a b
forall a b. (b -> a) -> Op a b
Op ((b -> a) -> Op a b) -> (b -> a) -> Op a b
forall a b. (a -> b) -> a -> b
$ a -> a
forall a. Floating a => a -> a
cosh (a -> a) -> (b -> a) -> b -> a
forall k (cat :: k -> k -> *) (b :: k) (c :: k) (a :: k).
Category cat =>
cat b c -> cat a b -> cat a c
. b -> a
f asinh :: Op a b -> Op a b
asinh (Op b -> a
f )=(b -> a) -> Op a b
forall a b. (b -> a) -> Op a b
Op ((b -> a) -> Op a b) -> (b -> a) -> Op a b
forall a b. (a -> b) -> a -> b
$ a -> a
forall a. Floating a => a -> a
asinh (a -> a) -> (b -> a) -> b -> a
forall k (cat :: k -> k -> *) (b :: k) (c :: k) (a :: k).
Category cat =>
cat b c -> cat a b -> cat a c
. b -> a
f atanh :: Op a b -> Op a b
atanh (Op b -> a
f )=(b -> a) -> Op a b
forall a b. (b -> a) -> Op a b
Op ((b -> a) -> Op a b) -> (b -> a) -> Op a b
forall a b. (a -> b) -> a -> b
$ a -> a
forall a. Floating a => a -> a
atanh (a -> a) -> (b -> a) -> b -> a
forall k (cat :: k -> k -> *) (b :: k) (c :: k) (a :: k).
Category cat =>
cat b c -> cat a b -> cat a c
. b -> a
f acosh :: Op a b -> Op a b
acosh (Op b -> a
f )=(b -> a) -> Op a b
forall a b. (b -> a) -> Op a b
Op ((b -> a) -> Op a b) -> (b -> a) -> Op a b
forall a b. (a -> b) -> a -> b
$ a -> a
forall a. Floating a => a -> a
acosh (a -> a) -> (b -> a) -> b -> a
forall k (cat :: k -> k -> *) (b :: k) (c :: k) (a :: k).
Category cat =>
cat b c -> cat a b -> cat a c
. b -> a
f Op b -> a
f ** :: Op a b -> Op a b -> Op a b
** Op b -> a
g =(b -> a) -> Op a b
forall a b. (b -> a) -> Op a b
Op ((b -> a) -> Op a b) -> (b -> a) -> Op a b
forall a b. (a -> b) -> a -> b
$ \b
a ->b -> a
f b
a a -> a -> a
forall a. Floating a => a -> a -> a
** b -> a
g b
a logBase :: Op a b -> Op a b -> Op a b
logBase (Op b -> a
f )(Op b -> a
g )=(b -> a) -> Op a b
forall a b. (b -> a) -> Op a b
Op ((b -> a) -> Op a b) -> (b -> a) -> Op a b
forall a b. (a -> b) -> a -> b
$ \b
a ->a -> a -> a
forall a. Floating a => a -> a -> a
logBase (b -> a
f b
a )(b -> a
g b
a )