```{-# LANGUAGE EmptyCase #-}
{-# LANGUAGE GeneralizedNewtypeDeriving #-}
{-# LANGUAGE StandaloneDeriving #-}
{-# LANGUAGE Trustworthy #-}
{-# LANGUAGE TypeOperators #-}

-----------------------------------------------------------------------------
-- |
-- Module      :  Data.Functor.Contravariant
-- Copyright   :  (C) 2007-2015 Edward Kmett
--
-- 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
----------------------------------------------------------------------------

module Data.Functor.Contravariant (
-- * Contravariant Functors
Contravariant(..)
, phantom

-- * Operators
, (>\$<), (>\$\$<), (\$<)

-- * Predicates
, Predicate(..)

-- * Comparisons
, Comparison(..)
, defaultComparison

-- * Equivalence Relations
, Equivalence(..)
, defaultEquivalence
, comparisonEquivalence

-- * Dual arrows
, Op(..)
) where

import Control.Applicative
import Control.Category
import Data.Function (on)

import Data.Functor.Product
import Data.Functor.Sum
import Data.Functor.Compose

import Data.Monoid (Alt(..))
import Data.Proxy
import GHC.Generics

import Prelude 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.

class Contravariant f where
contramap :: (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
\$< ()

infixl 4 >\$, \$<, >\$<, >\$\$<

-- | 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
contramap

deriving instance Contravariant f => Contravariant (Alt f)
deriving instance Contravariant f => Contravariant (Rec1 f)
deriving instance Contravariant f => Contravariant (M1 i c f)

instance Contravariant V1 where
contramap :: (a -> b) -> V1 b -> V1 a
contramap a -> b
_ V1 b
x = case V1 b
x of

instance Contravariant U1 where
contramap :: (a -> b) -> U1 b -> U1 a
contramap a -> b
_ U1 b
_ = U1 a
forall k (p :: k). U1 p
U1

instance Contravariant (K1 i c) where
contramap :: (a -> b) -> K1 i c b -> K1 i c a
contramap a -> b
_ (K1 c
c) = c -> K1 i c a
forall k i c (p :: k). c -> K1 i c p
K1 c
c

instance (Contravariant f, Contravariant g) => Contravariant (f :*: g) where
contramap :: (a -> b) -> (:*:) f g b -> (:*:) f g a
contramap a -> b
f (f b
xs :*: g b
ys) = (a -> b) -> f b -> f a
forall (f :: * -> *) a b. Contravariant f => (a -> b) -> f b -> f a
contramap a -> b
f f b
xs f a -> g a -> (:*:) f g a
forall k (f :: k -> *) (g :: k -> *) (p :: k).
f p -> g p -> (:*:) f g p
:*: (a -> b) -> g b -> g a
forall (f :: * -> *) a b. Contravariant f => (a -> b) -> f b -> f a
contramap a -> b
f g b
ys

instance (Functor f, Contravariant g) => Contravariant (f :.: g) where
contramap :: (a -> b) -> (:.:) f g b -> (:.:) f g a
contramap a -> b
f (Comp1 f (g b)
fg) = f (g a) -> (:.:) f g a
forall k2 k1 (f :: k2 -> *) (g :: k1 -> k2) (p :: k1).
f (g p) -> (:.:) f g p
Comp1 ((g b -> g a) -> f (g b) -> f (g a)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap ((a -> b) -> g b -> g a
forall (f :: * -> *) a b. Contravariant f => (a -> b) -> f b -> f a
contramap a -> b
f) f (g b)
fg)

instance (Contravariant f, Contravariant g) => Contravariant (f :+: g) where
contramap :: (a -> b) -> (:+:) f g b -> (:+:) f g a
contramap a -> b
f (L1 f b
xs) = f a -> (:+:) f g a
forall k (f :: k -> *) (g :: k -> *) (p :: k). f p -> (:+:) f g p
L1 ((a -> b) -> f b -> f a
forall (f :: * -> *) a b. Contravariant f => (a -> b) -> f b -> f a
contramap a -> b
f f b
xs)
contramap a -> b
f (R1 g b
ys) = g a -> (:+:) f g a
forall k (f :: k -> *) (g :: k -> *) (p :: k). g p -> (:+:) f g p
R1 ((a -> b) -> g b -> g a
forall (f :: * -> *) a b. Contravariant f => (a -> b) -> f b -> f a
contramap a -> b
f g b
ys)

instance (Contravariant f, Contravariant g) => Contravariant (Sum f g) where
contramap :: (a -> b) -> Sum f g b -> Sum f g a
contramap a -> b
f (InL f b
xs) = f a -> Sum f g a
forall k (f :: k -> *) (g :: k -> *) (a :: k). f a -> Sum f g a
InL ((a -> b) -> f b -> f a
forall (f :: * -> *) a b. Contravariant f => (a -> b) -> f b -> f a
contramap a -> b
f f b
xs)
contramap a -> b
f (InR g b
ys) = g a -> Sum f g a
forall k (f :: k -> *) (g :: k -> *) (a :: k). g a -> Sum f g a
InR ((a -> b) -> g b -> g a
forall (f :: * -> *) a b. Contravariant f => (a -> b) -> f b -> f a
contramap a -> b
f g b
ys)

instance (Contravariant f, Contravariant g)
=> Contravariant (Product f g) where
contramap :: (a -> b) -> Product f g b -> Product f g a
contramap a -> b
f (Pair f b
a g b
b) = f a -> g a -> Product f g a
forall k (f :: k -> *) (g :: k -> *) (a :: k).
f a -> g a -> Product f g a
Pair ((a -> b) -> f b -> f a
forall (f :: * -> *) a b. Contravariant f => (a -> b) -> f b -> f a
contramap a -> b
f f b
a) ((a -> b) -> g b -> g a
forall (f :: * -> *) a b. Contravariant f => (a -> b) -> f b -> f a
contramap a -> b
f g b
b)

instance Contravariant (Const a) where
contramap :: (a -> b) -> Const a b -> Const a a
contramap a -> b
_ (Const a
a) = a -> Const a a
forall k a (b :: k). a -> Const a b
Const a
a

instance (Functor f, Contravariant g) => Contravariant (Compose f g) where
contramap :: (a -> b) -> Compose f g b -> Compose f g a
contramap a -> b
f (Compose f (g b)
fga) = f (g a) -> Compose f g a
forall k k (f :: k -> *) (g :: k -> k) (a :: k).
f (g a) -> Compose f g a
Compose ((g b -> g a) -> f (g b) -> f (g a)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap ((a -> b) -> g b -> g a
forall (f :: * -> *) a b. Contravariant f => (a -> b) -> f b -> f a
contramap a -> b
f) f (g b)
fga)

instance Contravariant Proxy where
contramap :: (a -> b) -> Proxy b -> Proxy a
contramap a -> b
_ Proxy b
_ = Proxy a
forall k (t :: k). Proxy t
Proxy

newtype Predicate 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.
instance Contravariant Predicate where
contramap :: (a -> b) -> Predicate b -> Predicate a
contramap a -> b
f Predicate b
g = (a -> Bool) -> Predicate a
forall a. (a -> Bool) -> Predicate a
Predicate ((a -> Bool) -> Predicate a) -> (a -> Bool) -> Predicate a
forall a b. (a -> b) -> a -> b
\$ Predicate b -> b -> Bool
forall a. Predicate a -> a -> Bool
getPredicate Predicate b
g (b -> Bool) -> (a -> b) -> a -> Bool
forall k (cat :: k -> k -> *) (b :: k) (c :: k) (a :: k).
Category cat =>
cat b c -> cat a b -> cat a c
. a -> b
f

instance Semigroup (Predicate a) where
Predicate a -> Bool
p <> :: Predicate a -> Predicate a -> Predicate a
<> Predicate a -> Bool
q = (a -> Bool) -> Predicate a
forall a. (a -> Bool) -> Predicate a
Predicate ((a -> Bool) -> Predicate a) -> (a -> Bool) -> Predicate a
forall a b. (a -> b) -> a -> b
\$ \a
a -> a -> Bool
p a
a Bool -> Bool -> Bool
&& a -> Bool
q a
a

instance Monoid (Predicate a) where
mempty :: Predicate a
mempty = (a -> Bool) -> Predicate a
forall a. (a -> Bool) -> Predicate a
Predicate ((a -> Bool) -> Predicate a) -> (a -> Bool) -> Predicate a
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.
newtype Comparison a = Comparison { Comparison a -> a -> a -> Ordering
getComparison :: a -> a -> Ordering }

deriving instance Semigroup (Comparison a)
deriving instance Monoid (Comparison a)

-- | A 'Comparison' is a 'Contravariant' 'Functor', because 'contramap' can
-- apply its function argument to each input of the comparison function.
instance Contravariant Comparison where
contramap :: (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
\$ (b -> b -> Ordering) -> (a -> b) -> a -> a -> Ordering
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 :: Ord a => 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.
newtype Equivalence 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.
instance Contravariant Equivalence where
contramap :: (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
getEquivalence Equivalence b
g) a -> b
f

instance Semigroup (Equivalence a) where
Equivalence a -> a -> Bool
p <> :: Equivalence a -> Equivalence a -> Equivalence a
<> Equivalence a -> a -> Bool
q = (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
\$ \a
a a
b -> a -> a -> Bool
p a
a a
b Bool -> Bool -> Bool
&& a -> a -> Bool
q a
a a
b

instance Monoid (Equivalence a) where
mempty :: 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 :: Eq a => 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
p) = (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
\$ \a
a a
b -> a -> a -> Ordering
p a
a a
b Ordering -> Ordering -> Bool
forall a. Eq a => a -> a -> Bool
== Ordering
EQ

-- | Dual function arrows.
newtype Op a b = Op { Op a b -> b -> a
getOp :: b -> a }

deriving instance Semigroup a => Semigroup (Op a b)
deriving instance Monoid a => Monoid (Op a b)

instance Category Op where
id :: 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
. Op b -> a
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 =>
cat b c -> cat a b -> cat a c
. c -> b
f)

instance Contravariant (Op a) where
contramap :: (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
forall a b. Op a b -> b -> a
getOp Op a b
g (b -> a) -> (a -> b) -> a -> a
forall k (cat :: k -> k -> *) (b :: k) (c :: k) (a :: k).
Category cat =>
cat b c -> cat a b -> cat a c
. a -> b
f)

instance Num a => Num (Op a b) where
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. Num a => a -> a -> a
+ b -> a
g b
a
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. Num a => a -> a -> a
* b -> a
g b
a
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. Num a => a -> a -> a
- b -> a
g b
a
abs :: Op a b -> Op a b
abs (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. Num a => a -> a
abs (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
signum :: Op a b -> Op a b
signum (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. Num a => a -> a
signum (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
fromInteger :: Integer -> Op a b
fromInteger = (b -> a) -> Op a b
forall a b. (b -> a) -> Op a b
Op ((b -> a) -> Op a b) -> (Integer -> b -> a) -> Integer -> Op a b
forall k (cat :: k -> k -> *) (b :: k) (c :: k) (a :: k).
Category cat =>
cat b c -> cat a b -> cat a c
. a -> b -> a
forall a b. a -> b -> a
const (a -> b -> a) -> (Integer -> a) -> Integer -> b -> a
forall k (cat :: k -> k -> *) (b :: k) (c :: k) (a :: k).
Category cat =>
cat b c -> cat a b -> cat a c
. Integer -> a
forall a. Num a => Integer -> a
fromInteger

instance Fractional a => Fractional (Op a b) where
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. Fractional a => a -> a -> a
/ b -> a
g b
a
recip :: Op a b -> Op a b
recip (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. Fractional a => a -> a
recip (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
fromRational :: Rational -> Op a b
fromRational = (b -> a) -> Op a b
forall a b. (b -> a) -> Op a b
Op ((b -> a) -> Op a b) -> (Rational -> b -> a) -> Rational -> Op a b
forall k (cat :: k -> k -> *) (b :: k) (c :: k) (a :: k).
Category cat =>
cat b c -> cat a b -> cat a c
. a -> b -> a
forall a b. a -> b -> a
const (a -> b -> a) -> (Rational -> a) -> Rational -> b -> a
forall k (cat :: k -> k -> *) (b :: k) (c :: k) (a :: k).
Category cat =>
cat b c -> cat a b -> cat a c
. Rational -> a
forall a. Fractional a => Rational -> a
fromRational

instance Floating a => Floating (Op a b) where
pi :: Op a b
pi = (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 -> b -> a
forall a b. a -> b -> a
const a
forall a. Floating a => a
pi
exp :: Op a b -> Op a b
exp (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
exp (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
sqrt :: Op a b -> Op a b
sqrt (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
sqrt (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
log :: Op a b -> Op a b
log (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
log (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
sin :: Op a b -> Op a b
sin (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
sin (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
tan :: Op a b -> Op a b
tan (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
tan (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
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)
```