src/Data/Semigroup.hs

{-# LANGUAGE CPP #-}
#ifdef LANGUAGE_DeriveDataTypeable
{-# LANGUAGE DeriveDataTypeable #-}
#endif
#if defined(__GLASGOW_HASKELL__) && __GLASGOW_HASKELL__ >= 702
{-# LANGUAGE DefaultSignatures #-}
{-# LANGUAGE Trustworthy #-}
#endif
-----------------------------------------------------------------------------
-- |
-- Module : Data.Semigroup
-- Copyright : (C) 2011-2013 Edward Kmett,
-- License : BSD-style (see the file LICENSE)
--
-- Maintainer : Edward Kmett <ekmett@gmail.com>
-- Stability : provisional
-- Portability : portable
--
-- In mathematics, a semigroup is an algebraic structure consisting of a
-- set together with an associative binary operation. A semigroup
-- generalizes a monoid in that there might not exist an identity
-- element. It also (originally) generalized a group (a monoid with all
-- inverses) to a type where every element did not have to have an inverse,
-- thus the name semigroup.
--
-- The use of @(\<\>)@ in this module conflicts with an operator with the same
-- name that is being exported by Data.Monoid. However, this package
-- re-exports (most of) the contents of Data.Monoid, so to use semigroups
-- and monoids in the same package just
--
-- > import Data.Semigroup
--
----------------------------------------------------------------------------
module Data.Semigroup (
 Semigroup(..)
 -- * Semigroups
 , Min(..)
 , Max(..)
 , First(..)
 , Last(..)
 , WrappedMonoid(..)
 , timesN
 -- * Re-exported monoids from Data.Monoid
 , Monoid(..)
 , Dual(..)
 , Endo(..)
 , All(..)
 , Any(..)
 , Sum(..)
 , Product(..)
 -- * A better monoid for Maybe
 , Option(..)
 , option
 -- * Difference lists of a semigroup
 , diff
 , cycle1
 ) where

import Prelude hiding (foldr1)
import Data.Monoid (Monoid(..),Dual(..),Endo(..),All(..),Any(..),Sum(..),Product(..),Endo(..))
import Control.Applicative
import Control.Monad
import Control.Monad.Fix
import qualified Data.Monoid as Monoid
import Data.Foldable
import Data.Traversable
import Data.List.NonEmpty

import Numeric.Natural.Internal
import Data.Sequence (Seq, (><))
import Data.Set (Set)
import Data.IntSet (IntSet)
import Data.Map (Map)
import Data.IntMap (IntMap)

#ifndef BASE2
import Data.ByteString as Strict
import Data.ByteString.Lazy as Lazy
import qualified Data.Text as Strict
import qualified Data.Text.Lazy as Lazy
import Data.Hashable
import Data.HashMap.Lazy as Lazy
import Data.HashSet
#endif

#ifdef LANGUAGE_DeriveDataTypeable
import Data.Data
#endif

infixr 6 <>

class Semigroup a where
 -- | An associative operation.
 --
 -- > (a <> b) <> c = a <> (b <> c)
 --
 -- If @a@ is also a 'Monoid' we further require
 --
 -- > (<>) = mappend
 (<>) :: a -> a -> a
#if defined(__GLASGOW_HASKELL__) && __GLASGOW_HASKELL__ >= 702
 default (<>) :: Monoid a => a -> a -> a
 (<>) = mappend
#endif

 -- | Reduce a non-empty list with @\<\>@
 --
 -- The default definition should be sufficient, but this can be overridden for efficiency.
 --
 sconcat :: NonEmpty a -> a
 sconcat (a :| as) = go a as where
 go b (c:cs) = b <> go c cs
 go b [] = b

 -- | Repeat a value (n + 1) times.
 --
 -- > times1p n a = a <> a <> ... <> a -- using <> n times
 --
 -- The default definition uses peasant multiplication, exploiting associativity to only
 -- require /O(log n)/ uses of @\<\>@.
 --
 -- See also 'times'.

 times1p :: Whole n => n -> a -> a
 times1p y0 x0 = f x0 (1 Prelude.+ y0)
 where
 f x y
 | even y = f (x <> x) (y `quot` 2)
 | y == 1 = x
 | otherwise = g (x <> x) (unsafePred y `quot` 2) x
 g x y z
 | even y = g (x <> x) (y `quot` 2) z
 | y == 1 = x <> z
 | otherwise = g (x <> x) (unsafePred y `quot` 2) (x <> z)
 {-# INLINE times1p #-}

-- | A generalization of 'Data.List.cycle' to an arbitrary 'Semigroup'.
-- May fail to terminate for some values in some semigroups.
cycle1 :: Semigroup m => m -> m
cycle1 xs = xs' where xs' = xs <> xs'

instance Semigroup () where
 _ <> _ = ()
 sconcat _ = ()
 times1p _ _ = ()

instance Semigroup b => Semigroup (a -> b) where
 f <> g = \a -> f a <> g a
 times1p n f e = times1p n (f e)

instance Semigroup [a] where
 (<>) = (++)
 times1p n x = rep n where
 rep 0 = x
 rep i = x ++ rep (i - 1)

instance Semigroup a => Semigroup (Maybe a) where
 Nothing <> b = b
 a <> Nothing = a
 Just a <> Just b = Just (a <> b)

instance Semigroup (Either a b) where
 Left _ <> b = b
 a <> _ = a

instance (Semigroup a, Semigroup b) => Semigroup (a, b) where
 (a,b) <> (a',b') = (a<>a',b<>b')
 times1p n (a,b) = (times1p n a, times1p n b)

instance (Semigroup a, Semigroup b, Semigroup c) => Semigroup (a, b, c) where
 (a,b,c) <> (a',b',c') = (a<>a',b<>b',c<>c')
 times1p n (a,b,c) = (times1p n a, times1p n b, times1p n c)

instance (Semigroup a, Semigroup b, Semigroup c, Semigroup d) => Semigroup (a, b, c, d) where
 (a,b,c,d) <> (a',b',c',d') = (a<>a',b<>b',c<>c',d<>d')
 times1p n (a,b,c,d) = (times1p n a, times1p n b, times1p n c, times1p n d)

instance (Semigroup a, Semigroup b, Semigroup c, Semigroup d, Semigroup e) => Semigroup (a, b, c, d, e) where
 (a,b,c,d,e) <> (a',b',c',d',e') = (a<>a',b<>b',c<>c',d<>d',e<>e')
 times1p n (a,b,c,d,e) = (times1p n a, times1p n b, times1p n c, times1p n d, times1p n e)

instance Semigroup Ordering where
 LT <> _ = LT
 EQ <> y = y
 GT <> _ = GT

instance Semigroup a => Semigroup (Dual a) where
 Dual a <> Dual b = Dual (b <> a)
 times1p n (Dual a) = Dual (times1p n a)

instance Semigroup (Endo a) where
 Endo f <> Endo g = Endo (f . g)

instance Semigroup All where
 All a <> All b = All (a && b)
 times1p _ a = a

instance Semigroup Any where
 Any a <> Any b = Any (a || b)
 times1p _ a = a

instance Num a => Semigroup (Sum a) where
 Sum a <> Sum b = Sum (a + b)

instance Num a => Semigroup (Product a) where
 Product a <> Product b = Product (a * b)

instance Semigroup a => Semigroup (Const a b) where
 Const a <> Const b = Const (a <> b)

#if MIN_VERSION_base(3,0,0)
instance Semigroup (Monoid.First a) where
 Monoid.First Nothing <> b = b
 a <> _ = a
 times1p _ a = a

instance Semigroup (Monoid.Last a) where
 a <> Monoid.Last Nothing = a
 _ <> b = b
 times1p _ a = a
#endif

instance Semigroup (NonEmpty a) where
 (a :| as) <> ~(b :| bs) = a :| (as ++ b : bs)

newtype Min a = Min { getMin :: a } deriving
 ( Eq, Ord, Bounded, Show, Read
#ifdef LANGUAGE_DeriveDataTypeable
 , Data, Typeable
#endif
 )

instance Ord a => Semigroup (Min a) where
 Min a <> Min b = Min (a `min` b)
 times1p _ a = a

instance (Ord a, Bounded a) => Monoid (Min a) where
 mempty = maxBound
 mappend = (<>)

newtype Max a = Max { getMax :: a } deriving
 ( Eq, Ord, Bounded, Show, Read
#ifdef LANGUAGE_DeriveDataTypeable
 , Data, Typeable
#endif
 )

instance Ord a => Semigroup (Max a) where
 Max a <> Max b = Max (a `max` b)
 times1p _ a = a

instance (Ord a, Bounded a) => Monoid (Max a) where
 mempty = minBound
 mappend = (<>)

-- | Use @'Option' ('First' a)@ to get the behavior of 'Data.Monoid.First' from @Data.Monoid@.
newtype First a = First { getFirst :: a } deriving
 ( Eq, Ord, Bounded, Show, Read
#ifdef LANGUAGE_DeriveDataTypeable
 , Data
 , Typeable
#endif
 )

instance Semigroup (First a) where
 a <> _ = a
 times1p _ a = a

-- | Use @'Option' ('Last' a)@ to get the behavior of 'Data.Monoid.Last' from @Data.Monoid@
newtype Last a = Last { getLast :: a } deriving
 ( Eq, Ord, Bounded, Show, Read
#ifdef LANGUAGE_DeriveDataTypeable
 , Data, Typeable
#endif
 )

instance Semigroup (Last a) where
 _ <> b = b
 times1p _ a = a

-- (==)/XNOR on Bool forms a 'Semigroup', but has no good name

#ifndef BASE2
instance Semigroup Strict.ByteString where
 (<>) = mappend

instance Semigroup Lazy.ByteString where
 (<>) = mappend

instance Semigroup Strict.Text where
 (<>) = mappend

instance Semigroup Lazy.Text where
 (<>) = mappend

instance (Hashable k, Eq k) => Semigroup (Lazy.HashMap k a) where
 (<>) = mappend

instance (Hashable a, Eq a) => Semigroup (HashSet a) where
 (<>) = mappend
 times1p _ a = a
#endif

-- | Provide a Semigroup for an arbitrary Monoid.
newtype WrappedMonoid m = WrapMonoid
 { unwrapMonoid :: m } deriving
 ( Eq, Ord, Bounded, Show, Read
#ifdef LANGUAGE_DeriveDataTypeable
 , Data, Typeable
#endif
 )

instance Monoid m => Semigroup (WrappedMonoid m) where
 WrapMonoid a <> WrapMonoid b = WrapMonoid (a `mappend` b)

instance Monoid m => Monoid (WrappedMonoid m) where
 mempty = WrapMonoid mempty
 WrapMonoid a `mappend` WrapMonoid b = WrapMonoid (a `mappend` b)

-- | Repeat a value @n@ times.
--
-- > times n a = a <> a <> ... <> a -- using <> (n-1) times
--
-- Implemented using 'times1p'.
timesN :: (Whole n, Monoid a) => n -> a -> a
timesN n x | n == 0 = mempty
 | otherwise = unwrapMonoid . times1p (unsafePred n) . WrapMonoid $ x
{-# INLINE timesN #-}


-- | Option is effectively 'Maybe' with a better instance of 'Monoid', built off of an underlying 'Semigroup'
-- instead of an underlying 'Monoid'. Ideally, this type would not exist at all and we would just fix the 'Monoid' intance of 'Maybe'
newtype Option a = Option
 { getOption :: Maybe a } deriving
 ( Eq, Ord, Show, Read
#ifdef LANGUAGE_DeriveDataTypeable
 , Data, Typeable
#endif
 )

instance Functor Option where
 fmap f (Option a) = Option (fmap f a)

instance Applicative Option where
 pure a = Option (Just a)
 Option a <*> Option b = Option (a <*> b)

instance Monad Option where
 return = pure

 Option (Just a) >>= k = k a
 _ >>= _ = Option Nothing

 Option Nothing >> _ = Option Nothing
 _ >> b = b

instance Alternative Option where
 empty = Option Nothing
 Option Nothing <|> b = b
 a <|> _ = a

instance MonadPlus Option where
 mzero = Option Nothing
 mplus = (<|>)

instance MonadFix Option where
 mfix f = Option (mfix (getOption . f))

instance Foldable Option where
 foldMap f (Option (Just m)) = f m
 foldMap _ (Option Nothing) = mempty

instance Traversable Option where
 traverse f (Option (Just a)) = Option . Just <$> f a
 traverse _ (Option Nothing) = pure (Option Nothing)

-- | Fold an 'Option' case-wise, just like 'maybe'.
option :: b -> (a -> b) -> Option a -> b
option n j (Option m) = maybe n j m

instance Semigroup a => Semigroup (Option a) where
 Option a <> Option b = Option (a <> b)

instance Semigroup a => Monoid (Option a) where
 mempty = Option Nothing
 Option a `mappend` Option b = Option (a <> b)

-- | This lets you use a difference list of a Semigroup as a Monoid.
diff :: Semigroup m => m -> Endo m
diff = Endo . (<>)

instance Semigroup (Seq a) where
 (<>) = (><)

instance Semigroup IntSet where
 (<>) = mappend
 times1p _ a = a

instance Ord a => Semigroup (Set a) where
 (<>) = mappend
 times1p _ a = a

instance Semigroup (IntMap v) where
 (<>) = mappend
 times1p _ a = a

instance Ord k => Semigroup (Map k v) where
 (<>) = mappend
 times1p _ a = a