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{-# LANGUAGE CPP #-}
#ifdef LANGUAGE_DeriveDataTypeable
{-# LANGUAGE DeriveDataTypeable #-}
#endif
-----------------------------------------------------------------------------
-- |
-- Module : Data.Semigroup
-- Copyright : (C) 2011 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(..)
-- * 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)
#ifdef LANGUAGE_DeriveDataTypeable
import Data.Data
#endif
infixr 6 <>
class Semigroup a where
-- | An associative operation.
--
-- > (a <> b) <> c = a <> (b <> c)
(<>) :: a -> a -> a
-- | 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 @\<\>@.
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
(<>) = (++)
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 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)
#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'
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'
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
-- | 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)
-- | 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 = empty
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)
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 = empty
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
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