Safe Haskell	Safe
Language	Haskell2010

Papa.Base.Export.Data.Monoid

Synopsis

Documentation

class Monoid a where #

The class of monoids (types with an associative binary operation that has an identity). Instances should satisfy the following laws:

```
mappend mempty x = x
```
```
mappend x mempty = x
```

mappend x (mappend y z) = mappend (mappend x y) z

```
mconcat = foldr mappend mempty
```

The method names refer to the monoid of lists under concatenation, but there are many other instances.

Some types can be viewed as a monoid in more than one way, e.g. both addition and multiplication on numbers. In such cases we often define newtypes and make those instances of Monoid, e.g. Sum and Product.

Minimal complete definition

mempty, mappend

Methods

mempty :: a #

Identity of mappend

mappend :: a -> a -> a #

An associative operation

Instances

Monoid Ordering	Since: 2.1
Methods mempty :: Ordering # mappend :: Ordering -> Ordering -> Ordering # mconcat :: [Ordering] -> Ordering #
Monoid ()	Since: 2.1
Methods mempty :: () # mappend :: () -> () -> () # mconcat :: [()] -> () #
Monoid All	Since: 2.1
Methods mempty :: All # mappend :: All -> All -> All # mconcat :: [All] -> All #
Monoid Any	Since: 2.1
Methods mempty :: Any # mappend :: Any -> Any -> Any # mconcat :: [Any] -> Any #
Monoid [a]	Since: 2.1
Methods mempty :: [a] # mappend :: [a] -> [a] -> [a] # mconcat :: [[a]] -> [a] #
Monoid a => Monoid (Maybe a)	Lift a semigroup into `Maybe` forming a `Monoid` according to http://en.wikipedia.org/wiki/Monoid: "Any semigroup `S` may be turned into a monoid simply by adjoining an element `e` not in `S` and defining `ee = e` and `es = s = se` for all `s ∈ S`." Since there used to be no "Semigroup" typeclass providing just `mappend`, we use `Monoid` instead. Since: 2.1*
Methods mempty :: Maybe a # mappend :: Maybe a -> Maybe a -> Maybe a # mconcat :: [Maybe a] -> Maybe a #
Monoid a => Monoid (IO a)	Since: 4.9.0.0
Methods mempty :: IO a # mappend :: IO a -> IO a -> IO a # mconcat :: [IO a] -> IO a #
(Ord a, Bounded a) => Monoid (Min a)	Since: 4.9.0.0
Methods mempty :: Min a # mappend :: Min a -> Min a -> Min a # mconcat :: [Min a] -> Min a #
(Ord a, Bounded a) => Monoid (Max a)	Since: 4.9.0.0
Methods mempty :: Max a # mappend :: Max a -> Max a -> Max a # mconcat :: [Max a] -> Max a #
Monoid m => Monoid (WrappedMonoid m)	Since: 4.9.0.0
Methods mempty :: WrappedMonoid m # mappend :: WrappedMonoid m -> WrappedMonoid m -> WrappedMonoid m # mconcat :: [WrappedMonoid m] -> WrappedMonoid m #
Semigroup a => Monoid (Option a)	Since: 4.9.0.0
Methods mempty :: Option a # mappend :: Option a -> Option a -> Option a # mconcat :: [Option a] -> Option a #
Monoid a => Monoid (Dual a)	Since: 2.1
Methods mempty :: Dual a # mappend :: Dual a -> Dual a -> Dual a # mconcat :: [Dual a] -> Dual a #
Monoid (Endo a)	Since: 2.1
Methods mempty :: Endo a # mappend :: Endo a -> Endo a -> Endo a # mconcat :: [Endo a] -> Endo a #
Num a => Monoid (Sum a)	Since: 2.1
Methods mempty :: Sum a # mappend :: Sum a -> Sum a -> Sum a # mconcat :: [Sum a] -> Sum a #
Num a => Monoid (Product a)	Since: 2.1
Methods mempty :: Product a # mappend :: Product a -> Product a -> Product a # mconcat :: [Product a] -> Product a #
Monoid (First a)	Since: 2.1
Methods mempty :: First a # mappend :: First a -> First a -> First a # mconcat :: [First a] -> First a #
Monoid (Last a)	Since: 2.1
Methods mempty :: Last a # mappend :: Last a -> Last a -> Last a # mconcat :: [Last a] -> Last a #
Monoid b => Monoid (a -> b)	Since: 2.1
Methods mempty :: a -> b # mappend :: (a -> b) -> (a -> b) -> a -> b # mconcat :: [a -> b] -> a -> b #
(Monoid a, Monoid b) => Monoid (a, b)	Since: 2.1
Methods mempty :: (a, b) # mappend :: (a, b) -> (a, b) -> (a, b) # mconcat :: [(a, b)] -> (a, b) #
(Monoid a, Monoid b, Monoid c) => Monoid (a, b, c)	Since: 2.1
Methods mempty :: (a, b, c) # mappend :: (a, b, c) -> (a, b, c) -> (a, b, c) # mconcat :: [(a, b, c)] -> (a, b, c) #
Monoid a => Monoid (Const k a b)
Methods mempty :: Const k a b # mappend :: Const k a b -> Const k a b -> Const k a b # mconcat :: [Const k a b] -> Const k a b #
Alternative f => Monoid (Alt * f a)	Since: 4.8.0.0
Methods mempty :: Alt * f a # mappend :: Alt * f a -> Alt * f a -> Alt * f a # mconcat :: [Alt * f a] -> Alt * f a #
(Monoid a, Monoid b, Monoid c, Monoid d) => Monoid (a, b, c, d)	Since: 2.1
Methods mempty :: (a, b, c, d) # mappend :: (a, b, c, d) -> (a, b, c, d) -> (a, b, c, d) # mconcat :: [(a, b, c, d)] -> (a, b, c, d) #
(Monoid a, Monoid b, Monoid c, Monoid d, Monoid e) => Monoid (a, b, c, d, e)	Since: 2.1
Methods mempty :: (a, b, c, d, e) # mappend :: (a, b, c, d, e) -> (a, b, c, d, e) -> (a, b, c, d, e) # mconcat :: [(a, b, c, d, e)] -> (a, b, c, d, e) #

data Dual a :: * -> * #

The dual of a Monoid, obtained by swapping the arguments of mappend.

Instances

Monad Dual	Since: 4.8.0.0
Methods (>>=) :: Dual a -> (a -> Dual b) -> Dual b # (>>) :: Dual a -> Dual b -> Dual b # return :: a -> Dual a # fail :: String -> Dual a #
Functor Dual	Since: 4.8.0.0
Methods fmap :: (a -> b) -> Dual a -> Dual b # (<$) :: a -> Dual b -> Dual a #
Applicative Dual	Since: 4.8.0.0
Methods pure :: a -> Dual a # (<>) :: Dual (a -> b) -> Dual a -> Dual b # liftA2 :: (a -> b -> c) -> Dual a -> Dual b -> Dual c # (>) :: Dual a -> Dual b -> Dual b # (<*) :: Dual a -> Dual b -> Dual a #
Foldable Dual	Since: 4.8.0.0
Methods fold :: Monoid m => Dual m -> m # foldMap :: Monoid m => (a -> m) -> Dual a -> m # foldr :: (a -> b -> b) -> b -> Dual a -> b # foldr' :: (a -> b -> b) -> b -> Dual a -> b # foldl :: (b -> a -> b) -> b -> Dual a -> b # foldl' :: (b -> a -> b) -> b -> Dual a -> b # foldr1 :: (a -> a -> a) -> Dual a -> a # foldl1 :: (a -> a -> a) -> Dual a -> a # toList :: Dual a -> [a] # null :: Dual a -> Bool # length :: Dual a -> Int # elem :: Eq a => a -> Dual a -> Bool # maximum :: Ord a => Dual a -> a # minimum :: Ord a => Dual a -> a # sum :: Num a => Dual a -> a # product :: Num a => Dual a -> a #
Traversable Dual	Since: 4.8.0.0
Methods traverse :: Applicative f => (a -> f b) -> Dual a -> f (Dual b) # sequenceA :: Applicative f => Dual (f a) -> f (Dual a) # mapM :: Monad m => (a -> m b) -> Dual a -> m (Dual b) # sequence :: Monad m => Dual (m a) -> m (Dual a) #
Bounded a => Bounded (Dual a)
Methods minBound :: Dual a # maxBound :: Dual a #
Eq a => Eq (Dual a)
Methods (==) :: Dual a -> Dual a -> Bool # (/=) :: Dual a -> Dual a -> Bool #
Ord a => Ord (Dual a)
Methods compare :: Dual a -> Dual a -> Ordering # (<) :: Dual a -> Dual a -> Bool # (<=) :: Dual a -> Dual a -> Bool # (>) :: Dual a -> Dual a -> Bool # (>=) :: Dual a -> Dual a -> Bool # max :: Dual a -> Dual a -> Dual a # min :: Dual a -> Dual a -> Dual a #
Read a => Read (Dual a)
Methods readsPrec :: Int -> ReadS (Dual a) # readList :: ReadS [Dual a] # readPrec :: ReadPrec (Dual a) # readListPrec :: ReadPrec [Dual a] #
Show a => Show (Dual a)
Methods showsPrec :: Int -> Dual a -> ShowS # show :: Dual a -> String # showList :: [Dual a] -> ShowS #
Generic (Dual a)
Associated Types type Rep (Dual a) :: * -> * # Methods from :: Dual a -> Rep (Dual a) x # to :: Rep (Dual a) x -> Dual a #
Semigroup a => Semigroup (Dual a)	Since: 4.9.0.0
Methods (<>) :: Dual a -> Dual a -> Dual a # sconcat :: NonEmpty (Dual a) -> Dual a # stimes :: Integral b => b -> Dual a -> Dual a #
Monoid a => Monoid (Dual a)	Since: 2.1
Methods mempty :: Dual a # mappend :: Dual a -> Dual a -> Dual a # mconcat :: [Dual a] -> Dual a #
Generic1 * Dual
Associated Types type Rep1 Dual (f :: Dual -> ) :: k -> # Methods from1 :: f a -> Rep1 Dual f a # to1 :: Rep1 Dual f a -> f a #
type Rep (Dual a)
type Rep (Dual a) = D1 * (MetaData "Dual" "Data.Monoid" "base" True) (C1 * (MetaCons "Dual" PrefixI True) (S1 * (MetaSel (Just Symbol "getDual") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 * a)))
type Rep1 * Dual
type Rep1 * Dual = D1 * (MetaData "Dual" "Data.Monoid" "base" True) (C1 * (MetaCons "Dual" PrefixI True) (S1 * (MetaSel (Just Symbol "getDual") NoSourceUnpackedness NoSourceStrictness DecidedLazy) Par1))

data Endo a :: * -> * #

The monoid of endomorphisms under composition.

Instances

Generic (Endo a)
Associated Types type Rep (Endo a) :: * -> * # Methods from :: Endo a -> Rep (Endo a) x # to :: Rep (Endo a) x -> Endo a #
Semigroup (Endo a)	Since: 4.9.0.0
Methods (<>) :: Endo a -> Endo a -> Endo a # sconcat :: NonEmpty (Endo a) -> Endo a # stimes :: Integral b => b -> Endo a -> Endo a #
Monoid (Endo a)	Since: 2.1
Methods mempty :: Endo a # mappend :: Endo a -> Endo a -> Endo a # mconcat :: [Endo a] -> Endo a #
type Rep (Endo a)
type Rep (Endo a) = D1 * (MetaData "Endo" "Data.Monoid" "base" True) (C1 * (MetaCons "Endo" PrefixI True) (S1 * (MetaSel (Just Symbol "appEndo") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 * (a -> a))))

data All :: * #

Boolean monoid under conjunction (&&).

Instances

Bounded All
Methods minBound :: All # maxBound :: All #
Eq All
Methods (==) :: All -> All -> Bool # (/=) :: All -> All -> Bool #
Ord All
Methods compare :: All -> All -> Ordering # (<) :: All -> All -> Bool # (<=) :: All -> All -> Bool # (>) :: All -> All -> Bool # (>=) :: All -> All -> Bool # max :: All -> All -> All # min :: All -> All -> All #
Read All
Methods readsPrec :: Int -> ReadS All # readList :: ReadS [All] # readPrec :: ReadPrec All # readListPrec :: ReadPrec [All] #
Show All
Methods showsPrec :: Int -> All -> ShowS # show :: All -> String # showList :: [All] -> ShowS #
Generic All
Associated Types type Rep All :: * -> * # Methods from :: All -> Rep All x # to :: Rep All x -> All #
Semigroup All	Since: 4.9.0.0
Methods (<>) :: All -> All -> All # sconcat :: NonEmpty All -> All # stimes :: Integral b => b -> All -> All #
Monoid All	Since: 2.1
Methods mempty :: All # mappend :: All -> All -> All # mconcat :: [All] -> All #
type Rep All
type Rep All = D1 * (MetaData "All" "Data.Monoid" "base" True) (C1 * (MetaCons "All" PrefixI True) (S1 * (MetaSel (Just Symbol "getAll") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 * Bool)))

data Any :: * #

Boolean monoid under disjunction (||).

Instances

Bounded Any
Methods minBound :: Any # maxBound :: Any #
Eq Any
Methods (==) :: Any -> Any -> Bool # (/=) :: Any -> Any -> Bool #
Ord Any
Methods compare :: Any -> Any -> Ordering # (<) :: Any -> Any -> Bool # (<=) :: Any -> Any -> Bool # (>) :: Any -> Any -> Bool # (>=) :: Any -> Any -> Bool # max :: Any -> Any -> Any # min :: Any -> Any -> Any #
Read Any
Methods readsPrec :: Int -> ReadS Any # readList :: ReadS [Any] # readPrec :: ReadPrec Any # readListPrec :: ReadPrec [Any] #
Show Any
Methods showsPrec :: Int -> Any -> ShowS # show :: Any -> String # showList :: [Any] -> ShowS #
Generic Any
Associated Types type Rep Any :: * -> * # Methods from :: Any -> Rep Any x # to :: Rep Any x -> Any #
Semigroup Any	Since: 4.9.0.0
Methods (<>) :: Any -> Any -> Any # sconcat :: NonEmpty Any -> Any # stimes :: Integral b => b -> Any -> Any #
Monoid Any	Since: 2.1
Methods mempty :: Any # mappend :: Any -> Any -> Any # mconcat :: [Any] -> Any #
type Rep Any
type Rep Any = D1 * (MetaData "Any" "Data.Monoid" "base" True) (C1 * (MetaCons "Any" PrefixI True) (S1 * (MetaSel (Just Symbol "getAny") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 * Bool)))

data Sum a :: * -> * #

Monoid under addition.

Instances

Monad Sum	Since: 4.8.0.0
Methods (>>=) :: Sum a -> (a -> Sum b) -> Sum b # (>>) :: Sum a -> Sum b -> Sum b # return :: a -> Sum a # fail :: String -> Sum a #
Functor Sum	Since: 4.8.0.0
Methods fmap :: (a -> b) -> Sum a -> Sum b # (<$) :: a -> Sum b -> Sum a #
Applicative Sum	Since: 4.8.0.0
Methods pure :: a -> Sum a # (<>) :: Sum (a -> b) -> Sum a -> Sum b # liftA2 :: (a -> b -> c) -> Sum a -> Sum b -> Sum c # (>) :: Sum a -> Sum b -> Sum b # (<*) :: Sum a -> Sum b -> Sum a #
Foldable Sum	Since: 4.8.0.0
Methods fold :: Monoid m => Sum m -> m # foldMap :: Monoid m => (a -> m) -> Sum a -> m # foldr :: (a -> b -> b) -> b -> Sum a -> b # foldr' :: (a -> b -> b) -> b -> Sum a -> b # foldl :: (b -> a -> b) -> b -> Sum a -> b # foldl' :: (b -> a -> b) -> b -> Sum a -> b # foldr1 :: (a -> a -> a) -> Sum a -> a # foldl1 :: (a -> a -> a) -> Sum a -> a # toList :: Sum a -> [a] # null :: Sum a -> Bool # length :: Sum a -> Int # elem :: Eq a => a -> Sum a -> Bool # maximum :: Ord a => Sum a -> a # minimum :: Ord a => Sum a -> a # sum :: Num a => Sum a -> a # product :: Num a => Sum a -> a #
Traversable Sum	Since: 4.8.0.0
Methods traverse :: Applicative f => (a -> f b) -> Sum a -> f (Sum b) # sequenceA :: Applicative f => Sum (f a) -> f (Sum a) # mapM :: Monad m => (a -> m b) -> Sum a -> m (Sum b) # sequence :: Monad m => Sum (m a) -> m (Sum a) #
Bounded a => Bounded (Sum a)
Methods minBound :: Sum a # maxBound :: Sum a #
Eq a => Eq (Sum a)
Methods (==) :: Sum a -> Sum a -> Bool # (/=) :: Sum a -> Sum a -> Bool #
Num a => Num (Sum a)
Methods (+) :: Sum a -> Sum a -> Sum a # (-) :: Sum a -> Sum a -> Sum a # (*) :: Sum a -> Sum a -> Sum a # negate :: Sum a -> Sum a # abs :: Sum a -> Sum a # signum :: Sum a -> Sum a # fromInteger :: Integer -> Sum a #
Ord a => Ord (Sum a)
Methods compare :: Sum a -> Sum a -> Ordering # (<) :: Sum a -> Sum a -> Bool # (<=) :: Sum a -> Sum a -> Bool # (>) :: Sum a -> Sum a -> Bool # (>=) :: Sum a -> Sum a -> Bool # max :: Sum a -> Sum a -> Sum a # min :: Sum a -> Sum a -> Sum a #
Read a => Read (Sum a)
Methods readsPrec :: Int -> ReadS (Sum a) # readList :: ReadS [Sum a] # readPrec :: ReadPrec (Sum a) # readListPrec :: ReadPrec [Sum a] #
Show a => Show (Sum a)
Methods showsPrec :: Int -> Sum a -> ShowS # show :: Sum a -> String # showList :: [Sum a] -> ShowS #
Generic (Sum a)
Associated Types type Rep (Sum a) :: * -> * # Methods from :: Sum a -> Rep (Sum a) x # to :: Rep (Sum a) x -> Sum a #
Num a => Semigroup (Sum a)	Since: 4.9.0.0
Methods (<>) :: Sum a -> Sum a -> Sum a # sconcat :: NonEmpty (Sum a) -> Sum a # stimes :: Integral b => b -> Sum a -> Sum a #
Num a => Monoid (Sum a)	Since: 2.1
Methods mempty :: Sum a # mappend :: Sum a -> Sum a -> Sum a # mconcat :: [Sum a] -> Sum a #
Generic1 * Sum
Associated Types type Rep1 Sum (f :: Sum -> ) :: k -> # Methods from1 :: f a -> Rep1 Sum f a # to1 :: Rep1 Sum f a -> f a #
type Rep (Sum a)
type Rep (Sum a) = D1 * (MetaData "Sum" "Data.Monoid" "base" True) (C1 * (MetaCons "Sum" PrefixI True) (S1 * (MetaSel (Just Symbol "getSum") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 * a)))
type Rep1 * Sum
type Rep1 * Sum = D1 * (MetaData "Sum" "Data.Monoid" "base" True) (C1 * (MetaCons "Sum" PrefixI True) (S1 * (MetaSel (Just Symbol "getSum") NoSourceUnpackedness NoSourceStrictness DecidedLazy) Par1))

data Product a :: * -> * #

Monoid under multiplication.

Instances

Monad Product	Since: 4.8.0.0
Methods (>>=) :: Product a -> (a -> Product b) -> Product b # (>>) :: Product a -> Product b -> Product b # return :: a -> Product a # fail :: String -> Product a #
Functor Product	Since: 4.8.0.0
Methods fmap :: (a -> b) -> Product a -> Product b # (<$) :: a -> Product b -> Product a #
Applicative Product	Since: 4.8.0.0
Methods pure :: a -> Product a # (<>) :: Product (a -> b) -> Product a -> Product b # liftA2 :: (a -> b -> c) -> Product a -> Product b -> Product c # (>) :: Product a -> Product b -> Product b # (<*) :: Product a -> Product b -> Product a #
Foldable Product	Since: 4.8.0.0
Methods fold :: Monoid m => Product m -> m # foldMap :: Monoid m => (a -> m) -> Product a -> m # foldr :: (a -> b -> b) -> b -> Product a -> b # foldr' :: (a -> b -> b) -> b -> Product a -> b # foldl :: (b -> a -> b) -> b -> Product a -> b # foldl' :: (b -> a -> b) -> b -> Product a -> b # foldr1 :: (a -> a -> a) -> Product a -> a # foldl1 :: (a -> a -> a) -> Product a -> a # toList :: Product a -> [a] # null :: Product a -> Bool # length :: Product a -> Int # elem :: Eq a => a -> Product a -> Bool # maximum :: Ord a => Product a -> a # minimum :: Ord a => Product a -> a # sum :: Num a => Product a -> a # product :: Num a => Product a -> a #
Traversable Product	Since: 4.8.0.0
Methods traverse :: Applicative f => (a -> f b) -> Product a -> f (Product b) # sequenceA :: Applicative f => Product (f a) -> f (Product a) # mapM :: Monad m => (a -> m b) -> Product a -> m (Product b) # sequence :: Monad m => Product (m a) -> m (Product a) #
Bounded a => Bounded (Product a)
Methods minBound :: Product a # maxBound :: Product a #
Eq a => Eq (Product a)
Methods (==) :: Product a -> Product a -> Bool # (/=) :: Product a -> Product a -> Bool #
Num a => Num (Product a)
Methods (+) :: Product a -> Product a -> Product a # (-) :: Product a -> Product a -> Product a # (*) :: Product a -> Product a -> Product a # negate :: Product a -> Product a # abs :: Product a -> Product a # signum :: Product a -> Product a # fromInteger :: Integer -> Product a #
Ord a => Ord (Product a)
Methods compare :: Product a -> Product a -> Ordering # (<) :: Product a -> Product a -> Bool # (<=) :: Product a -> Product a -> Bool # (>) :: Product a -> Product a -> Bool # (>=) :: Product a -> Product a -> Bool # max :: Product a -> Product a -> Product a # min :: Product a -> Product a -> Product a #
Read a => Read (Product a)
Methods readsPrec :: Int -> ReadS (Product a) # readList :: ReadS [Product a] # readPrec :: ReadPrec (Product a) # readListPrec :: ReadPrec [Product a] #
Show a => Show (Product a)
Methods showsPrec :: Int -> Product a -> ShowS # show :: Product a -> String # showList :: [Product a] -> ShowS #
Generic (Product a)
Associated Types type Rep (Product a) :: * -> * # Methods from :: Product a -> Rep (Product a) x # to :: Rep (Product a) x -> Product a #
Num a => Semigroup (Product a)	Since: 4.9.0.0
Methods (<>) :: Product a -> Product a -> Product a # sconcat :: NonEmpty (Product a) -> Product a # stimes :: Integral b => b -> Product a -> Product a #
Num a => Monoid (Product a)	Since: 2.1
Methods mempty :: Product a # mappend :: Product a -> Product a -> Product a # mconcat :: [Product a] -> Product a #
Generic1 * Product
Associated Types type Rep1 Product (f :: Product -> ) :: k -> # Methods from1 :: f a -> Rep1 Product f a # to1 :: Rep1 Product f a -> f a #
type Rep (Product a)
type Rep (Product a) = D1 * (MetaData "Product" "Data.Monoid" "base" True) (C1 * (MetaCons "Product" PrefixI True) (S1 * (MetaSel (Just Symbol "getProduct") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 * a)))
type Rep1 * Product
type Rep1 * Product = D1 * (MetaData "Product" "Data.Monoid" "base" True) (C1 * (MetaCons "Product" PrefixI True) (S1 * (MetaSel (Just Symbol "getProduct") NoSourceUnpackedness NoSourceStrictness DecidedLazy) Par1))