Copyright  (c) Andy Gill 2001 (c) Oregon Graduate Institute of Science and Technology 2001 

License  BSDstyle (see the file libraries/base/LICENSE) 
Maintainer  libraries@haskell.org 
Stability  stable 
Portability  portable 
Safe Haskell  Trustworthy 
Language  Haskell2010 
A type a
is a Monoid
if it provides an associative function (<>
)
that lets you combine any two values of type a
into one, and a neutral
element (mempty
) such that
a <> mempty == mempty <> a == a
A Monoid
is a Semigroup
with the added requirement of a neutral element.
Thus any Monoid
is a Semigroup
, but not the other way around.
Examples
The Sum
monoid is defined by the numerical addition operator and `0` as neutral element:
>>>
mempty :: Sum Int
Sum {getSum = 0}>>>
Sum 1 <> Sum 2 <> Sum 3 <> Sum 4 :: Sum Int
Sum {getSum = 10}
We can combine multiple values in a list into a single value using the mconcat
function.
Note that we have to specify the type here since Int
is a monoid under several different
operations:
>>>
mconcat [1,2,3,4] :: Sum Int
Sum {getSum = 10}>>>
mconcat [] :: Sum Int
Sum {getSum = 0}
Another valid monoid instance of Int
is Product
It is defined by multiplication
and `1` as neutral element:
>>>
Product 1 <> Product 2 <> Product 3 <> Product 4 :: Product Int
Product {getProduct = 24}>>>
mconcat [1,2,3,4] :: Product Int
Product {getProduct = 24}>>>
mconcat [] :: Product Int
Product {getProduct = 1}
Synopsis
 class Semigroup a => Monoid a where
 (<>) :: Semigroup a => a > a > a
 newtype Dual a = Dual {
 getDual :: a
 newtype Endo a = Endo {
 appEndo :: a > a
 newtype All = All {}
 newtype Any = Any {}
 newtype Sum a = Sum {
 getSum :: a
 newtype Product a = Product {
 getProduct :: a
 newtype First a = First {}
 newtype Last a = Last {}
 newtype Alt f a = Alt {
 getAlt :: f a
 newtype Ap f a = Ap {
 getAp :: f a
Monoid
typeclass
class Semigroup a => Monoid a where Source #
The class of monoids (types with an associative binary operation that has an identity). Instances should satisfy the following:
 Right identity
x
<>
mempty
= x Left identity
mempty
<>
x = x Associativity
x
(<>
(y<>
z) = (x<>
y)<>
zSemigroup
law) Concatenation
mconcat
=foldr
(<>
)mempty
You can alternatively define mconcat
instead of mempty
, in which case the
laws are:
 Unit
mconcat
(pure
x) = x Multiplication
mconcat
(join
xss) =mconcat
(fmap
mconcat
xss) Subclass
mconcat
(toList
xs) =sconcat
xs
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 newtype
s and make those instances
of Monoid
, e.g. Sum
and Product
.
NOTE: Semigroup
is a superclass of Monoid
since base4.11.0.0.
Identity of mappend
Examples
>>>
"Hello world" <> mempty
"Hello world"
>>>
mempty <> [1, 2, 3]
[1,2,3]
mappend :: a > a > a Source #
An associative operation
NOTE: This method is redundant and has the default
implementation
since base4.11.0.0.
Should it be implemented manually, since mappend
= (<>
)mappend
is a synonym for
(<>
), it is expected that the two functions are defined the same
way. In a future GHC release mappend
will be removed from Monoid
.
Fold a list using the monoid.
For most types, the default definition for mconcat
will be
used, but the function is included in the class definition so
that an optimized version can be provided for specific types.
>>>
mconcat ["Hello", " ", "Haskell", "!"]
"Hello Haskell!"
Instances
Monoid ByteArray Source #  Since: base4.17.0.0 
Monoid All Source #  Since: base2.1 
Monoid Any Source #  Since: base2.1 
Monoid Event Source #  Since: base4.4.0.0 
Monoid Lifetime Source # 
Since: base4.8.0.0 
Monoid Ordering Source #  Since: base2.1 
Monoid () Source #  Since: base2.1 
FiniteBits a => Monoid (And a) Source #  This constraint is arguably too strong. However,
as some types (such as Since: base4.16 
FiniteBits a => Monoid (Iff a) Source #  This constraint is arguably
too strong. However, as some types (such as Since: base4.16 
Bits a => Monoid (Ior a) Source #  Since: base4.16 
Bits a => Monoid (Xor a) Source #  Since: base4.16 
Monoid (Comparison a) Source # 
mempty :: Comparison a mempty = Comparison _ _ > EQ 
Defined in Data.Functor.Contravariant mempty :: Comparison a Source # mappend :: Comparison a > Comparison a > Comparison a Source # mconcat :: [Comparison a] > Comparison a Source #  
Monoid (Equivalence a) Source # 
mempty :: Equivalence a mempty = Equivalence _ _ > True 
Defined in Data.Functor.Contravariant mempty :: Equivalence a Source # mappend :: Equivalence a > Equivalence a > Equivalence a Source # mconcat :: [Equivalence a] > Equivalence a Source #  
Monoid (Predicate a) Source # 
mempty :: Predicate a mempty = _ > True 
Monoid a => Monoid (Identity a) Source #  Since: base4.9.0.0 
Monoid (First a) Source #  Since: base2.1 
Monoid (Last a) Source #  Since: base2.1 
Monoid a => Monoid (Down a) Source #  Since: base4.11.0.0 
(Ord a, Bounded a) => Monoid (Max a) Source #  Since: base4.9.0.0 
(Ord a, Bounded a) => Monoid (Min a) Source #  Since: base4.9.0.0 
Monoid m => Monoid (WrappedMonoid m) Source #  Since: base4.9.0.0 
Defined in Data.Semigroup mempty :: WrappedMonoid m Source # mappend :: WrappedMonoid m > WrappedMonoid m > WrappedMonoid m Source # mconcat :: [WrappedMonoid m] > WrappedMonoid m Source #  
Monoid a => Monoid (Dual a) Source #  Since: base2.1 
Monoid (Endo a) Source #  Since: base2.1 
Num a => Monoid (Product a) Source #  Since: base2.1 
Num a => Monoid (Sum a) Source #  Since: base2.1 
Monoid a => Monoid (STM a) Source #  Since: base4.17.0.0 
(Generic a, Monoid (Rep a ())) => Monoid (Generically a) Source #  Since: base4.17.0.0 
Defined in GHC.Generics mempty :: Generically a Source # mappend :: Generically a > Generically a > Generically a Source # mconcat :: [Generically a] > Generically a Source #  
Monoid p => Monoid (Par1 p) Source #  Since: base4.12.0.0 
Monoid a => Monoid (IO a) Source #  Since: base4.9.0.0 
Semigroup a => Monoid (Maybe a) Source #  Lift a semigroup into Since 4.11.0: constraint on inner Since: base2.1 
Monoid a => Monoid (a) Source #  Since: base4.15 
Monoid [a] Source #  Since: base2.1 
Monoid a => Monoid (Op a b) Source # 
mempty :: Op a b mempty = Op _ > mempty 
Monoid (Proxy s) Source #  Since: base4.7.0.0 
Monoid (U1 p) Source #  Since: base4.12.0.0 
Monoid a => Monoid (ST s a) Source #  Since: base4.11.0.0 
(Monoid a, Monoid b) => Monoid (a, b) Source #  Since: base2.1 
Monoid b => Monoid (a > b) Source #  Since: base2.1 
Monoid a => Monoid (Const a b) Source #  Since: base4.9.0.0 
(Applicative f, Monoid a) => Monoid (Ap f a) Source #  Since: base4.12.0.0 
Alternative f => Monoid (Alt f a) Source #  Since: base4.8.0.0 
Monoid (f p) => Monoid (Rec1 f p) Source #  Since: base4.12.0.0 
(Monoid a, Monoid b, Monoid c) => Monoid (a, b, c) Source #  Since: base2.1 
(Monoid (f a), Monoid (g a)) => Monoid (Product f g a) Source #  Since: base4.16.0.0 
(Monoid (f p), Monoid (g p)) => Monoid ((f :*: g) p) Source #  Since: base4.12.0.0 
Monoid c => Monoid (K1 i c p) Source #  Since: base4.12.0.0 
(Monoid a, Monoid b, Monoid c, Monoid d) => Monoid (a, b, c, d) Source #  Since: base2.1 
Monoid (f (g a)) => Monoid (Compose f g a) Source #  Since: base4.16.0.0 
Monoid (f (g p)) => Monoid ((f :.: g) p) Source #  Since: base4.12.0.0 
Monoid (f p) => Monoid (M1 i c f p) Source #  Since: base4.12.0.0 
(Monoid a, Monoid b, Monoid c, Monoid d, Monoid e) => Monoid (a, b, c, d, e) Source #  Since: base2.1 
(<>) :: Semigroup a => a > a > a infixr 6 Source #
An associative operation.
Examples
>>>
[1,2,3] <> [4,5,6]
[1,2,3,4,5,6]
>>>
Just [1, 2, 3] <> Just [4, 5, 6]
Just [1,2,3,4,5,6]
>>>
putStr "Hello, " <> putStrLn "World!"
Hello, World!
The dual of a Monoid
, obtained by swapping the arguments of mappend
.
 The dual of a Monoid
, obtained by swapping the arguments of (<>)
.
Dual a <> Dual b == Dual (b <> a)
Examples
>>>
Dual "Hello" <> Dual "World"
Dual {getDual = "WorldHello"}
>>>
Dual (Dual "Hello") <> Dual (Dual "World")
Dual {getDual = Dual {getDual = "HelloWorld"}}
Instances
MonadFix Dual Source #  Since: base4.8.0.0 
MonadZip Dual Source #  Since: base4.8.0.0 
Foldable Dual Source #  Since: base4.8.0.0 
Defined in Data.Foldable fold :: Monoid m => Dual m > m Source # foldMap :: Monoid m => (a > m) > Dual a > m Source # foldMap' :: Monoid m => (a > m) > Dual a > m Source # foldr :: (a > b > b) > b > Dual a > b Source # foldr' :: (a > b > b) > b > Dual a > b Source # foldl :: (b > a > b) > b > Dual a > b Source # foldl' :: (b > a > b) > b > Dual a > b Source # foldr1 :: (a > a > a) > Dual a > a Source # foldl1 :: (a > a > a) > Dual a > a Source # toList :: Dual a > [a] Source # null :: Dual a > Bool Source # length :: Dual a > Int Source # elem :: Eq a => a > Dual a > Bool Source # maximum :: Ord a => Dual a > a Source # minimum :: Ord a => Dual a > a Source #  
Foldable1 Dual Source #  Since: base4.18.0.0 
Defined in Data.Foldable1 fold1 :: Semigroup m => Dual m > m Source # foldMap1 :: Semigroup m => (a > m) > Dual a > m Source # foldMap1' :: Semigroup m => (a > m) > Dual a > m Source # toNonEmpty :: Dual a > NonEmpty a Source # maximum :: Ord a => Dual a > a Source # minimum :: Ord a => Dual a > a Source # foldrMap1 :: (a > b) > (a > b > b) > Dual a > b Source # foldlMap1' :: (a > b) > (b > a > b) > Dual a > b Source # foldlMap1 :: (a > b) > (b > a > b) > Dual a > b Source # foldrMap1' :: (a > b) > (a > b > b) > Dual a > b Source #  
Traversable Dual Source #  Since: base4.8.0.0 
Applicative Dual Source #  Since: base4.8.0.0 
Functor Dual Source #  Since: base4.8.0.0 
Monad Dual Source #  Since: base4.8.0.0 
Generic1 Dual Source #  
Data a => Data (Dual a) Source #  Since: base4.8.0.0 
Defined in Data.Data gfoldl :: (forall d b. Data d => c (d > b) > d > c b) > (forall g. g > c g) > Dual a > c (Dual a) Source # gunfold :: (forall b r. Data b => c (b > r) > c r) > (forall r. r > c r) > Constr > c (Dual a) Source # toConstr :: Dual a > Constr Source # dataTypeOf :: Dual a > DataType Source # dataCast1 :: Typeable t => (forall d. Data d => c (t d)) > Maybe (c (Dual a)) Source # dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) > Maybe (c (Dual a)) Source # gmapT :: (forall b. Data b => b > b) > Dual a > Dual a Source # gmapQl :: (r > r' > r) > r > (forall d. Data d => d > r') > Dual a > r Source # gmapQr :: forall r r'. (r' > r > r) > r > (forall d. Data d => d > r') > Dual a > r Source # gmapQ :: (forall d. Data d => d > u) > Dual a > [u] Source # gmapQi :: Int > (forall d. Data d => d > u) > Dual a > u Source # gmapM :: Monad m => (forall d. Data d => d > m d) > Dual a > m (Dual a) Source # gmapMp :: MonadPlus m => (forall d. Data d => d > m d) > Dual a > m (Dual a) Source # gmapMo :: MonadPlus m => (forall d. Data d => d > m d) > Dual a > m (Dual a) Source #  
Monoid a => Monoid (Dual a) Source #  Since: base2.1 
Semigroup a => Semigroup (Dual a) Source #  Since: base4.9.0.0 
Bounded a => Bounded (Dual a) Source #  Since: base2.1 
Generic (Dual a) Source #  
Read a => Read (Dual a) Source #  Since: base2.1 
Show a => Show (Dual a) Source #  Since: base2.1 
Eq a => Eq (Dual a) Source #  Since: base2.1 
Ord a => Ord (Dual a) Source #  Since: base2.1 
Defined in Data.Semigroup.Internal  
type Rep1 Dual Source #  Since: base4.7.0.0 
Defined in Data.Semigroup.Internal  
type Rep (Dual a) Source #  Since: base4.7.0.0 
Defined in Data.Semigroup.Internal 
The monoid of endomorphisms under composition.
Endo f <> Endo g == Endo (f . g)
Examples
>>>
let computation = Endo ("Hello, " ++) <> Endo (++ "!")
>>>
appEndo computation "Haskell"
"Hello, Haskell!"
>>>
let computation = Endo (*3) <> Endo (+1)
>>>
appEndo computation 1
6
Bool
wrappers
Boolean monoid under conjunction (&&)
.
All x <> All y = All (x && y)
Examples
>>>
All True <> mempty <> All False)
All {getAll = False}
>>>
mconcat (map (\x > All (even x)) [2,4,6,7,8])
All {getAll = False}
>>>
All True <> mempty
All {getAll = True}
Instances
Data All Source #  Since: base4.8.0.0 
Defined in Data.Data gfoldl :: (forall d b. Data d => c (d > b) > d > c b) > (forall g. g > c g) > All > c All Source # gunfold :: (forall b r. Data b => c (b > r) > c r) > (forall r. r > c r) > Constr > c All Source # toConstr :: All > Constr Source # dataTypeOf :: All > DataType Source # dataCast1 :: Typeable t => (forall d. Data d => c (t d)) > Maybe (c All) Source # dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) > Maybe (c All) Source # gmapT :: (forall b. Data b => b > b) > All > All Source # gmapQl :: (r > r' > r) > r > (forall d. Data d => d > r') > All > r Source # gmapQr :: forall r r'. (r' > r > r) > r > (forall d. Data d => d > r') > All > r Source # gmapQ :: (forall d. Data d => d > u) > All > [u] Source # gmapQi :: Int > (forall d. Data d => d > u) > All > u Source # gmapM :: Monad m => (forall d. Data d => d > m d) > All > m All Source # gmapMp :: MonadPlus m => (forall d. Data d => d > m d) > All > m All Source # gmapMo :: MonadPlus m => (forall d. Data d => d > m d) > All > m All Source #  
Monoid All Source #  Since: base2.1 
Semigroup All Source #  Since: base4.9.0.0 
Bounded All Source #  Since: base2.1 
Generic All Source #  
Read All Source #  Since: base2.1 
Show All Source #  Since: base2.1 
Eq All Source #  Since: base2.1 
Ord All Source #  Since: base2.1 
type Rep All Source #  Since: base4.7.0.0 
Defined in Data.Semigroup.Internal 
Boolean monoid under disjunction ()
.
Any x <> Any y = Any (x  y)
Examples
>>>
Any True <> mempty <> Any False
Any {getAny = True}
>>>
mconcat (map (\x > Any (even x)) [2,4,6,7,8])
Any {getAny = True}
>>>
Any False <> mempty
Any {getAny = False}
Instances
Data Any Source #  Since: base4.8.0.0 
Defined in Data.Data gfoldl :: (forall d b. Data d => c (d > b) > d > c b) > (forall g. g > c g) > Any > c Any Source # gunfold :: (forall b r. Data b => c (b > r) > c r) > (forall r. r > c r) > Constr > c Any Source # toConstr :: Any > Constr Source # dataTypeOf :: Any > DataType Source # dataCast1 :: Typeable t => (forall d. Data d => c (t d)) > Maybe (c Any) Source # dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) > Maybe (c Any) Source # gmapT :: (forall b. Data b => b > b) > Any > Any Source # gmapQl :: (r > r' > r) > r > (forall d. Data d => d > r') > Any > r Source # gmapQr :: forall r r'. (r' > r > r) > r > (forall d. Data d => d > r') > Any > r Source # gmapQ :: (forall d. Data d => d > u) > Any > [u] Source # gmapQi :: Int > (forall d. Data d => d > u) > Any > u Source # gmapM :: Monad m => (forall d. Data d => d > m d) > Any > m Any Source # gmapMp :: MonadPlus m => (forall d. Data d => d > m d) > Any > m Any Source # gmapMo :: MonadPlus m => (forall d. Data d => d > m d) > Any > m Any Source #  
Monoid Any Source #  Since: base2.1 
Semigroup Any Source #  Since: base4.9.0.0 
Bounded Any Source #  Since: base2.1 
Generic Any Source #  
Read Any Source #  Since: base2.1 
Show Any Source #  Since: base2.1 
Eq Any Source #  Since: base2.1 
Ord Any Source #  Since: base2.1 
type Rep Any Source #  Since: base4.7.0.0 
Defined in Data.Semigroup.Internal 
Num
wrappers
Monoid under addition.
Sum a <> Sum b = Sum (a + b)
Examples
>>>
Sum 1 <> Sum 2 <> mempty
Sum {getSum = 3}
>>>
mconcat [ Sum n  n < [3 .. 9]]
Sum {getSum = 42}
Instances
MonadFix Sum Source #  Since: base4.8.0.0 
MonadZip Sum Source #  Since: base4.8.0.0 
Foldable Sum Source #  Since: base4.8.0.0 
Defined in Data.Foldable fold :: Monoid m => Sum m > m Source # foldMap :: Monoid m => (a > m) > Sum a > m Source # foldMap' :: Monoid m => (a > m) > Sum a > m Source # foldr :: (a > b > b) > b > Sum a > b Source # foldr' :: (a > b > b) > b > Sum a > b Source # foldl :: (b > a > b) > b > Sum a > b Source # foldl' :: (b > a > b) > b > Sum a > b Source # foldr1 :: (a > a > a) > Sum a > a Source # foldl1 :: (a > a > a) > Sum a > a Source # toList :: Sum a > [a] Source # null :: Sum a > Bool Source # length :: Sum a > Int Source # elem :: Eq a => a > Sum a > Bool Source # maximum :: Ord a => Sum a > a Source # minimum :: Ord a => Sum a > a Source #  
Foldable1 Sum Source #  Since: base4.18.0.0 
Defined in Data.Foldable1 fold1 :: Semigroup m => Sum m > m Source # foldMap1 :: Semigroup m => (a > m) > Sum a > m Source # foldMap1' :: Semigroup m => (a > m) > Sum a > m Source # toNonEmpty :: Sum a > NonEmpty a Source # maximum :: Ord a => Sum a > a Source # minimum :: Ord a => Sum a > a Source # foldrMap1 :: (a > b) > (a > b > b) > Sum a > b Source # foldlMap1' :: (a > b) > (b > a > b) > Sum a > b Source # foldlMap1 :: (a > b) > (b > a > b) > Sum a > b Source # foldrMap1' :: (a > b) > (a > b > b) > Sum a > b Source #  
Traversable Sum Source #  Since: base4.8.0.0 
Applicative Sum Source #  Since: base4.8.0.0 
Functor Sum Source #  Since: base4.8.0.0 
Monad Sum Source #  Since: base4.8.0.0 
Generic1 Sum Source #  
Data a => Data (Sum a) Source #  Since: base4.8.0.0 
Defined in Data.Data gfoldl :: (forall d b. Data d => c (d > b) > d > c b) > (forall g. g > c g) > Sum a > c (Sum a) Source # gunfold :: (forall b r. Data b => c (b > r) > c r) > (forall r. r > c r) > Constr > c (Sum a) Source # toConstr :: Sum a > Constr Source # dataTypeOf :: Sum a > DataType Source # dataCast1 :: Typeable t => (forall d. Data d => c (t d)) > Maybe (c (Sum a)) Source # dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) > Maybe (c (Sum a)) Source # gmapT :: (forall b. Data b => b > b) > Sum a > Sum a Source # gmapQl :: (r > r' > r) > r > (forall d. Data d => d > r') > Sum a > r Source # gmapQr :: forall r r'. (r' > r > r) > r > (forall d. Data d => d > r') > Sum a > r Source # gmapQ :: (forall d. Data d => d > u) > Sum a > [u] Source # gmapQi :: Int > (forall d. Data d => d > u) > Sum a > u Source # gmapM :: Monad m => (forall d. Data d => d > m d) > Sum a > m (Sum a) Source # gmapMp :: MonadPlus m => (forall d. Data d => d > m d) > Sum a > m (Sum a) Source # gmapMo :: MonadPlus m => (forall d. Data d => d > m d) > Sum a > m (Sum a) Source #  
Num a => Monoid (Sum a) Source #  Since: base2.1 
Num a => Semigroup (Sum a) Source #  Since: base4.9.0.0 
Bounded a => Bounded (Sum a) Source #  Since: base2.1 
Generic (Sum a) Source #  
Num a => Num (Sum a) Source #  Since: base4.7.0.0 
Read a => Read (Sum a) Source #  Since: base2.1 
Show a => Show (Sum a) Source #  Since: base2.1 
Eq a => Eq (Sum a) Source #  Since: base2.1 
Ord a => Ord (Sum a) Source #  Since: base2.1 
Defined in Data.Semigroup.Internal  
type Rep1 Sum Source #  Since: base4.7.0.0 
Defined in Data.Semigroup.Internal  
type Rep (Sum a) Source #  Since: base4.7.0.0 
Defined in Data.Semigroup.Internal 
Monoid under multiplication.
Product x <> Product y == Product (x * y)
Examples
>>>
Product 3 <> Product 4 <> mempty
Product {getProduct = 12}
>>>
mconcat [ Product n  n < [2 .. 10]]
Product {getProduct = 3628800}
Product  

Instances
MonadFix Product Source #  Since: base4.8.0.0 
MonadZip Product Source #  Since: base4.8.0.0 
Foldable Product Source #  Since: base4.8.0.0 
Defined in Data.Foldable fold :: Monoid m => Product m > m Source # foldMap :: Monoid m => (a > m) > Product a > m Source # foldMap' :: Monoid m => (a > m) > Product a > m Source # foldr :: (a > b > b) > b > Product a > b Source # foldr' :: (a > b > b) > b > Product a > b Source # foldl :: (b > a > b) > b > Product a > b Source # foldl' :: (b > a > b) > b > Product a > b Source # foldr1 :: (a > a > a) > Product a > a Source # foldl1 :: (a > a > a) > Product a > a Source # toList :: Product a > [a] Source # null :: Product a > Bool Source # length :: Product a > Int Source # elem :: Eq a => a > Product a > Bool Source # maximum :: Ord a => Product a > a Source # minimum :: Ord a => Product a > a Source #  
Foldable1 Product Source #  Since: base4.18.0.0 
Defined in Data.Foldable1 fold1 :: Semigroup m => Product m > m Source # foldMap1 :: Semigroup m => (a > m) > Product a > m Source # foldMap1' :: Semigroup m => (a > m) > Product a > m Source # toNonEmpty :: Product a > NonEmpty a Source # maximum :: Ord a => Product a > a Source # minimum :: Ord a => Product a > a Source # head :: Product a > a Source # last :: Product a > a Source # foldrMap1 :: (a > b) > (a > b > b) > Product a > b Source # foldlMap1' :: (a > b) > (b > a > b) > Product a > b Source # foldlMap1 :: (a > b) > (b > a > b) > Product a > b Source # foldrMap1' :: (a > b) > (a > b > b) > Product a > b Source #  
Traversable Product Source #  Since: base4.8.0.0 
Defined in Data.Traversable  
Applicative Product Source #  Since: base4.8.0.0 
Defined in Data.Semigroup.Internal  
Functor Product Source #  Since: base4.8.0.0 
Monad Product Source #  Since: base4.8.0.0 
Generic1 Product Source #  
Data a => Data (Product a) Source #  Since: base4.8.0.0 
Defined in Data.Data gfoldl :: (forall d b. Data d => c (d > b) > d > c b) > (forall g. g > c g) > Product a > c (Product a) Source # gunfold :: (forall b r. Data b => c (b > r) > c r) > (forall r. r > c r) > Constr > c (Product a) Source # toConstr :: Product a > Constr Source # dataTypeOf :: Product a > DataType Source # dataCast1 :: Typeable t => (forall d. Data d => c (t d)) > Maybe (c (Product a)) Source # dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) > Maybe (c (Product a)) Source # gmapT :: (forall b. Data b => b > b) > Product a > Product a Source # gmapQl :: (r > r' > r) > r > (forall d. Data d => d > r') > Product a > r Source # gmapQr :: forall r r'. (r' > r > r) > r > (forall d. Data d => d > r') > Product a > r Source # gmapQ :: (forall d. Data d => d > u) > Product a > [u] Source # gmapQi :: Int > (forall d. Data d => d > u) > Product a > u Source # gmapM :: Monad m => (forall d. Data d => d > m d) > Product a > m (Product a) Source # gmapMp :: MonadPlus m => (forall d. Data d => d > m d) > Product a > m (Product a) Source # gmapMo :: MonadPlus m => (forall d. Data d => d > m d) > Product a > m (Product a) Source #  
Num a => Monoid (Product a) Source #  Since: base2.1 
Num a => Semigroup (Product a) Source #  Since: base4.9.0.0 
Bounded a => Bounded (Product a) Source #  Since: base2.1 
Generic (Product a) Source #  
Num a => Num (Product a) Source #  Since: base4.7.0.0 
Defined in Data.Semigroup.Internal (+) :: Product a > Product a > Product a Source # () :: Product a > Product a > Product a Source # (*) :: Product a > Product a > Product a Source # negate :: Product a > Product a Source # abs :: Product a > Product a Source # signum :: Product a > Product a Source # fromInteger :: Integer > Product a Source #  
Read a => Read (Product a) Source #  Since: base2.1 
Show a => Show (Product a) Source #  Since: base2.1 
Eq a => Eq (Product a) Source #  Since: base2.1 
Ord a => Ord (Product a) Source #  Since: base2.1 
Defined in Data.Semigroup.Internal  
type Rep1 Product Source #  Since: base4.7.0.0 
Defined in Data.Semigroup.Internal  
type Rep (Product a) Source #  Since: base4.7.0.0 
Defined in Data.Semigroup.Internal 
Maybe
wrappers
To implement find
or findLast
on any Foldable
:
findLast :: Foldable t => (a > Bool) > t a > Maybe a findLast pred = getLast . foldMap (x > if pred x then Last (Just x) else Last Nothing)
Much of Map
s interface can be implemented with
alter
. Some of the rest can be implemented with a new
alterF
function and either First
or Last
:
alterF :: (Functor f, Ord k) => (Maybe a > f (Maybe a)) > k > Map k a > f (Map k a) instance Monoid a => Functor ((,) a)  from Data.Functor
insertLookupWithKey :: Ord k => (k > v > v > v) > k > v
> Map k v > (Maybe v, Map k v)
insertLookupWithKey combine key value =
Arrow.first getFirst . alterF
doChange key
where
doChange Nothing = (First Nothing, Just value)
doChange (Just oldValue) =
(First (Just oldValue),
Just (combine key value oldValue))
Maybe monoid returning the leftmost nonNothing
value.
is isomorphic to First
a
, but precedes it
historically.Alt
Maybe
a
Beware that Data.Monoid.
First
is different from
Data.Semigroup.
First
. The former returns the first nonNothing
,
so Data.Monoid.First Nothing <> x = x
. The latter simply returns the first value,
thus Data.Semigroup.First Nothing <> x = Data.Semigroup.First Nothing
.
Examples
>>>
First (Just "hello") <> First Nothing <> First (Just "world")
First {getFirst = Just "hello"}
>>>
First Nothing <> mempty
First {getFirst = Nothing}
Instances
MonadFix First Source #  Since: base4.8.0.0 
MonadZip First Source #  Since: base4.8.0.0 
Foldable First Source #  Since: base4.8.0.0 
Defined in Data.Foldable fold :: Monoid m => First m > m Source # foldMap :: Monoid m => (a > m) > First a > m Source # foldMap' :: Monoid m => (a > m) > First a > m Source # foldr :: (a > b > b) > b > First a > b Source # foldr' :: (a > b > b) > b > First a > b Source # foldl :: (b > a > b) > b > First a > b Source # foldl' :: (b > a > b) > b > First a > b Source # foldr1 :: (a > a > a) > First a > a Source # foldl1 :: (a > a > a) > First a > a Source # toList :: First a > [a] Source # null :: First a > Bool Source # length :: First a > Int Source # elem :: Eq a => a > First a > Bool Source # maximum :: Ord a => First a > a Source # minimum :: Ord a => First a > a Source #  
Traversable First Source #  Since: base4.8.0.0 
Applicative First Source #  Since: base4.8.0.0 
Functor First Source #  Since: base4.8.0.0 
Monad First Source #  Since: base4.8.0.0 
Generic1 First Source #  
Data a => Data (First a) Source #  Since: base4.8.0.0 
Defined in Data.Data gfoldl :: (forall d b. Data d => c (d > b) > d > c b) > (forall g. g > c g) > First a > c (First a) Source # gunfold :: (forall b r. Data b => c (b > r) > c r) > (forall r. r > c r) > Constr > c (First a) Source # toConstr :: First a > Constr Source # dataTypeOf :: First a > DataType Source # dataCast1 :: Typeable t => (forall d. Data d => c (t d)) > Maybe (c (First a)) Source # dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) > Maybe (c (First a)) Source # gmapT :: (forall b. Data b => b > b) > First a > First a Source # gmapQl :: (r > r' > r) > r > (forall d. Data d => d > r') > First a > r Source # gmapQr :: forall r r'. (r' > r > r) > r > (forall d. Data d => d > r') > First a > r Source # gmapQ :: (forall d. Data d => d > u) > First a > [u] Source # gmapQi :: Int > (forall d. Data d => d > u) > First a > u Source # gmapM :: Monad m => (forall d. Data d => d > m d) > First a > m (First a) Source # gmapMp :: MonadPlus m => (forall d. Data d => d > m d) > First a > m (First a) Source # gmapMo :: MonadPlus m => (forall d. Data d => d > m d) > First a > m (First a) Source #  
Monoid (First a) Source #  Since: base2.1 
Semigroup (First a) Source #  Since: base4.9.0.0 
Generic (First a) Source #  
Read a => Read (First a) Source #  Since: base2.1 
Show a => Show (First a) Source #  Since: base2.1 
Eq a => Eq (First a) Source #  Since: base2.1 
Ord a => Ord (First a) Source #  Since: base2.1 
type Rep1 First Source #  Since: base4.7.0.0 
Defined in Data.Monoid  
type Rep (First a) Source #  Since: base4.7.0.0 
Defined in Data.Monoid 
Maybe monoid returning the rightmost nonNothing
value.
is isomorphic to Last
a
, and thus to
Dual
(First
a)Dual
(Alt
Maybe
a)
Data.Semigroup.
Last
. The former returns the last nonNothing
,
so x <> Data.Monoid.Last Nothing = x
. The latter simply returns the last value,
thus x <> Data.Semigroup.Last Nothing = Data.Semigroup.Last Nothing
.
Examples
>>>
Last (Just "hello") <> Last Nothing <> Last (Just "world")
Last {getLast = Just "world"}
>>>
Last Nothing <> mempty
Last {getLast = Nothing}
Instances
MonadFix Last Source #  Since: base4.8.0.0 
MonadZip Last Source #  Since: base4.8.0.0 
Foldable Last Source #  Since: base4.8.0.0 
Defined in Data.Foldable fold :: Monoid m => Last m > m Source # foldMap :: Monoid m => (a > m) > Last a > m Source # foldMap' :: Monoid m => (a > m) > Last a > m Source # foldr :: (a > b > b) > b > Last a > b Source # foldr' :: (a > b > b) > b > Last a > b Source # foldl :: (b > a > b) > b > Last a > b Source # foldl' :: (b > a > b) > b > Last a > b Source # foldr1 :: (a > a > a) > Last a > a Source # foldl1 :: (a > a > a) > Last a > a Source # toList :: Last a > [a] Source # null :: Last a > Bool Source # length :: Last a > Int Source # elem :: Eq a => a > Last a > Bool Source # maximum :: Ord a => Last a > a Source # minimum :: Ord a => Last a > a Source #  
Traversable Last Source #  Since: base4.8.0.0 
Applicative Last Source #  Since: base4.8.0.0 
Functor Last Source #  Since: base4.8.0.0 
Monad Last Source #  Since: base4.8.0.0 
Generic1 Last Source #  
Data a => Data (Last a) Source #  Since: base4.8.0.0 
Defined in Data.Data gfoldl :: (forall d b. Data d => c (d > b) > d > c b) > (forall g. g > c g) > Last a > c (Last a) Source # gunfold :: (forall b r. Data b => c (b > r) > c r) > (forall r. r > c r) > Constr > c (Last a) Source # toConstr :: Last a > Constr Source # dataTypeOf :: Last a > DataType Source # dataCast1 :: Typeable t => (forall d. Data d => c (t d)) > Maybe (c (Last a)) Source # dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) > Maybe (c (Last a)) Source # gmapT :: (forall b. Data b => b > b) > Last a > Last a Source # gmapQl :: (r > r' > r) > r > (forall d. Data d => d > r') > Last a > r Source # gmapQr :: forall r r'. (r' > r > r) > r > (forall d. Data d => d > r') > Last a > r Source # gmapQ :: (forall d. Data d => d > u) > Last a > [u] Source # gmapQi :: Int > (forall d. Data d => d > u) > Last a > u Source # gmapM :: Monad m => (forall d. Data d => d > m d) > Last a > m (Last a) Source # gmapMp :: MonadPlus m => (forall d. Data d => d > m d) > Last a > m (Last a) Source # gmapMo :: MonadPlus m => (forall d. Data d => d > m d) > Last a > m (Last a) Source #  
Monoid (Last a) Source #  Since: base2.1 
Semigroup (Last a) Source #  Since: base4.9.0.0 
Generic (Last a) Source #  
Read a => Read (Last a) Source #  Since: base2.1 
Show a => Show (Last a) Source #  Since: base2.1 
Eq a => Eq (Last a) Source #  Since: base2.1 
Ord a => Ord (Last a) Source #  Since: base2.1 
Defined in Data.Monoid  
type Rep1 Last Source #  Since: base4.7.0.0 
Defined in Data.Monoid  
type Rep (Last a) Source #  Since: base4.7.0.0 
Defined in Data.Monoid 
Alternative
wrapper
Monoid under <>
.
Alt l <> Alt r == Alt (l <> r)
Examples
>>>
Alt (Just 12) <> Alt (Just 24)
Alt {getAlt = Just 12}
>>>
Alt Nothing <> Alt (Just 24)
Alt {getAlt = Just 24}
Since: base4.8.0.0
Instances
Generic1 (Alt f :: k > Type) Source #  
MonadFix f => MonadFix (Alt f) Source #  Since: base4.8.0.0 
MonadZip f => MonadZip (Alt f) Source #  Since: base4.8.0.0 
Foldable f => Foldable (Alt f) Source #  Since: base4.12.0.0 
Defined in Data.Foldable fold :: Monoid m => Alt f m > m Source # foldMap :: Monoid m => (a > m) > Alt f a > m Source # foldMap' :: Monoid m => (a > m) > Alt f a > m Source # foldr :: (a > b > b) > b > Alt f a > b Source # foldr' :: (a > b > b) > b > Alt f a > b Source # foldl :: (b > a > b) > b > Alt f a > b Source # foldl' :: (b > a > b) > b > Alt f a > b Source # foldr1 :: (a > a > a) > Alt f a > a Source # foldl1 :: (a > a > a) > Alt f a > a Source # toList :: Alt f a > [a] Source # null :: Alt f a > Bool Source # length :: Alt f a > Int Source # elem :: Eq a => a > Alt f a > Bool Source # maximum :: Ord a => Alt f a > a Source # minimum :: Ord a => Alt f a > a Source #  
Foldable1 f => Foldable1 (Alt f) Source #  Since: base4.18.0.0 
Defined in Data.Foldable1 fold1 :: Semigroup m => Alt f m > m Source # foldMap1 :: Semigroup m => (a > m) > Alt f a > m Source # foldMap1' :: Semigroup m => (a > m) > Alt f a > m Source # toNonEmpty :: Alt f a > NonEmpty a Source # maximum :: Ord a => Alt f a > a Source # minimum :: Ord a => Alt f a > a Source # foldrMap1 :: (a > b) > (a > b > b) > Alt f a > b Source # foldlMap1' :: (a > b) > (b > a > b) > Alt f a > b Source # foldlMap1 :: (a > b) > (b > a > b) > Alt f a > b Source # foldrMap1' :: (a > b) > (a > b > b) > Alt f a > b Source #  
Contravariant f => Contravariant (Alt f) Source #  
Traversable f => Traversable (Alt f) Source #  Since: base4.12.0.0 
Alternative f => Alternative (Alt f) Source #  Since: base4.8.0.0 
Applicative f => Applicative (Alt f) Source #  Since: base4.8.0.0 
Functor f => Functor (Alt f) Source #  Since: base4.8.0.0 
Monad f => Monad (Alt f) Source #  Since: base4.8.0.0 
MonadPlus f => MonadPlus (Alt f) Source #  Since: base4.8.0.0 
(Data (f a), Data a, Typeable f) => Data (Alt f a) Source #  Since: base4.8.0.0 
Defined in Data.Data gfoldl :: (forall d b. Data d => c (d > b) > d > c b) > (forall g. g > c g) > Alt f a > c (Alt f a) Source # gunfold :: (forall b r. Data b => c (b > r) > c r) > (forall r. r > c r) > Constr > c (Alt f a) Source # toConstr :: Alt f a > Constr Source # dataTypeOf :: Alt f a > DataType Source # dataCast1 :: Typeable t => (forall d. Data d => c (t d)) > Maybe (c (Alt f a)) Source # dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) > Maybe (c (Alt f a)) Source # gmapT :: (forall b. Data b => b > b) > Alt f a > Alt f a Source # gmapQl :: (r > r' > r) > r > (forall d. Data d => d > r') > Alt f a > r Source # gmapQr :: forall r r'. (r' > r > r) > r > (forall d. Data d => d > r') > Alt f a > r Source # gmapQ :: (forall d. Data d => d > u) > Alt f a > [u] Source # gmapQi :: Int > (forall d. Data d => d > u) > Alt f a > u Source # gmapM :: Monad m => (forall d. Data d => d > m d) > Alt f a > m (Alt f a) Source # gmapMp :: MonadPlus m => (forall d. Data d => d > m d) > Alt f a > m (Alt f a) Source # gmapMo :: MonadPlus m => (forall d. Data d => d > m d) > Alt f a > m (Alt f a) Source #  
Alternative f => Monoid (Alt f a) Source #  Since: base4.8.0.0 
Alternative f => Semigroup (Alt f a) Source #  Since: base4.9.0.0 
Enum (f a) => Enum (Alt f a) Source #  Since: base4.8.0.0 
Defined in Data.Semigroup.Internal succ :: Alt f a > Alt f a Source # pred :: Alt f a > Alt f a Source # toEnum :: Int > Alt f a Source # fromEnum :: Alt f a > Int Source # enumFrom :: Alt f a > [Alt f a] Source # enumFromThen :: Alt f a > Alt f a > [Alt f a] Source # enumFromTo :: Alt f a > Alt f a > [Alt f a] Source # enumFromThenTo :: Alt f a > Alt f a > Alt f a > [Alt f a] Source #  
Generic (Alt f a) Source #  
Num (f a) => Num (Alt f a) Source #  Since: base4.8.0.0 
Defined in Data.Semigroup.Internal  
Read (f a) => Read (Alt f a) Source #  Since: base4.8.0.0 
Show (f a) => Show (Alt f a) Source #  Since: base4.8.0.0 
Eq (f a) => Eq (Alt f a) Source #  Since: base4.8.0.0 
Ord (f a) => Ord (Alt f a) Source #  Since: base4.8.0.0 
Defined in Data.Semigroup.Internal  
type Rep1 (Alt f :: k > Type) Source #  Since: base4.8.0.0 
Defined in Data.Semigroup.Internal  
type Rep (Alt f a) Source #  Since: base4.8.0.0 
Defined in Data.Semigroup.Internal 
Applicative
wrapper
This data type witnesses the lifting of a Monoid
into an
Applicative
pointwise.
Examples
>>>
Ap (Just [1, 2, 3]) <> Ap Nothing
Ap {getAp = Nothing}
>>>
Ap [Sum 10, Sum 20] <> Ap [Sum 1, Sum 2]
Ap {getAp = [Sum {getSum = 11},Sum {getSum = 12},Sum {getSum = 21},Sum {getSum = 22}]}
Since: base4.12.0.0
Instances
Generic1 (Ap f :: k > Type) Source #  
MonadFail f => MonadFail (Ap f) Source #  Since: base4.12.0.0 
MonadFix f => MonadFix (Ap f) Source #  Since: base4.12.0.0 
Foldable f => Foldable (Ap f) Source #  Since: base4.12.0.0 
Defined in Data.Foldable fold :: Monoid m => Ap f m > m Source # foldMap :: Monoid m => (a > m) > Ap f a > m Source # foldMap' :: Monoid m => (a > m) > Ap f a > m Source # foldr :: (a > b > b) > b > Ap f a > b Source # foldr' :: (a > b > b) > b > Ap f a > b Source # foldl :: (b > a > b) > b > Ap f a > b Source # foldl' :: (b > a > b) > b > Ap f a > b Source # foldr1 :: (a > a > a) > Ap f a > a Source # foldl1 :: (a > a > a) > Ap f a > a Source # toList :: Ap f a > [a] Source # null :: Ap f a > Bool Source # length :: Ap f a > Int Source # elem :: Eq a => a > Ap f a > Bool Source # maximum :: Ord a => Ap f a > a Source # minimum :: Ord a => Ap f a > a Source #  
Foldable1 f => Foldable1 (Ap f) Source #  Since: base4.18.0.0 
Defined in Data.Foldable1 fold1 :: Semigroup m => Ap f m > m Source # foldMap1 :: Semigroup m => (a > m) > Ap f a > m Source # foldMap1' :: Semigroup m => (a > m) > Ap f a > m Source # toNonEmpty :: Ap f a > NonEmpty a Source # maximum :: Ord a => Ap f a > a Source # minimum :: Ord a => Ap f a > a Source # foldrMap1 :: (a > b) > (a > b > b) > Ap f a > b Source # foldlMap1' :: (a > b) > (b > a > b) > Ap f a > b Source # foldlMap1 :: (a > b) > (b > a > b) > Ap f a > b Source # foldrMap1' :: (a > b) > (a > b > b) > Ap f a > b Source #  
Traversable f => Traversable (Ap f) Source #  Since: base4.12.0.0 
Alternative f => Alternative (Ap f) Source #  Since: base4.12.0.0 
Applicative f => Applicative (Ap f) Source #  Since: base4.12.0.0 
Functor f => Functor (Ap f) Source #  Since: base4.12.0.0 
Monad f => Monad (Ap f) Source #  Since: base4.12.0.0 
MonadPlus f => MonadPlus (Ap f) Source #  Since: base4.12.0.0 
(Data (f a), Data a, Typeable f) => Data (Ap f a) Source #  Since: base4.12.0.0 
Defined in Data.Data gfoldl :: (forall d b. Data d => c (d > b) > d > c b) > (forall g. g > c g) > Ap f a > c (Ap f a) Source # gunfold :: (forall b r. Data b => c (b > r) > c r) > (forall r. r > c r) > Constr > c (Ap f a) Source # toConstr :: Ap f a > Constr Source # dataTypeOf :: Ap f a > DataType Source # dataCast1 :: Typeable t => (forall d. Data d => c (t d)) > Maybe (c (Ap f a)) Source # dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) > Maybe (c (Ap f a)) Source # gmapT :: (forall b. Data b => b > b) > Ap f a > Ap f a Source # gmapQl :: (r > r' > r) > r > (forall d. Data d => d > r') > Ap f a > r Source # gmapQr :: forall r r'. (r' > r > r) > r > (forall d. Data d => d > r') > Ap f a > r Source # gmapQ :: (forall d. Data d => d > u) > Ap f a > [u] Source # gmapQi :: Int > (forall d. Data d => d > u) > Ap f a > u Source # gmapM :: Monad m => (forall d. Data d => d > m d) > Ap f a > m (Ap f a) Source # gmapMp :: MonadPlus m => (forall d. Data d => d > m d) > Ap f a > m (Ap f a) Source # gmapMo :: MonadPlus m => (forall d. Data d => d > m d) > Ap f a > m (Ap f a) Source #  
(Applicative f, Monoid a) => Monoid (Ap f a) Source #  Since: base4.12.0.0 
(Applicative f, Semigroup a) => Semigroup (Ap f a) Source #  Since: base4.12.0.0 
(Applicative f, Bounded a) => Bounded (Ap f a) Source #  Since: base4.12.0.0 
Enum (f a) => Enum (Ap f a) Source #  Since: base4.12.0.0 
Defined in Data.Monoid succ :: Ap f a > Ap f a Source # pred :: Ap f a > Ap f a Source # toEnum :: Int > Ap f a Source # fromEnum :: Ap f a > Int Source # enumFrom :: Ap f a > [Ap f a] Source # enumFromThen :: Ap f a > Ap f a > [Ap f a] Source # enumFromTo :: Ap f a > Ap f a > [Ap f a] Source # enumFromThenTo :: Ap f a > Ap f a > Ap f a > [Ap f a] Source #  
Generic (Ap f a) Source #  
(Applicative f, Num a) => Num (Ap f a) Source #  Note that even if the underlying Commutativity:
Additive inverse:
Distributivity:
Since: base4.12.0.0 
Read (f a) => Read (Ap f a) Source #  Since: base4.12.0.0 
Show (f a) => Show (Ap f a) Source #  Since: base4.12.0.0 
Eq (f a) => Eq (Ap f a) Source #  Since: base4.12.0.0 
Ord (f a) => Ord (Ap f a) Source #  Since: base4.12.0.0 
Defined in Data.Monoid  
type Rep1 (Ap f :: k > Type) Source #  Since: base4.12.0.0 
Defined in Data.Monoid  
type Rep (Ap f a) Source #  Since: base4.12.0.0 
Defined in Data.Monoid 