Copyright | (c) Andy Gill 2001 (c) Oregon Graduate Institute of Science and Technology 2001 |
---|---|
License | BSD-style (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 :: k -> Type) (a :: k) = Alt {
- getAlt :: f a
- newtype Ap (f :: k -> Type) (a :: k) = 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 base-4.11.0.0.
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 base-4.11.0.0.
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 All Source # | @since base-2.01 |
Monoid Any Source # | @since base-2.01 |
Monoid Event Source # | @since base-4.4.0.0 |
Monoid Lifetime Source # |
@since base-4.8.0.0 |
Monoid ExceptionContext Source # | |
Defined in GHC.Internal.Exception.Context | |
Monoid Ordering Source # | @since base-2.01 |
Monoid () Source # | @since base-2.01 |
Monoid a => Monoid (STM a) Source # | @since base-4.17.0.0 |
FiniteBits a => Monoid (And a) Source # | This constraint is arguably too strong. However,
as some types (such as @since base-4.16 |
FiniteBits a => Monoid (Iff a) Source # | This constraint is arguably
too strong. However, as some types (such as @since base-4.16 |
Bits a => Monoid (Ior a) Source # | @since base-4.16 |
Bits a => Monoid (Xor a) Source # | @since base-4.16 |
Monoid a => Monoid (Identity a) Source # | @since base-4.9.0.0 |
Ord a => Monoid (Max a) Source # | @since base-4.8.0.0 |
Ord a => Monoid (Min a) Source # | @since base-4.8.0.0 |
Monoid (First a) Source # | @since base-2.01 |
Monoid (Last a) Source # | @since base-2.01 |
Monoid a => Monoid (Down a) Source # | @since base-4.11.0.0 |
Monoid a => Monoid (Dual a) Source # | @since base-2.01 |
Monoid (Endo a) Source # | @since base-2.01 |
Num a => Monoid (Product a) Source # | @since base-2.01 |
Num a => Monoid (Sum a) Source # | @since base-2.01 |
(Generic a, Monoid (Rep a ())) => Monoid (Generically a) Source # | @since base-4.17.0.0 |
Defined in GHC.Internal.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 base-4.12.0.0 |
Monoid a => Monoid (IO a) Source # | @since base-4.9.0.0 |
Semigroup a => Monoid (Maybe a) Source # | Lift a semigroup into Since 4.11.0: constraint on inner @since base-2.01 |
Monoid a => Monoid (Solo a) Source # | @since base-4.15 |
Monoid [a] Source # | @since base-2.01 |
Monoid (Proxy s) Source # | @since base-4.7.0.0 |
Monoid (U1 p) Source # | @since base-4.12.0.0 |
Monoid a => Monoid (ST s a) Source # | @since base-4.11.0.0 |
(Monoid a, Monoid b) => Monoid (a, b) Source # | @since base-2.01 |
Monoid b => Monoid (a -> b) Source # | @since base-2.01 |
Monoid a => Monoid (Const a b) Source # | @since base-4.9.0.0 |
(Applicative f, Monoid a) => Monoid (Ap f a) Source # | @since base-4.12.0.0 |
Alternative f => Monoid (Alt f a) Source # | @since base-4.8.0.0 |
Monoid (f p) => Monoid (Rec1 f p) Source # | @since base-4.12.0.0 |
(Monoid a, Monoid b, Monoid c) => Monoid (a, b, c) Source # | @since base-2.01 |
(Monoid (f p), Monoid (g p)) => Monoid ((f :*: g) p) Source # | @since base-4.12.0.0 |
Monoid c => Monoid (K1 i c p) Source # | @since base-4.12.0.0 |
(Monoid a, Monoid b, Monoid c, Monoid d) => Monoid (a, b, c, d) Source # | @since base-2.01 |
Monoid (f (g p)) => Monoid ((f :.: g) p) Source # | @since base-4.12.0.0 |
Monoid (f p) => Monoid (M1 i c f p) Source # | @since base-4.12.0.0 |
(Monoid a, Monoid b, Monoid c, Monoid d, Monoid e) => Monoid (a, b, c, d, e) Source # | @since base-2.01 |
(<>) :: 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 (<>)
.
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
Applicative Dual Source # | @since base-4.8.0.0 | ||||
Functor Dual Source # | @since base-4.8.0.0 | ||||
Monad Dual Source # | @since base-4.8.0.0 | ||||
MonadFix Dual Source # | @since base-4.8.0.0 | ||||
Foldable Dual Source # | @since base-4.8.0.0 | ||||
Defined in GHC.Internal.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 # | |||||
Traversable Dual Source # | @since base-4.8.0.0 | ||||
Generic1 Dual Source # | |||||
Defined in GHC.Internal.Data.Semigroup.Internal
| |||||
Monoid a => Monoid (Dual a) Source # | @since base-2.01 | ||||
Semigroup a => Semigroup (Dual a) Source # | @since base-4.9.0.0 | ||||
Data a => Data (Dual a) Source # | @since base-4.8.0.0 | ||||
Defined in GHC.Internal.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 # | |||||
Bounded a => Bounded (Dual a) Source # | @since base-2.01 | ||||
Generic (Dual a) Source # | |||||
Defined in GHC.Internal.Data.Semigroup.Internal
| |||||
Read a => Read (Dual a) Source # | @since base-2.01 | ||||
Show a => Show (Dual a) Source # | @since base-2.01 | ||||
Eq a => Eq (Dual a) Source # | @since base-2.01 | ||||
Ord a => Ord (Dual a) Source # | @since base-2.01 | ||||
Defined in GHC.Internal.Data.Semigroup.Internal | |||||
type Rep1 Dual Source # | @since base-4.7.0.0 | ||||
Defined in GHC.Internal.Data.Semigroup.Internal | |||||
type Rep (Dual a) Source # | @since base-4.7.0.0 | ||||
Defined in GHC.Internal.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
Instances
Monoid (Endo a) Source # | @since base-2.01 | ||||
Semigroup (Endo a) Source # | @since base-4.9.0.0 | ||||
Generic (Endo a) Source # | |||||
Defined in GHC.Internal.Data.Semigroup.Internal
| |||||
type Rep (Endo a) Source # | @since base-4.7.0.0 | ||||
Defined in GHC.Internal.Data.Semigroup.Internal |
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
Monoid All Source # | @since base-2.01 | ||||
Semigroup All Source # | @since base-4.9.0.0 | ||||
Data All Source # | @since base-4.8.0.0 | ||||
Defined in GHC.Internal.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 # | |||||
Bounded All Source # | @since base-2.01 | ||||
Generic All Source # | |||||
Defined in GHC.Internal.Data.Semigroup.Internal
| |||||
Read All Source # | @since base-2.01 | ||||
Show All Source # | @since base-2.01 | ||||
Eq All Source # | @since base-2.01 | ||||
Ord All Source # | @since base-2.01 | ||||
type Rep All Source # | @since base-4.7.0.0 | ||||
Defined in GHC.Internal.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
Monoid Any Source # | @since base-2.01 | ||||
Semigroup Any Source # | @since base-4.9.0.0 | ||||
Data Any Source # | @since base-4.8.0.0 | ||||
Defined in GHC.Internal.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 # | |||||
Bounded Any Source # | @since base-2.01 | ||||
Generic Any Source # | |||||
Defined in GHC.Internal.Data.Semigroup.Internal
| |||||
Read Any Source # | @since base-2.01 | ||||
Show Any Source # | @since base-2.01 | ||||
Eq Any Source # | @since base-2.01 | ||||
Ord Any Source # | @since base-2.01 | ||||
type Rep Any Source # | @since base-4.7.0.0 | ||||
Defined in GHC.Internal.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
Applicative Sum Source # | @since base-4.8.0.0 | ||||
Functor Sum Source # | @since base-4.8.0.0 | ||||
Monad Sum Source # | @since base-4.8.0.0 | ||||
MonadFix Sum Source # | @since base-4.8.0.0 | ||||
Foldable Sum Source # | @since base-4.8.0.0 | ||||
Defined in GHC.Internal.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 # | |||||
Traversable Sum Source # | @since base-4.8.0.0 | ||||
Generic1 Sum Source # | |||||
Defined in GHC.Internal.Data.Semigroup.Internal
| |||||
Num a => Monoid (Sum a) Source # | @since base-2.01 | ||||
Num a => Semigroup (Sum a) Source # | @since base-4.9.0.0 | ||||
Data a => Data (Sum a) Source # | @since base-4.8.0.0 | ||||
Defined in GHC.Internal.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 # | |||||
Bounded a => Bounded (Sum a) Source # | @since base-2.01 | ||||
Generic (Sum a) Source # | |||||
Defined in GHC.Internal.Data.Semigroup.Internal
| |||||
Num a => Num (Sum a) Source # | @since base-4.7.0.0 | ||||
Defined in GHC.Internal.Data.Semigroup.Internal | |||||
Read a => Read (Sum a) Source # | @since base-2.01 | ||||
Show a => Show (Sum a) Source # | @since base-2.01 | ||||
Eq a => Eq (Sum a) Source # | @since base-2.01 | ||||
Ord a => Ord (Sum a) Source # | @since base-2.01 | ||||
Defined in GHC.Internal.Data.Semigroup.Internal | |||||
type Rep1 Sum Source # | @since base-4.7.0.0 | ||||
Defined in GHC.Internal.Data.Semigroup.Internal | |||||
type Rep (Sum a) Source # | @since base-4.7.0.0 | ||||
Defined in GHC.Internal.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
Applicative Product Source # | @since base-4.8.0.0 | ||||
Defined in GHC.Internal.Data.Semigroup.Internal | |||||
Functor Product Source # | @since base-4.8.0.0 | ||||
Monad Product Source # | @since base-4.8.0.0 | ||||
MonadFix Product Source # | @since base-4.8.0.0 | ||||
Foldable Product Source # | @since base-4.8.0.0 | ||||
Defined in GHC.Internal.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 # | |||||
Traversable Product Source # | @since base-4.8.0.0 | ||||
Defined in GHC.Internal.Data.Traversable | |||||
Generic1 Product Source # | |||||
Defined in GHC.Internal.Data.Semigroup.Internal
| |||||
Num a => Monoid (Product a) Source # | @since base-2.01 | ||||
Num a => Semigroup (Product a) Source # | @since base-4.9.0.0 | ||||
Data a => Data (Product a) Source # | @since base-4.8.0.0 | ||||
Defined in GHC.Internal.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 # | |||||
Bounded a => Bounded (Product a) Source # | @since base-2.01 | ||||
Generic (Product a) Source # | |||||
Defined in GHC.Internal.Data.Semigroup.Internal
| |||||
Num a => Num (Product a) Source # | @since base-4.7.0.0 | ||||
Defined in GHC.Internal.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 base-2.01 | ||||
Show a => Show (Product a) Source # | @since base-2.01 | ||||
Eq a => Eq (Product a) Source # | @since base-2.01 | ||||
Ord a => Ord (Product a) Source # | @since base-2.01 | ||||
Defined in GHC.Internal.Data.Semigroup.Internal | |||||
type Rep1 Product Source # | @since base-4.7.0.0 | ||||
Defined in GHC.Internal.Data.Semigroup.Internal | |||||
type Rep (Product a) Source # | @since base-4.7.0.0 | ||||
Defined in GHC.Internal.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 GHC.Internal.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 non-Nothing
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 non-Nothing
,
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
Applicative First Source # | @since base-4.8.0.0 | ||||
Functor First Source # | @since base-4.8.0.0 | ||||
Monad First Source # | @since base-4.8.0.0 | ||||
MonadFix First Source # | @since base-4.8.0.0 | ||||
Foldable First Source # | @since base-4.8.0.0 | ||||
Defined in GHC.Internal.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 base-4.8.0.0 | ||||
Defined in GHC.Internal.Data.Traversable | |||||
Generic1 First Source # | |||||
Defined in GHC.Internal.Data.Monoid
| |||||
Monoid (First a) Source # | @since base-2.01 | ||||
Semigroup (First a) Source # | @since base-4.9.0.0 | ||||
Data a => Data (First a) Source # | @since base-4.8.0.0 | ||||
Defined in GHC.Internal.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 # | |||||
Generic (First a) Source # | |||||
Defined in GHC.Internal.Data.Monoid
| |||||
Read a => Read (First a) Source # | @since base-2.01 | ||||
Show a => Show (First a) Source # | @since base-2.01 | ||||
Eq a => Eq (First a) Source # | @since base-2.01 | ||||
Ord a => Ord (First a) Source # | @since base-2.01 | ||||
Defined in GHC.Internal.Data.Monoid | |||||
type Rep1 First Source # | @since base-4.7.0.0 | ||||
Defined in GHC.Internal.Data.Monoid | |||||
type Rep (First a) Source # | @since base-4.7.0.0 | ||||
Defined in GHC.Internal.Data.Monoid |
Maybe monoid returning the rightmost non-Nothing
value.
is isomorphic to Last
a
, and thus to
Dual
(First
a)Dual
(Alt
Maybe
a)
Data.Semigroup.
Last
. The former returns the last non-Nothing
,
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
Applicative Last Source # | @since base-4.8.0.0 | ||||
Functor Last Source # | @since base-4.8.0.0 | ||||
Monad Last Source # | @since base-4.8.0.0 | ||||
MonadFix Last Source # | @since base-4.8.0.0 | ||||
Foldable Last Source # | @since base-4.8.0.0 | ||||
Defined in GHC.Internal.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 base-4.8.0.0 | ||||
Generic1 Last Source # | |||||
Defined in GHC.Internal.Data.Monoid
| |||||
Monoid (Last a) Source # | @since base-2.01 | ||||
Semigroup (Last a) Source # | @since base-4.9.0.0 | ||||
Data a => Data (Last a) Source # | @since base-4.8.0.0 | ||||
Defined in GHC.Internal.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 # | |||||
Generic (Last a) Source # | |||||
Defined in GHC.Internal.Data.Monoid
| |||||
Read a => Read (Last a) Source # | @since base-2.01 | ||||
Show a => Show (Last a) Source # | @since base-2.01 | ||||
Eq a => Eq (Last a) Source # | @since base-2.01 | ||||
Ord a => Ord (Last a) Source # | @since base-2.01 | ||||
Defined in GHC.Internal.Data.Monoid | |||||
type Rep1 Last Source # | @since base-4.7.0.0 | ||||
Defined in GHC.Internal.Data.Monoid | |||||
type Rep (Last a) Source # | @since base-4.7.0.0 | ||||
Defined in GHC.Internal.Data.Monoid |
Alternative
wrapper
newtype Alt (f :: k -> Type) (a :: k) Source #
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 base-4.8.0.0
Instances
Generic1 (Alt f :: k -> Type) Source # | |||||
Defined in GHC.Internal.Data.Semigroup.Internal
| |||||
Alternative f => Alternative (Alt f) Source # | @since base-4.8.0.0 | ||||
Applicative f => Applicative (Alt f) Source # | @since base-4.8.0.0 | ||||
Functor f => Functor (Alt f) Source # | @since base-4.8.0.0 | ||||
Monad f => Monad (Alt f) Source # | @since base-4.8.0.0 | ||||
MonadPlus f => MonadPlus (Alt f) Source # | @since base-4.8.0.0 | ||||
MonadFix f => MonadFix (Alt f) Source # | @since base-4.8.0.0 | ||||
Foldable f => Foldable (Alt f) Source # | @since base-4.12.0.0 | ||||
Defined in GHC.Internal.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 # | |||||
Traversable f => Traversable (Alt f) Source # | @since base-4.12.0.0 | ||||
Defined in GHC.Internal.Data.Traversable | |||||
Alternative f => Monoid (Alt f a) Source # | @since base-4.8.0.0 | ||||
Alternative f => Semigroup (Alt f a) Source # | @since base-4.9.0.0 | ||||
(Data (f a), Data a, Typeable f) => Data (Alt f a) Source # | @since base-4.8.0.0 | ||||
Defined in GHC.Internal.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 # | |||||
Enum (f a) => Enum (Alt f a) Source # | @since base-4.8.0.0 | ||||
Defined in GHC.Internal.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 # | |||||
Defined in GHC.Internal.Data.Semigroup.Internal
| |||||
Num (f a) => Num (Alt f a) Source # | @since base-4.8.0.0 | ||||
Defined in GHC.Internal.Data.Semigroup.Internal | |||||
Read (f a) => Read (Alt f a) Source # | @since base-4.8.0.0 | ||||
Show (f a) => Show (Alt f a) Source # | @since base-4.8.0.0 | ||||
Eq (f a) => Eq (Alt f a) Source # | @since base-4.8.0.0 | ||||
Ord (f a) => Ord (Alt f a) Source # | @since base-4.8.0.0 | ||||
Defined in GHC.Internal.Data.Semigroup.Internal | |||||
type Rep1 (Alt f :: k -> Type) Source # | @since base-4.8.0.0 | ||||
Defined in GHC.Internal.Data.Semigroup.Internal | |||||
type Rep (Alt f a) Source # | @since base-4.8.0.0 | ||||
Defined in GHC.Internal.Data.Semigroup.Internal |
Applicative
wrapper
newtype Ap (f :: k -> Type) (a :: k) Source #
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 base-4.12.0.0
Instances
Generic1 (Ap f :: k -> Type) Source # | |||||
Defined in GHC.Internal.Data.Monoid
| |||||
Alternative f => Alternative (Ap f) Source # | @since base-4.12.0.0 | ||||
Applicative f => Applicative (Ap f) Source # | @since base-4.12.0.0 | ||||
Functor f => Functor (Ap f) Source # | @since base-4.12.0.0 | ||||
Monad f => Monad (Ap f) Source # | @since base-4.12.0.0 | ||||
MonadPlus f => MonadPlus (Ap f) Source # | @since base-4.12.0.0 | ||||
MonadFail f => MonadFail (Ap f) Source # | @since base-4.12.0.0 | ||||
MonadFix f => MonadFix (Ap f) Source # | @since base-4.12.0.0 | ||||
Foldable f => Foldable (Ap f) Source # | @since base-4.12.0.0 | ||||
Defined in GHC.Internal.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 # | |||||
Traversable f => Traversable (Ap f) Source # | @since base-4.12.0.0 | ||||
Defined in GHC.Internal.Data.Traversable | |||||
(Applicative f, Monoid a) => Monoid (Ap f a) Source # | @since base-4.12.0.0 | ||||
(Applicative f, Semigroup a) => Semigroup (Ap f a) Source # | @since base-4.12.0.0 | ||||
(Data (f a), Data a, Typeable f) => Data (Ap f a) Source # | @since base-4.12.0.0 | ||||
Defined in GHC.Internal.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, Bounded a) => Bounded (Ap f a) Source # | @since base-4.12.0.0 | ||||
Enum (f a) => Enum (Ap f a) Source # | @since base-4.12.0.0 | ||||
Defined in GHC.Internal.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 # | |||||
Defined in GHC.Internal.Data.Monoid
| |||||
(Applicative f, Num a) => Num (Ap f a) Source # | Note that even if the underlying Commutativity:
Additive inverse:
Distributivity:
@since base-4.12.0.0 | ||||
Defined in GHC.Internal.Data.Monoid | |||||
Read (f a) => Read (Ap f a) Source # | @since base-4.12.0.0 | ||||
Show (f a) => Show (Ap f a) Source # | @since base-4.12.0.0 | ||||
Eq (f a) => Eq (Ap f a) Source # | @since base-4.12.0.0 | ||||
Ord (f a) => Ord (Ap f a) Source # | @since base-4.12.0.0 | ||||
Defined in GHC.Internal.Data.Monoid | |||||
type Rep1 (Ap f :: k -> Type) Source # | @since base-4.12.0.0 | ||||
Defined in GHC.Internal.Data.Monoid | |||||
type Rep (Ap f a) Source # | @since base-4.12.0.0 | ||||
Defined in GHC.Internal.Data.Monoid |