Copyright	(C) 2011-2015 Edward Kmett
License	BSD-style (see the file LICENSE)
Maintainer	Edward Kmett <ekmett@gmail.com>
Stability	provisional
Portability	portable
Safe Haskell	Safe
Language	Haskell2010

Data.Bifunctor.Apply

Contents

Biappliable bifunctors

Description

Synopsis

class (forall a. Functor (p a)) => Bifunctor (p :: Type -> Type -> Type) where
- bimap :: (a -> b) -> (c -> d) -> p a c -> p b d
- first :: (a -> b) -> p a c -> p b c
- second :: (b -> c) -> p a b -> p a c
class Bifunctor p => Biapply (p :: Type -> Type -> Type) where
- (<<.>>) :: p (a -> b) (c -> d) -> p a c -> p b d
- (.>>) :: p a b -> p c d -> p c d
- (<<.) :: p a b -> p c d -> p a b
(<<$>>) :: (a -> b) -> a -> b
(<<..>>) :: Biapply p => p a c -> p (a -> b) (c -> d) -> p b d
bilift2 :: Biapply w => (a -> b -> c) -> (d -> e -> f) -> w a d -> w b e -> w c f
bilift3 :: Biapply w => (a -> b -> c -> d) -> (e -> f -> g -> h) -> w a e -> w b f -> w c g -> w d h

Biappliable bifunctors

class (forall a. Functor (p a)) => Bifunctor (p :: Type -> Type -> Type) where #

A bifunctor is a type constructor that takes two type arguments and is a functor in both arguments. That is, unlike with Functor, a type constructor such as Either does not need to be partially applied for a Bifunctor instance, and the methods in this class permit mapping functions over the Left value or the Right value, or both at the same time.

Formally, the class Bifunctor represents a bifunctor from Hask -> Hask.

Intuitively it is a bifunctor where both the first and second arguments are covariant.

You can define a Bifunctor by either defining bimap or by defining both first and second. A partially applied Bifunctor must be a Functor and the second method must agree with fmap. From this it follows that:

second id = id

If you supply bimap, you should ensure that:

bimap id id ≡ id

If you supply first and second, ensure:

first id ≡ id
second id ≡ id

If you supply both, you should also ensure:

bimap f g ≡ first f . second g

These ensure by parametricity:

bimap  (f . g) (h . i) ≡ bimap f h . bimap g i
first  (f . g) ≡ first  f . first  g
second (f . g) ≡ second f . second g

Since 4.18.0.0 Functor is a superclass of 'Bifunctor.

Since: base-4.8.0.0

Minimal complete definition

bimap | first, second

Methods

bimap :: (a -> b) -> (c -> d) -> p a c -> p b d #

Map over both arguments at the same time.

bimap f g ≡ first f . second g

Examples

Expand

>>> bimap toUpper (+1) ('j', 3)
('J',4)

>>> bimap toUpper (+1) (Left 'j')
Left 'J'

>>> bimap toUpper (+1) (Right 3)
Right 4

first :: (a -> b) -> p a c -> p b c #

Map covariantly over the first argument.

first f ≡ bimap f id

Examples

Expand

>>> first toUpper ('j', 3)
('J',3)

>>> first toUpper (Left 'j')
Left 'J'

second :: (b -> c) -> p a b -> p a c #

Map covariantly over the second argument.

second ≡ bimap id

Examples

Expand

>>> second (+1) ('j', 3)
('j',4)

>>> second (+1) (Right 3)
Right 4

Instances

Instances details

Bifunctor Either	Since: base-4.8.0.0
Instance details Defined in Data.Bifunctor Methods bimap :: (a -> b) -> (c -> d) -> Either a c -> Either b d # first :: (a -> b) -> Either a c -> Either b c # second :: (b -> c) -> Either a b -> Either a c #
Bifunctor Arg	Since: base-4.9.0.0
Instance details Defined in Data.Semigroup Methods bimap :: (a -> b) -> (c -> d) -> Arg a c -> Arg b d # first :: (a -> b) -> Arg a c -> Arg b c # second :: (b -> c) -> Arg a b -> Arg a c #
Bifunctor (,)	Class laws for tuples hold only up to laziness. Both `first` `id` and `second` `id` are lazier than `id` (and `fmap` `id`): `>>>` first id (undefined :: (Int, Word)) `seq` ()() `>>>` second id (undefined :: (Int, Word)) `seq` ()() `>>>` id (undefined :: (Int, Word)) `seq` ()*** Exception: Prelude.undefined Since: base-4.8.0.0
Instance details Defined in Data.Bifunctor Methods bimap :: (a -> b) -> (c -> d) -> (a, c) -> (b, d) # first :: (a -> b) -> (a, c) -> (b, c) # second :: (b -> c) -> (a, b) -> (a, c) #
Bifunctor (Const :: Type -> Type -> Type)	Since: base-4.8.0.0
Instance details Defined in Data.Bifunctor Methods bimap :: (a -> b) -> (c -> d) -> Const a c -> Const b d # first :: (a -> b) -> Const a c -> Const b c # second :: (b -> c) -> Const a b -> Const a c #
Bifunctor bi => Bifunctor (Biap bi)
Instance details Defined in Data.Bifunctor.Biap Methods bimap :: (a -> b) -> (c -> d) -> Biap bi a c -> Biap bi b d # first :: (a -> b) -> Biap bi a c -> Biap bi b c # second :: (b -> c) -> Biap bi a b -> Biap bi a c #
Bifunctor (Tagged :: Type -> Type -> Type)
Instance details Defined in Data.Tagged Methods bimap :: (a -> b) -> (c -> d) -> Tagged a c -> Tagged b d # first :: (a -> b) -> Tagged a c -> Tagged b c # second :: (b -> c) -> Tagged a b -> Tagged a c #
Bifunctor (Constant :: Type -> Type -> Type)
Instance details Defined in Data.Functor.Constant Methods bimap :: (a -> b) -> (c -> d) -> Constant a c -> Constant b d # first :: (a -> b) -> Constant a c -> Constant b c # second :: (b -> c) -> Constant a b -> Constant a c #
Bifunctor ((,,) x1)	Since: base-4.8.0.0
Instance details Defined in Data.Bifunctor Methods bimap :: (a -> b) -> (c -> d) -> (x1, a, c) -> (x1, b, d) # first :: (a -> b) -> (x1, a, c) -> (x1, b, c) # second :: (b -> c) -> (x1, a, b) -> (x1, a, c) #
Bifunctor (K1 i :: Type -> Type -> Type)	Since: base-4.9.0.0
Instance details Defined in Data.Bifunctor Methods bimap :: (a -> b) -> (c -> d) -> K1 i a c -> K1 i b d # first :: (a -> b) -> K1 i a c -> K1 i b c # second :: (b -> c) -> K1 i a b -> K1 i a c #
Bifunctor ((,,,) x1 x2)	Since: base-4.8.0.0
Instance details Defined in Data.Bifunctor Methods bimap :: (a -> b) -> (c -> d) -> (x1, x2, a, c) -> (x1, x2, b, d) # first :: (a -> b) -> (x1, x2, a, c) -> (x1, x2, b, c) # second :: (b -> c) -> (x1, x2, a, b) -> (x1, x2, a, c) #
Functor f => Bifunctor (Clown f :: Type -> Type -> Type)
Instance details Defined in Data.Bifunctor.Clown Methods bimap :: (a -> b) -> (c -> d) -> Clown f a c -> Clown f b d # first :: (a -> b) -> Clown f a c -> Clown f b c # second :: (b -> c) -> Clown f a b -> Clown f a c #
Bifunctor p => Bifunctor (Flip p)
Instance details Defined in Data.Bifunctor.Flip Methods bimap :: (a -> b) -> (c -> d) -> Flip p a c -> Flip p b d # first :: (a -> b) -> Flip p a c -> Flip p b c # second :: (b -> c) -> Flip p a b -> Flip p a c #
Functor g => Bifunctor (Joker g :: Type -> Type -> Type)
Instance details Defined in Data.Bifunctor.Joker Methods bimap :: (a -> b) -> (c -> d) -> Joker g a c -> Joker g b d # first :: (a -> b) -> Joker g a c -> Joker g b c # second :: (b -> c) -> Joker g a b -> Joker g a c #
Bifunctor p => Bifunctor (WrappedBifunctor p)
Instance details Defined in Data.Bifunctor.Wrapped Methods bimap :: (a -> b) -> (c -> d) -> WrappedBifunctor p a c -> WrappedBifunctor p b d # first :: (a -> b) -> WrappedBifunctor p a c -> WrappedBifunctor p b c # second :: (b -> c) -> WrappedBifunctor p a b -> WrappedBifunctor p a c #
Bifunctor ((,,,,) x1 x2 x3)	Since: base-4.8.0.0
Instance details Defined in Data.Bifunctor Methods bimap :: (a -> b) -> (c -> d) -> (x1, x2, x3, a, c) -> (x1, x2, x3, b, d) # first :: (a -> b) -> (x1, x2, x3, a, c) -> (x1, x2, x3, b, c) # second :: (b -> c) -> (x1, x2, x3, a, b) -> (x1, x2, x3, a, c) #
(Bifunctor f, Bifunctor g) => Bifunctor (Product f g)
Instance details Defined in Data.Bifunctor.Product Methods bimap :: (a -> b) -> (c -> d) -> Product f g a c -> Product f g b d # first :: (a -> b) -> Product f g a c -> Product f g b c # second :: (b -> c) -> Product f g a b -> Product f g a c #
(Bifunctor p, Bifunctor q) => Bifunctor (Sum p q)
Instance details Defined in Data.Bifunctor.Sum Methods bimap :: (a -> b) -> (c -> d) -> Sum p q a c -> Sum p q b d # first :: (a -> b) -> Sum p q a c -> Sum p q b c # second :: (b -> c) -> Sum p q a b -> Sum p q a c #
Bifunctor ((,,,,,) x1 x2 x3 x4)	Since: base-4.8.0.0
Instance details Defined in Data.Bifunctor Methods bimap :: (a -> b) -> (c -> d) -> (x1, x2, x3, x4, a, c) -> (x1, x2, x3, x4, b, d) # first :: (a -> b) -> (x1, x2, x3, x4, a, c) -> (x1, x2, x3, x4, b, c) # second :: (b -> c) -> (x1, x2, x3, x4, a, b) -> (x1, x2, x3, x4, a, c) #
(Functor f, Bifunctor p) => Bifunctor (Tannen f p)
Instance details Defined in Data.Bifunctor.Tannen Methods bimap :: (a -> b) -> (c -> d) -> Tannen f p a c -> Tannen f p b d # first :: (a -> b) -> Tannen f p a c -> Tannen f p b c # second :: (b -> c) -> Tannen f p a b -> Tannen f p a c #
Bifunctor ((,,,,,,) x1 x2 x3 x4 x5)	Since: base-4.8.0.0
Instance details Defined in Data.Bifunctor Methods bimap :: (a -> b) -> (c -> d) -> (x1, x2, x3, x4, x5, a, c) -> (x1, x2, x3, x4, x5, b, d) # first :: (a -> b) -> (x1, x2, x3, x4, x5, a, c) -> (x1, x2, x3, x4, x5, b, c) # second :: (b -> c) -> (x1, x2, x3, x4, x5, a, b) -> (x1, x2, x3, x4, x5, a, c) #
(Bifunctor p, Functor f, Functor g) => Bifunctor (Biff p f g)
Instance details Defined in Data.Bifunctor.Biff Methods bimap :: (a -> b) -> (c -> d) -> Biff p f g a c -> Biff p f g b d # first :: (a -> b) -> Biff p f g a c -> Biff p f g b c # second :: (b -> c) -> Biff p f g a b -> Biff p f g a c #

class Bifunctor p => Biapply (p :: Type -> Type -> Type) where #

Minimal complete definition

(<<.>>)

Methods

(<<.>>) :: p (a -> b) (c -> d) -> p a c -> p b d infixl 4 #

(.>>) :: p a b -> p c d -> p c d infixl 4 #

a .> b ≡ const id <$> a <.> b

(<<.) :: p a b -> p c d -> p a b infixl 4 #

a <. b ≡ const <$> a <.> b

Instances

Instances details

Biapply Arg #
Instance details Defined in Data.Functor.Bind.Class Methods (<<.>>) :: Arg (a -> b) (c -> d) -> Arg a c -> Arg b d # (.>>) :: Arg a b -> Arg c d -> Arg c d # (<<.) :: Arg a b -> Arg c d -> Arg a b #
Biapply (,) #
Instance details Defined in Data.Functor.Bind.Class Methods (<<.>>) :: (a -> b, c -> d) -> (a, c) -> (b, d) # (.>>) :: (a, b) -> (c, d) -> (c, d) # (<<.) :: (a, b) -> (c, d) -> (a, b) #
Biapply (Const :: Type -> Type -> Type) #
Instance details Defined in Data.Functor.Bind.Class Methods (<<.>>) :: Const (a -> b) (c -> d) -> Const a c -> Const b d # (.>>) :: Const a b -> Const c d -> Const c d # (<<.) :: Const a b -> Const c d -> Const a b #
Biapply (Tagged :: Type -> Type -> Type) #
Instance details Defined in Data.Functor.Bind.Class Methods (<<.>>) :: Tagged (a -> b) (c -> d) -> Tagged a c -> Tagged b d # (.>>) :: Tagged a b -> Tagged c d -> Tagged c d # (<<.) :: Tagged a b -> Tagged c d -> Tagged a b #
Semigroup x => Biapply ((,,) x) #
Instance details Defined in Data.Functor.Bind.Class Methods (<<.>>) :: (x, a -> b, c -> d) -> (x, a, c) -> (x, b, d) # (.>>) :: (x, a, b) -> (x, c, d) -> (x, c, d) # (<<.) :: (x, a, b) -> (x, c, d) -> (x, a, b) #
(Semigroup x, Semigroup y) => Biapply ((,,,) x y) #
Instance details Defined in Data.Functor.Bind.Class Methods (<<.>>) :: (x, y, a -> b, c -> d) -> (x, y, a, c) -> (x, y, b, d) # (.>>) :: (x, y, a, b) -> (x, y, c, d) -> (x, y, c, d) # (<<.) :: (x, y, a, b) -> (x, y, c, d) -> (x, y, a, b) #
Apply f => Biapply (Clown f :: Type -> Type -> Type) #
Instance details Defined in Data.Functor.Bind.Class Methods (<<.>>) :: Clown f (a -> b) (c -> d) -> Clown f a c -> Clown f b d # (.>>) :: Clown f a b -> Clown f c d -> Clown f c d # (<<.) :: Clown f a b -> Clown f c d -> Clown f a b #
Biapply p => Biapply (Flip p) #
Instance details Defined in Data.Functor.Bind.Class Methods (<<.>>) :: Flip p (a -> b) (c -> d) -> Flip p a c -> Flip p b d # (.>>) :: Flip p a b -> Flip p c d -> Flip p c d # (<<.) :: Flip p a b -> Flip p c d -> Flip p a b #
Apply g => Biapply (Joker g :: Type -> Type -> Type) #
Instance details Defined in Data.Functor.Bind.Class Methods (<<.>>) :: Joker g (a -> b) (c -> d) -> Joker g a c -> Joker g b d # (.>>) :: Joker g a b -> Joker g c d -> Joker g c d # (<<.) :: Joker g a b -> Joker g c d -> Joker g a b #
Biapply p => Biapply (WrappedBifunctor p) #
Instance details Defined in Data.Functor.Bind.Class Methods (<<.>>) :: WrappedBifunctor p (a -> b) (c -> d) -> WrappedBifunctor p a c -> WrappedBifunctor p b d # (.>>) :: WrappedBifunctor p a b -> WrappedBifunctor p c d -> WrappedBifunctor p c d # (<<.) :: WrappedBifunctor p a b -> WrappedBifunctor p c d -> WrappedBifunctor p a b #
(Semigroup x, Semigroup y, Semigroup z) => Biapply ((,,,,) x y z) #
Instance details Defined in Data.Functor.Bind.Class Methods (<<.>>) :: (x, y, z, a -> b, c -> d) -> (x, y, z, a, c) -> (x, y, z, b, d) # (.>>) :: (x, y, z, a, b) -> (x, y, z, c, d) -> (x, y, z, c, d) # (<<.) :: (x, y, z, a, b) -> (x, y, z, c, d) -> (x, y, z, a, b) #
(Biapply p, Biapply q) => Biapply (Product p q) #
Instance details Defined in Data.Functor.Bind.Class Methods (<<.>>) :: Product p q (a -> b) (c -> d) -> Product p q a c -> Product p q b d # (.>>) :: Product p q a b -> Product p q c d -> Product p q c d # (<<.) :: Product p q a b -> Product p q c d -> Product p q a b #
(Apply f, Biapply p) => Biapply (Tannen f p) #
Instance details Defined in Data.Functor.Bind.Class Methods (<<.>>) :: Tannen f p (a -> b) (c -> d) -> Tannen f p a c -> Tannen f p b d # (.>>) :: Tannen f p a b -> Tannen f p c d -> Tannen f p c d # (<<.) :: Tannen f p a b -> Tannen f p c d -> Tannen f p a b #
(Biapply p, Apply f, Apply g) => Biapply (Biff p f g) #
Instance details Defined in Data.Functor.Bind.Class Methods (<<.>>) :: Biff p f g (a -> b) (c -> d) -> Biff p f g a c -> Biff p f g b d # (.>>) :: Biff p f g a b -> Biff p f g c d -> Biff p f g c d # (<<.) :: Biff p f g a b -> Biff p f g c d -> Biff p f g a b #

(<<$>>) :: (a -> b) -> a -> b infixl 4 #

(<<..>>) :: Biapply p => p a c -> p (a -> b) (c -> d) -> p b d infixl 4 #

bilift2 :: Biapply w => (a -> b -> c) -> (d -> e -> f) -> w a d -> w b e -> w c f #

Lift binary functions

bilift3 :: Biapply w => (a -> b -> c -> d) -> (e -> f -> g -> h) -> w a e -> w b f -> w c g -> w d h #

Lift ternary functions