semigroupoids-5.3.3: Semigroupoids: Category sans id

Copyright(C) 2011-2015 Edward Kmett
LicenseBSD-style (see the file LICENSE)
MaintainerEdward Kmett <ekmett@gmail.com>
Stabilityprovisional
Portabilityportable
Safe HaskellSafe
LanguageHaskell98

Data.Bifunctor.Apply

Contents

Description

 
Synopsis

Biappliable bifunctors

class 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.

If you supply bimap, you should ensure that:

bimap id idid

If you supply first and second, ensure:

first idid
second idid

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: 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.

secondbimap id

Examples

Expand
>>> second (+1) ('j', 3)
('j',4)
>>> second (+1) (Right 3)
Right 4
Instances
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 (,)

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 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 ((,,) 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 (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 (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 (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) #

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 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 #

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 (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 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 ((,,,,,) 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) #

(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 ((,,,,,,) 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) #

(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 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 where Source #

Minimal complete definition

(<<.>>)

Methods

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

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

a .> b ≡ const id <$> a <.> b

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

a <. b ≡ const <$> a <.> b
Instances
Biapply (,) Source # 
Instance details

Defined in Data.Functor.Bind.Class

Methods

(<<.>>) :: (a -> b, c -> d) -> (a, c) -> (b, d) Source #

(.>>) :: (a, b) -> (c, d) -> (c, d) Source #

(<<.) :: (a, b) -> (c, d) -> (a, b) Source #

Biapply Arg Source # 
Instance details

Defined in Data.Functor.Bind.Class

Methods

(<<.>>) :: Arg (a -> b) (c -> d) -> Arg a c -> Arg b d Source #

(.>>) :: Arg a b -> Arg c d -> Arg c d Source #

(<<.) :: Arg a b -> Arg c d -> Arg a b Source #

Semigroup x => Biapply ((,,) x) Source # 
Instance details

Defined in Data.Functor.Bind.Class

Methods

(<<.>>) :: (x, a -> b, c -> d) -> (x, a, c) -> (x, b, d) Source #

(.>>) :: (x, a, b) -> (x, c, d) -> (x, c, d) Source #

(<<.) :: (x, a, b) -> (x, c, d) -> (x, a, b) Source #

Biapply (Const :: Type -> Type -> Type) Source # 
Instance details

Defined in Data.Functor.Bind.Class

Methods

(<<.>>) :: Const (a -> b) (c -> d) -> Const a c -> Const b d Source #

(.>>) :: Const a b -> Const c d -> Const c d Source #

(<<.) :: Const a b -> Const c d -> Const a b Source #

Biapply (Tagged :: Type -> Type -> Type) Source # 
Instance details

Defined in Data.Functor.Bind.Class

Methods

(<<.>>) :: Tagged (a -> b) (c -> d) -> Tagged a c -> Tagged b d Source #

(.>>) :: Tagged a b -> Tagged c d -> Tagged c d Source #

(<<.) :: Tagged a b -> Tagged c d -> Tagged a b Source #

(Semigroup x, Semigroup y) => Biapply ((,,,) x y) Source # 
Instance details

Defined in Data.Functor.Bind.Class

Methods

(<<.>>) :: (x, y, a -> b, c -> d) -> (x, y, a, c) -> (x, y, b, d) Source #

(.>>) :: (x, y, a, b) -> (x, y, c, d) -> (x, y, c, d) Source #

(<<.) :: (x, y, a, b) -> (x, y, c, d) -> (x, y, a, b) Source #

(Semigroup x, Semigroup y, Semigroup z) => Biapply ((,,,,) x y z) Source # 
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) Source #

(.>>) :: (x, y, z, a, b) -> (x, y, z, c, d) -> (x, y, z, c, d) Source #

(<<.) :: (x, y, z, a, b) -> (x, y, z, c, d) -> (x, y, z, a, b) Source #

Biapply p => Biapply (WrappedBifunctor p) Source # 
Instance details

Defined in Data.Functor.Bind.Class

Apply g => Biapply (Joker g :: Type -> Type -> Type) Source # 
Instance details

Defined in Data.Functor.Bind.Class

Methods

(<<.>>) :: Joker g (a -> b) (c -> d) -> Joker g a c -> Joker g b d Source #

(.>>) :: Joker g a b -> Joker g c d -> Joker g c d Source #

(<<.) :: Joker g a b -> Joker g c d -> Joker g a b Source #

Biapply p => Biapply (Flip p) Source # 
Instance details

Defined in Data.Functor.Bind.Class

Methods

(<<.>>) :: Flip p (a -> b) (c -> d) -> Flip p a c -> Flip p b d Source #

(.>>) :: Flip p a b -> Flip p c d -> Flip p c d Source #

(<<.) :: Flip p a b -> Flip p c d -> Flip p a b Source #

Apply f => Biapply (Clown f :: Type -> Type -> Type) Source # 
Instance details

Defined in Data.Functor.Bind.Class

Methods

(<<.>>) :: Clown f (a -> b) (c -> d) -> Clown f a c -> Clown f b d Source #

(.>>) :: Clown f a b -> Clown f c d -> Clown f c d Source #

(<<.) :: Clown f a b -> Clown f c d -> Clown f a b Source #

(Biapply p, Biapply q) => Biapply (Product p q) Source # 
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 Source #

(.>>) :: Product p q a b -> Product p q c d -> Product p q c d Source #

(<<.) :: Product p q a b -> Product p q c d -> Product p q a b Source #

(Apply f, Biapply p) => Biapply (Tannen f p) Source # 
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 Source #

(.>>) :: Tannen f p a b -> Tannen f p c d -> Tannen f p c d Source #

(<<.) :: Tannen f p a b -> Tannen f p c d -> Tannen f p a b Source #

(Biapply p, Apply f, Apply g) => Biapply (Biff p f g) Source # 
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 Source #

(.>>) :: Biff p f g a b -> Biff p f g c d -> Biff p f g c d Source #

(<<.) :: Biff p f g a b -> Biff p f g c d -> Biff p f g a b Source #

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

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

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

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 Source #

Lift ternary functions