semigroupoids-5.0.0.1: Semigroupoids: Category sans id

Copyright (C) 2011-2015 Edward Kmett BSD-style (see the file LICENSE) Edward Kmett provisional portable Safe-Inferred Haskell98

Data.Bifunctor.Apply

Contents

Description

Synopsis

• class Bifunctor p 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 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 Bifunctor p where

Minimal definition either `bimap` or `first` and `second`

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` `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
```

Minimal complete definition

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`

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

Map covariantly over the first argument.

``first` f ≡ `bimap` f `id``

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

Map covariantly over the second argument.

``second` ≡ `bimap` `id``

Instances

 Bifunctor Either Bifunctor (,) Bifunctor Const Bifunctor Arg Bifunctor ((,,) x) Bifunctor p => Bifunctor (WrappedBifunctor p) Functor g => Bifunctor (Joker g) Bifunctor p => Bifunctor (Flip p) Functor f => Bifunctor (Clown f) Bifunctor (Tagged *) Bifunctor ((,,,) x y) (Functor f, Bifunctor p) => Bifunctor (Tannen f p) (Bifunctor f, Bifunctor g) => Bifunctor (Product f g) Bifunctor ((,,,,) x y z) (Bifunctor p, Functor f, Functor g) => Bifunctor (Biff p f g) Bifunctor ((,,,,,) x y z w) Bifunctor ((,,,,,,) x y z w v)

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 (,) Biapply Const Biapply Arg Semigroup x => Biapply ((,,) x) Biapply p => Biapply (WrappedBifunctor p) Apply g => Biapply (Joker g) Biapply p => Biapply (Flip p) Apply f => Biapply (Clown f) Biapply (Tagged *) (Semigroup x, Semigroup y) => Biapply ((,,,) x y) (Apply f, Biapply p) => Biapply (Tannen f p) (Biapply p, Biapply q) => Biapply (Product p q) (Semigroup x, Semigroup y, Semigroup z) => Biapply ((,,,,) x y z) (Biapply p, Apply f, Apply g) => Biapply (Biff p f g)

(<<\$>>) :: (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