mirror of
https://github.com/idris-lang/Idris2.git
synced 2024-12-21 02:31:50 +03:00
129 lines
4.5 KiB
Idris
129 lines
4.5 KiB
Idris
module Control.Arrow
|
|
|
|
import Control.Category
|
|
import Data.Either
|
|
import Data.Morphisms
|
|
|
|
|
|
infixr 5 <++>
|
|
infixr 3 ***
|
|
infixr 3 &&&
|
|
infixr 2 +++
|
|
infixr 2 \|/
|
|
|
|
public export
|
|
interface Category arr => Arrow (0 arr : Type -> Type -> Type) where
|
|
||| Converts a function from input to output into a arrow computation.
|
|
arrow : (a -> b) -> arr a b
|
|
||| Converts an arrow from `a` to `b` into an arrow on pairs, that applies
|
|
||| its argument to the first component and leaves the second component
|
|
||| untouched, thus saving its value across a computation.
|
|
first : arr a b -> arr (a, c) (b, c)
|
|
||| Converts an arrow from `a` to `b` into an arrow on pairs, that applies
|
|
||| its argument to the second component and leaves the first component
|
|
||| untouched, thus saving its value across a computation.
|
|
second : arr a b -> arr (c, a) (c, b)
|
|
second f = arrow {arr = arr} swap >>> first f >>> arrow {arr = arr} swap
|
|
where
|
|
swap : (x, y) -> (y, x)
|
|
swap (a, b) = (b, a)
|
|
||| A combinator which processes both components of a pair.
|
|
(***) : arr a b -> arr a' b' -> arr (a, a') (b, b')
|
|
f *** g = first f >>> second g
|
|
||| A combinator which builds a pair from the results of two arrows.
|
|
(&&&) : arr a b -> arr a b' -> arr a (b, b')
|
|
f &&& g = arrow dup >>> f *** g
|
|
|
|
public export
|
|
implementation Arrow Morphism where
|
|
arrow f = Mor f
|
|
first (Mor f) = Mor $ \(a, b) => (f a, b)
|
|
second (Mor f) = Mor $ \(a, b) => (a, f b)
|
|
(Mor f) *** (Mor g) = Mor $ \(a, b) => (f a, g b)
|
|
(Mor f) &&& (Mor g) = Mor $ \a => (f a, g a)
|
|
|
|
public export
|
|
implementation Monad m => Arrow (Kleislimorphism m) where
|
|
arrow f = Kleisli (pure . f)
|
|
first (Kleisli f) = Kleisli $ \(a, b) => do x <- f a
|
|
pure (x, b)
|
|
|
|
second (Kleisli f) = Kleisli $ \(a, b) => do x <- f b
|
|
pure (a, x)
|
|
|
|
(Kleisli f) *** (Kleisli g) = Kleisli $ \(a, b) => do x <- f a
|
|
y <- g b
|
|
pure (x, y)
|
|
|
|
(Kleisli f) &&& (Kleisli g) = Kleisli $ \a => do x <- f a
|
|
y <- g a
|
|
pure (x, y)
|
|
|
|
public export
|
|
interface Arrow arr => ArrowZero (0 arr : Type -> Type -> Type) where
|
|
zeroArrow : arr a b
|
|
|
|
public export
|
|
interface ArrowZero arr => ArrowPlus (0 arr : Type -> Type -> Type) where
|
|
(<++>) : arr a b -> arr a b -> arr a b
|
|
|
|
public export
|
|
interface Arrow arr => ArrowChoice (0 arr : Type -> Type -> Type) where
|
|
left : arr a b -> arr (Either a c) (Either b c)
|
|
|
|
right : arr a b -> arr (Either c a) (Either c b)
|
|
right f = arrow mirror >>> left f >>> arrow mirror
|
|
|
|
|
|
(+++) : arr a b -> arr c d -> arr (Either a c) (Either b d)
|
|
f +++ g = left f >>> right g
|
|
|
|
(\|/) : arr a b -> arr c b -> arr (Either a c) b
|
|
f \|/ g = f +++ g >>> arrow fromEither
|
|
|
|
public export
|
|
implementation Monad m => ArrowChoice (Kleislimorphism m) where
|
|
left f = f +++ (arrow id)
|
|
right f = (arrow id) +++ f
|
|
f +++ g = (f >>> (arrow Left)) \|/ (g >>> (arrow Right))
|
|
(Kleisli f) \|/ (Kleisli g) = Kleisli (either f g)
|
|
|
|
public export
|
|
interface Arrow arr => ArrowApply (0 arr : Type -> Type -> Type) where
|
|
app : arr (arr a b, a) b
|
|
|
|
public export
|
|
implementation Monad m => ArrowApply (Kleislimorphism m) where
|
|
app = Kleisli $ \(Kleisli f, x) => f x
|
|
|
|
public export
|
|
data ArrowMonad : (Type -> Type -> Type) -> Type -> Type where
|
|
MkArrowMonad : (runArrowMonad : arr Unit a) -> ArrowMonad arr a
|
|
|
|
public export
|
|
runArrowMonad : ArrowMonad arr a -> arr Unit a
|
|
runArrowMonad (MkArrowMonad a) = a
|
|
|
|
public export
|
|
implementation Arrow a => Functor (ArrowMonad a) where
|
|
map f (MkArrowMonad m) = MkArrowMonad $ m >>> arrow f
|
|
|
|
public export
|
|
implementation Arrow a => Applicative (ArrowMonad a) where
|
|
pure x = MkArrowMonad $ arrow $ \_ => x
|
|
(MkArrowMonad f) <*> (MkArrowMonad x) = MkArrowMonad $ f &&& x >>> arrow (uncurry id)
|
|
|
|
public export
|
|
implementation ArrowApply a => Monad (ArrowMonad a) where
|
|
(MkArrowMonad m) >>= f =
|
|
MkArrowMonad $ m >>> (arrow $ \x => (runArrowMonad (f x), ())) >>> app
|
|
|
|
public export
|
|
interface Arrow arr => ArrowLoop (0 arr : Type -> Type -> Type) where
|
|
loop : arr (Pair a c) (Pair b c) -> arr a b
|
|
|
|
||| Applying a binary operator to the results of two arrow computations.
|
|
public export
|
|
liftA2 : Arrow arr => (a -> b -> c) -> arr d a -> arr d b -> arr d c
|
|
liftA2 op f g = (f &&& g) >>> arrow (\(a, b) => a `op` b)
|