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