# Type composition

### From HaskellWiki

## 1 Introduction

I'd like to get some forms of type composition into a standard library. Below is my first shot at it. I'm using these definitions in a new version of Phooey.

Comments & suggestions, please. Conal 23:16, 8 March 2007 (UTC)

## 2 Code

{-# OPTIONS -fglasgow-exts -cpp #-} ---------------------------------------------------------------------- -- Various type constructor compositions and instances for them. -- References: -- [1] \"Applicative Programming with Effects\" -- <http://www.soi.city.ac.uk/~ross/papers/Applicative.html> ---------------------------------------------------------------------- module Control.Compose ( Cofunctor(..) , Compose(..), onComp , StaticArrow(..) , Flip(..) , ArrowAp(..) , App(..) ) where import Control.Applicative import Control.Arrow hiding (pure) import Data.Monoid -- | Often useful for /acceptors/ (consumers, sinks) of values. class Cofunctor acc where cofmap :: (a -> b) -> (acc b -> acc a) -- | Composition of type constructors: unary & unary. Called \"g . f\" in -- [1], section 5, but GHC won't parse that, nor will it parse any infix -- type operators in an export list. Haddock won't parse any type infixes -- at all. newtype Compose g f a = Comp { unComp :: g (f a) } -- | Apply a function within the 'Comp' constructor. onComp :: (g (f a) -> g' (f' a')) -> ((Compose g f) a -> (Compose g' f') a') onComp h (Comp gfa) = Comp (h gfa) instance (Functor g, Functor f) => Functor (Compose g f) where fmap h (Comp gf) = Comp (fmap (fmap h) gf) instance (Applicative g, Applicative f) => Applicative (Compose g f) where pure = Comp . pure . pure Comp getf <*> Comp getx = Comp (liftA2 (<*>) getf getx) -- instance (Functor g, Cofunctor f) => Cofunctor (Compose g f) where -- cofmap h (Comp gf) = Comp (fmap (cofmap h) gf) -- Or this alternative. Having both yields "Duplicate instance -- declarations". instance (Cofunctor g, Functor f) => Cofunctor (Compose g f) where cofmap h (Comp gf) = Comp (cofmap (fmap h) gf) -- standard Monoid instance for Applicative applied to Monoid instance (Applicative (Compose g f), Monoid a) => Monoid (Compose g f a) where { mempty = pure mempty; mappend = (*>) } -- | Composition of type constructors: unary with binary. newtype StaticArrow f (~>) a b = Static { unStatic :: f (a ~> b) } instance (Applicative f, Arrow (~>)) => Arrow (StaticArrow f (~>)) where arr = Static . pure . arr Static g >>> Static h = Static (liftA2 (>>>) g h) first (Static g) = Static (liftA first g) -- For instance, /\ a b. f (a -> m b) =~ StaticArrow f Kleisli m -- | Composition of type constructors: binary with unary. newtype ArrowAp (~>) f a b = ArrowAp {unArrowAp :: f a ~> f b} instance (Arrow (~>), Applicative f) => Arrow (ArrowAp (~>) f) where arr = ArrowAp . arr . liftA ArrowAp g >>> ArrowAp h = ArrowAp (g >>> h) first (ArrowAp a) = ArrowAp (arr splitA >>> first a >>> arr mergeA) instance (ArrowLoop (~>), Applicative f) => ArrowLoop (ArrowAp (~>) f) where -- loop :: UI (b,d) (c,d) -> UI b c loop (ArrowAp k) = ArrowAp (loop (arr mergeA >>> k >>> arr splitA)) -- Wolfgang Jeltsch pointed out a problem with these definitions: 'splitA' -- and 'mergeA' are not inverses. The definition of 'first', e.g., -- violates the \"extension\" law and causes repeated execution. Look for -- a reformulation or a clarification of required properties of the -- applicative functor @f@. -- -- See also "Arrows and Computation", which notes that the following type -- is "almost an arrow" (http://www.soi.city.ac.uk/~ross/papers/fop.html). -- -- > newtype ListMap i o = LM ([i] -> [o]) mergeA :: Applicative f => (f a, f b) -> f (a,b) mergeA ~(fa,fb) = liftA2 (,) fa fb splitA :: Applicative f => f (a,b) -> (f a, f b) splitA fab = (liftA fst fab, liftA snd fab) -- | Flip type arguments newtype Flip (~>) b a = Flip (a ~> b) instance Arrow (~>) => Cofunctor (Flip (~>) b) where cofmap h (Flip f) = Flip (arr h >>> f) -- | Type application newtype App f a = App { unApp :: f a } -- Example: App IO () instance (Applicative f, Monoid m) => Monoid (App f m) where mempty = App (pure mempty) App a `mappend` App b = App (a *> b) {- -- We can also drop the App constructor, but then we overlap with many -- other instances, like [a]. instance (Applicative f, Monoid a) => Monoid (f a) where mempty = pure mempty mappend = (*>) -}