# 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, first draft

{-# OPTIONS -fglasgow-exts #-} -- Various type constructor compositions and instances for them. -- References: -- [1] [http://www.soi.city.ac.uk/~ross/papers/Applicative.html Applicative Programming with Effects] module Control.Compose ((:.:)(..), (:.::)(..), (::.:)(..), 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. newtype (g :.: f) a = T_T { runT_T :: g (f a) } instance (Functor g, Functor f) => Functor (g :.: f) where fmap f (T_T m) = T_T (fmap (fmap f) m) instance (Applicative g, Applicative f) => Applicative (g :.: f) where pure = T_T . pure . pure T_T getf <*> T_T getx = T_T (liftA2 (<*>) getf getx) -- standard Monoid instance for Applicative applied to Monoid instance (Applicative (f :.: g), Monoid a) => Monoid ((f :.: g) a) where { mempty = pure mempty; mappend = (*>) } instance (Functor g, Cofunctor f) => Cofunctor (g :.: f) where cofmap h (T_T gf) = T_T (fmap (cofmap h) gf) -- Or this alternative. Having both yields "Duplicate instance -- declarations". How to decide between these instances? -- instance (Cofunctor g, Functor f) => Cofunctor (g :.: f) where -- cofmap h (T_T gf) = T_T (cofmap (fmap h) gf) -- | Composition of type constructors: unary & binary. Called -- "StaticArrow" in [1], section 6. newtype (f :.:: (~>)) a b = T_TT { runT_TT :: f (a ~> b) } instance (Applicative f, Arrow (~>)) => Arrow (f :.:: (~>)) where arr = T_TT . pure . arr T_TT g >>> T_TT h = T_TT (liftA2 (>>>) g h) first (T_TT g) = T_TT (liftA first g) -- For instance, /\ a b. f (a -> m b) =~ f :.:: Kleisli m -- | Composition of type constructors: unary & binary. -- 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@. newtype ((~>) ::.: f) a b = TT_T {runTT_T :: f a ~> f b} instance (Arrow (~>), Applicative f) => Arrow ((~>) ::.: f) where arr = TT_T . arr . liftA TT_T g >>> TT_T h = TT_T (g >>> h) first (TT_T a) = TT_T (arr splitA >>> first a >>> arr mergeA) instance (ArrowLoop (~>), Applicative f) => ArrowLoop ((~>) ::.: f) where -- loop :: UI (b,d) (c,d) -> UI b c loop (TT_T k) = TT_T (loop (arr mergeA >>> k >>> arr splitA)) 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 { runApp :: 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 = (*>) -}