How to model mixins / multiple interfaces in Haskell?

Question

I came across this question on modeling inheritance in Haskell and it reminded me that I have a little more complicated version of the same problem. I\'ll adopt the example from

Answer 1

Though I still suspect we should think about the whole think about the whole thing another way, less OO-inspired, here's another possible solution. I shall keep to the Monsters example, though a 2D graphics program seems indeed a better example.

{-# LANGUAGE TypeFamilies, MultiParamTypeClasses, DeriveFunctor, FlexibleContexts #-}

import Control.Monad.Identity

class (Functor f, Functor (PropT f p)) => AttachProp f p where
  type PropT f p :: * -> *
  attachProp :: p -> f o -> PropT f p o
  detachProp :: PropT f p o -> (p, f o)

fmapProp :: (AttachProp f p, AttachProp f p')
  => f o -- dummy parameter (unevaluated), because type-functions aren't injective
         -> (p -> p') -> PropT f p o -> PropT f p' o
fmapProp q f pt = let (p, fo) = detachProp pt
                  in attachProp (f p) $ fo `asTypeOf` q


data R3Phys = R3Phys { position, momentum :: Vec3 }
data Colour = Colour

data Physical a = Physical R3Phys a deriving (Functor)
data Coloured a = Coloured Colour a deriving (Functor)
data PhysColoured a = PhysColoured Colour R3Phys a deriving (Functor)

instance AttachProp Identity R3Phys where
  type PropT Identity R3Phys = Physical
  attachProp rp = Physical rp . runIdentity
  detachProp (Physical rp o) = (rp, Identity o)
instance AttachProp Identity Colour where
  type PropT Identity Colour = Coloured
  attachProp c = Coloured c . runIdentity
  detachProp (Coloured c o) = (c, Identity o)
instance AttachProp Coloured R3Phys where
  type PropT Coloured R3Phys = PhysColoured
  attachProp rp (Coloured c o) = PhysColoured c rp o
  detachProp (PhysColoured c rp o) = (rp, Coloured c o)
instance AttachProp Physical Colour where
  type PropT Physical Colour = PhysColoured
  attachProp c (Physical rp o) = PhysColoured c rp o
  detachProp (PhysColoured c rp o) = (c, Physical rp o)

Note that PropT (PropT Identity R3Phys) Colour a and PropT (PropT Identity Colour) R3Phys a are the same type, namely PhysColoured a. Of course, we need again O (n²) instances for n mixins. Could easily be done with Template Haskell, though obviously you should think twice if you want that.