Derive instances (Applicative, Monad, ...) for structures lifted over functors

I'll start small: Given that we know how to define various instances for Product GHC could do it automatically.

data P f g a = f a ::: g a deriving (Functor, Applicative, Alternative)

{-
instance (Applicative f, Applicative g) => Applicative (P f g) where
  pure x = pure x ::: pure x
  (f:::g) <*> (x:::y) = (f <*> x) ::: (g <*> y)
-}

And for specific constructors as well

data Q a = [a] :*: Maybe a deriving (Functor, Applicative, Alternative)

{-
instance Applicative Q where
  pure x = [x] :*: Just x
  (f:*:g) <*> (x:*:y) = (f <*> x) :*: (g <*> y)
-}

Alternative

Use GeneralizedNewtypeDeriving

newtype Q a = Q (Product [] Maybe a)
  deriving (Functor, Applicative, Alternative)

pattern (:*:) :: [a] -> Maybe a -> Q a
pattern a :*: b = Q (Pair a b)

Future Work

This should work for a combination of various things, using Const _ deprives us of Alternative

newtype U e a = U (([] `Product` Maybe `Product` Const e) a)
  deriving (Functor, Applicative)

using sums where one summand is identity gives us Applicative / Alternative

-- data Lift f a = Pure a | Other (f a)
import Control.Applicative.Lift

data V a = V ((Lift [] `Product` Maybe) a) 
  deriving (Functor, Applicative, Alternative)

I want to be able to write this directly

data U e a = U [a] (Maybe a) (Const e a)
  deriving (Functor, Applicative)

data V a
  = VL a   (Maybe a)
  | VR [a] (Maybe a)
  deriving (Functor, Applicative, Alternative)

Future, Future Work ==

left-Kan extension

data Lan g h a where
  Lan :: (g b -> a) -> h b -> Lan g h a
  deriving (Functor, Applicative)

codensity

data Endo a = Endo (a -> a)

newtype CodEndo a = CE (forall xx. (a -> Endo xx) -> Endo xx)
  deriving (Functor, Applicative, Monad)

and comonad

data Rose a = a :< [Rose a]
  deriving (Functor, Applicative, Monad, Comonad, ...)

Trac metadata

Trac field	Value
Version	8.0.1
Type	Bug
TypeOfFailure	OtherFailure
Priority	normal
Resolution	Unresolved
Component	Compiler
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