If you have a Functor f with an instance Monoid a => Monoid (f a), f is Applicative!

liftedMonoid.hs

{-# LANGUAGE TypeOperators, ScopedTypeVariables, TupleSections #-}

import Data.Constraint
import Data.Monoid
import Control.Applicative


pureDefault :: forall f a. Functor f 
   => (Monoid () :- Monoid (f ())) 
   -> a -> f a
pureDefault (Sub Dict) a = a <$ (mempty :: f ())

apDefault :: forall f a b. Functor f 
   => (Monoid (First (a -> b), First a) :- Monoid (f (First (a -> b), First a))) 
   -> f (a -> b) -> f a -> f b
apDefault (Sub Dict) fab fa = fmap (\(First (Just f), First (Just a)) -> f a) $ 
  fmap (\f -> (First (Just f), mempty)) fab `mappend` fmap (\a -> (mempty, First (Just a))) fa


-- Example (Dual is just the identity applicative)

pfDual :: Monoid a :- Monoid (Dual a)
pfDual = Sub Dict

instance Functor Dual where
  fmap f (Dual a) = Dual (f a)

instance Applicative Dual where
  pure = pureDefault pfDual
  (<*>) = apDefault pfDual