recursive programming in applicative functors?

[Ross Paterson]

On Thu, Mar 15, 2007 at 09:34:21AM -0700, Conal Elliott wrote:
> Is there a fixpoint operator for applicative functors, like mfix for
> MonadFix and loop for ArrowLoop?

Hmm, there could be an afix :: (a -> f a) -> f a; I wonder what axioms
it would satisfy.  These ones for mfix would carry over:

purity
	afix (pure . h) = pure (fix h)
sliding
	afix (fmap h . f) = fmap h (afix (f . h)), for strict h.
nesting
	afix (\x -> afix (\y -> f x y)) = afix (\x -> f x x)

but there'd be no way to express tightening laws.  Perhaps

	afix (\(x, y) -> pair (f x) (g y)) = pair (afix f) (afix g)
	  where pair a b = (,) <$> a <*> b

Do you have any non-monadic examples in mind?  I guess zip-lists are
one possibility.