traverseBia :: (Traversable t, Biapplicative p)
            => (a -> p b c) -> t a -> p (t b) (t c)
traverseBia = inline (traverseBiaWith traverse)
 --------
traverseBiaWith :: forall p a b c s t. Biapplicative p
  => (forall f x. Applicative f => (a -> f x) -> s -> f (t x))
  -> (a -> p b c) -> s -> p (t b) (t c)
traverseBiaWith trav p s = smash p (trav One s)
-------
smash :: forall p t a b c. Biapplicative p
      => (a -> p b c)
      -> (forall x. Mag a x (t x))
      -> p (t b) (t c)
smash p m = go m m
  where
    go :: forall x y. Mag a b x -> Mag a c y -> p x y
    go (Pure t) (Pure u) = bipure t u
    go (Map f x) (Map g y) = bimap f g (go x y)
    go (Ap fs xs) (Ap gs ys) = go fs gs <<*>> go xs ys
#if MIN_VERSION_base(4,10,0)
    go (LiftA2 f xs ys) (LiftA2 g zs ws) = biliftA2 f g (go xs zs) (go ys ws)
#endif
    go (One x) (One _) = p x
    go _ _ = impossibleError

---- and then the magma is 

-- This is used to reify a traversal for 'traverseBia'. It's a somewhat
-- bogus 'Functor' and 'Applicative' closely related to 'Magma' from the
-- @lens@ package. Valid traversals don't use (<$), (<*), or (*>), so
-- we leave them out. We offer all the rest of the Functor and Applicative
-- operations to improve performance: we generally want to keep the structure
-- as small as possible. We might even consider using RULES to widen lifts
-- when we can:
--
--   liftA2 f x y <*> z ==> liftA3 f x y z,
--
-- etc., up to the pointer tagging limit. But we do need to be careful. I don't
-- *think* GHC will ever inline the traversal into the go function (because that
-- would duplicate work), but if it did, and if different RULES fired for the
-- two copies, everything would break horribly.
--
-- Note: if it's necessary for some reason, we *could* relax GADTs to
-- ExistentialQuantification by changing the type of One to
--
--   One :: (b -> c) -> a -> Mag a b c
--
-- where the function will always end up being id. But we allocate a *lot*
-- of One constructors, so this would definitely be bad for performance.
data Mag a b t where
  Pure :: t -> Mag a b t
  Map :: (x -> t) -> Mag a b x -> Mag a b t
  Ap :: Mag a b (t -> u) -> Mag a b t -> Mag a b u
#if MIN_VERSION_base(4,10,0)
  LiftA2 :: (t -> u -> v) -> Mag a b t -> Mag a b u -> Mag a b v
#endif
  One :: a -> Mag a b b

instance Functor (Mag a b) where
  fmap = Map

instance Applicative (Mag a b) where
  pure = Pure
  (<*>) = Ap
#if MIN_VERSION_base(4,10,0)
  liftA2 = LiftA2
#endif