[Haskell-cafe] gathering position information lazily using traverse

[Jan Christiansen]

Dear Applicative experts,

I am seeking advice on Applicative instances and their use in  
traverse. Consider the following Applicative instance.

   newtype Proj a = Proj { unProj :: [Bool] -> a }

   instance Functor Proj where
     fmap g (Proj f) = Proj (g . f)

   instance Applicative Proj where
     pure = Proj . const
     Proj f <*> Proj x = Proj (\p -> f (False:p) (x (True:p)))

In fact, this is not an Applicative instance as it does not satisfy  
the laws. On basis of this "instance" I have defined the following  
function.

   gshape :: Traversable t => t a -> t [Bool]
   gshape x = unProj (traverse (const (Proj reverse)) x) []

The idea is simply to replace every polymorphic component by an  
identifier that identifies the position of the component in the data  
structure. That is, provided with the identifier I want to be able to  
project to the corresponding component. In this case this identifier  
is a path in the "idiomatic term" from the root to the component.

I can define a correct Applicative instance if I add an additional  
constructor, which represents pure. I did not prove that it satisfies  
all laws but I think it does.

   data Proj a = Pure a | Proj ([Bool] -> a)

   instance Functor Proj where
     fmap g (Pure x) = Pure (g x)
     fmap g (Proj f) = Proj (g . f)

   instance Applicative Proj where
     pure x = Pure x
     Pure f <*> Pure x = Pure (f x)
     Pure f <*> Proj x = Proj (\p -> f (x p))
     Proj f <*> Pure x = Proj (\p -> f p x)
     Proj f <*> Proj x = Proj (\p -> f (False:p) (x (True:p)))

My problem is that this correct instance is too strict for my purpose.  
It is important that gshape operates correctly on partial data, that  
is, even if its argument is partial all the components should be  
replaced correctly. For example, we have

   gshape (Node _|_ 0 (Leaf 1))) = Node _|_ [False,True] (Leaf [True])


If the applicative instance performs pattern matching, like the latter  
instance, then gshape is too strict. Therefore I suspect that there is  
no correct Applicative instance that satisfies my needs but I am not  
at all certain.

Thanks, Jan