Re: [Haskell-cafe] gathering position information lazily using traverse

On 10.09.2010, at 23:58, Felipe Lessa wrote:

Hmmmm...

On Fri, Sep 10, 2010 at 6:47 PM, Jan Christiansen wrote:

...

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

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

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

...

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)))

(pure f) <*> Proj x === Pure f <*> Proj x === Proj (\p -> f (x p))

(Proj f) <*> (pure x) === Proj f <*> Pure x === Proj (\p -> f p x)

Was this difference intended?

Yes, this is intended. This difference is the reason why the former instance does not satisfy the Applicative laws while the latter does. The first instance provides every subterm of an idiomatic term with a position. Even a "pure" term is provided with a position although it does not use it. The latter instance does not provide a "pure" term with a position as it does not need one. Therefore, the second instance simply passes position p to a subterm if the other subterm is pure. In the example for the first instance we can observe that we unnecessarily extend the position with True and False respectively.