A new case for the pointed functor class

There has been some debate for some time as to whether there should be a superclass for Applicative with just the pure function should exist:

-- Natural law:
-- fmap f . pure = pure . f
class Functor f => Pointed f where
  pure :: a -> f a

The charge laid against this class is that there are no laws for this single function beyond the single law that is naturally implied. Compare this to a more reasonable class

-- Natural laws:
-- distRight . right . fmap f = fmap (right f) . distRight
-- distRight . left f = fmap (left f) . distRight
--
-- Other laws:
-- 1. either absurd (fmap Right) = distRight :: Either Void (f a) -> f (Either Void a)
-- 2. fmap assocL . distRight . right distRight . assocR = distRight :: Either (Either a b) (f c) -> f (Either (Either a b) c)
--  where
--   assocL :: Either a (Either b c) -> Either (Either a b) c
--   assocL (Left a) = Left (Left a)
--   assocL (Right (Left a)) = Left (Right a)
--   assocL (Right (Right a)) = Right a
--   assocR :: Either (Either a b) c -> Either a (Either b c)
--   assocR (Left (Left a)) = Left a
--   assocR (Left (Right a)) = Right (Left a)
--   assocR (Right a) = Right (Right a)
class Functor f => StrongSum f where
  distRight :: Either a (f b) -> f (Either a b)

  distLeft :: Either (f a) b -> f (Either a b)
  distLeft = fmap switch . distRight . switch
   where
    switch :: Either a b -> Either b a
    switch = Right ||| Left

StrongSum is a honest class with two additional real laws in addition to its natural laws. No one would object to creating and using such a class.

What if I told you that Pointed and StrongSum are in fact the same class?

class Functor f => Pointed f where
  pure :: a -> f a
  pure = fmap (id ||| absurd) . distRight . Left

  distRight : Either a (f b) -> f (Either a b)
  distRight = either (fmap Left . pure) (fmap Right)

Theorem 1. If we define pure, then the distRight automatically satisfies the 2 required laws.

Proof. Law 1:

either absurd (fmap Right)
= { extensionality of absurd }
either (fmap Left . pure) (fmap Right)
= { definition in terms of pure }
distRight :: Either Void (f a) -> f (Either Void a)

Law 2: fmap assocL . either (fmap Left . pure) (fmap Right) . right (either (fmap Left . pure) (fmap Right)) . assocR = either (fmap Left . pure) (fmap Right)

case 1: Right x

fmap assocL . either (fmap Left . pure) (fmap Right) . right (either (fmap Left . pure) (fmap Right)) . assocR $ Right x
=
fmap assocL . either (fmap Left . pure) (fmap Right) . right (either (fmap Left . pure) (fmap Right)) $ Right (Right x)
=
fmap assocL . either (fmap Left . pure) (fmap Right) $ Right (either (fmap Left . pure) (fmap Right) (Right x))
=
fmap assocL . either (fmap Left . pure) (fmap Right) $ Right (fmap Right x)
=
fmap assocL $ fmap Right (fmap Right x)
=
fmap (assocL . Right . Right) x
=
fmap Right x
=
either (fmap Left . pure) (fmap Right) $ Right x

case 2: Left (Right x)

fmap assocL . either (fmap Left . pure) (fmap Right) . right (either (fmap Left . pure) (fmap Right)) . assocR $ Left (Right x)
=
fmap assocL . either (fmap Left . pure) (fmap Right) . right (either (fmap Left . pure) (fmap Right)) $ Right (Left x)
=
fmap assocL . either (fmap Left . pure) (fmap Right) $ Right (either (fmap Left . pure) (fmap Right) (Left x))
=
fmap assocL . either (fmap Left . pure) (fmap Right) $ Right (fmap Left (pure x))
=
fmap assocL $ fmap Right (fmap Left (pure x))
=
fmap (assocL . Right . Left) (pure x)
=
fmap (Left . Right) (pure x)
=
fmap Left . fmap Right . pure $ x
=
fmap Left . pure . Right $ x
=
fmap Left . pure $ Right x
=
either (fmap Left . pure) (fmap Right) $ Left (Right x)

case 3: Left (Left x)

fmap assocL . either (fmap Left . pure) (fmap Right) . right (either (fmap Left . pure) (fmap Right)) . assocR $ Left (Left x)
=
fmap assocL . either (fmap Left . pure) (fmap Right) . right (either (fmap Left . pure) (fmap Right)) $ Left x
=
fmap assocL . either (fmap Left . pure) (fmap Right) $ Left x
=
fmap assocL . fmap Left . pure $ x
=
fmap (assocL . Left) . pure $ x
=
fmap (Left . Left) . pure $ x
=
fmap Left . fmap Left . pure $ x
=
fmap Left . pure . Left $ x
=
fmap Left . pure $ Left x
=
either (fmap Left . pure) (fmap Right) $ Left (Left x)

Theorem 2. If we define pure, then pure = fmap (id ||| absurd) . distRight . Left where distRight = either (fmap Left . pure) (fmap Right)

Proof.

fmap (id ||| absurd) . distRight . Left
=
fmap (id ||| absurd) . either (fmap Left . pure) (fmap Right) . Left
=
fmap (id ||| absurd) . fmap Left . pure
=
fmap ((id ||| absurd) . Left) . pure
=
fmap id . pure
=
pure

Theorem 3. If we define distRight such that distRight satisfies its two laws then distRight = either (fmap Left . pure) (fmap Right) where pure = fmap (id ||| absurd) . distRight . Left.

Proof. either (fmap Left . fmap (id ||| absurd) . distRight . Left) (fmap Right) = distRight

case 1: Left x

either (fmap Left . fmap (id ||| absurd) . distRight . Left) (fmap Right) $ Left x
=
fmap Left . fmap (id ||| absurd) . distRight $ Left x
=
fmap (Left . (id ||| absurd)) $ distRight (Left x)
=
fmap (Left ||| (Left . absurd)) $ distRight (Left x)
=
fmap (Left ||| absurd) $ distRight (Left x)
= { extensionality of absurd }
fmap (Left ||| Right) $ distRight (Left x)
=
fmap id $ distRight (Left x)
=
distRight (Left x)

case 2: Right x

either (fmap Left . fmap (id ||| absurd) . distRight . Left) (fmap Right) $ Right x
=
fmap Right x
=
fmap (left absurd . Right) x
=
fmap (left absurd) $ fmap Right x
=
fmap (left absurd) . either absurd (fmap Right) $ Right x
= { Law 1 for distRight }
fmap (left absurd) . distRight $ Right x
= { Natural law for distRight }
distRight . left absurd $ Right x
=
distRight $ Right x

Interestingly we only ever used the first law for distRight. By composing these proofs together we should be able to show that the second law for distRight holds whenever the first law does.

In conclusion we have seen that the StrongSum class and the Pointed class are equivalent classes. The pure function contains the heart of what a law abiding distRight function is, such that whenever we have a law abiding distRight function it can be written in terms of pure and every instance of pure yields a law abiding distRight function. Given that I think people would welcome a StrongSum class, it only makes sense to equally welcome the Pointed class.