[Haskell-cafe] Pointed, but not Applicative

[Sebastian Fischer]

On Wed, Aug 31, 2011 at 6:13 AM, Ryan Ingram <ryani.spam at gmail.com> wrote:
> technically it violates 'fmap id' == 'id' [...]
>
> If you add this FList law, though, you're OK:
>
> runFList fl as = runFList fl [] ++ as

I think the idea of functional lists is that the monoids of 'lists'
and 'functions on lists' are isomorphic with isomorphisms toFList and
toList:

    toFList [] = id
    toFList (xs++ys) = toFList xs . toFList ys

    toList id = []
    toList (f . g) = toList f ++ toList g

These can be defined as:

    toFList = (++)
    toList = ($[])

Eliding newtypes, runFList is just the identity function so we need to check

    f l = toList f ++ l

to verify your law. This is the same as

    f = toFList (toList f)

which (once we know that toList and toFList are isomorphisms) does
indeed hold because:

    toFList . toList = id
    toList . toFList = id

Sebastian