[Haskell-cafe] Type of scramblings

[Roman Cheplyaka]

I am reading through Oleg's "Eliminating translucent existentials"[1].

[1]: http://okmij.org/ftp/Computation/Existentials.html#eliminating-translucent

He draws a distinction between
  
  forall a . [a] -> [a]

and

  forall a . [a]^n -> [a]

as types of "scramblings". This is something I'm struggling to understand.

First of all, I think here we're talking about total functions, otherwise
there's no point in introducing dependent types.

There are of course more total functions of type `[a]^n -> [a]` than of type
`[a] -> [a]`, in the sense that any term of the latter type can be assigned the
former type. But, on the other hand, any total function `f :: [a]^n -> [a]`
has an "equivalent" total function

  g :: [a] -> [a]
  g xs | length xs == n = f xs
       | otherwise = xs

(The condition `length xs == n` can be replaced by a similar condition that also
works for infinite lists.)

The functions `f` and `g` are equivalent in the sense that for any list `xs` of
length `n` `f xs === g xs`. Thus, even though it seems that we allow more total
functions by replacing `[a]` with `[a]^n`, that doesn't buy us any additional
expressiveness.

What am I missing?

Roman