How to “perfectly” override a dict?

How can I make as “perfect” a subclass of dict as possible? The end goal is to have a simple dict in which the keys are lowercase.

It would seem that there should be some tiny set of primitives I can override to make this work, but according to all my research and attempts it seem like this isn’t the case:

• If I override __getitem__/__setitem__, then get/set don’t work. How can I make them work? Surely I don’t need to implement them individually?

• Am I preventing pickling from working, and do I need to implement __setstate__ etc?

• Do I need repr, update and __init__?

• Should I just use mutablemapping (it seems one shouldn’t use UserDict
  or DictMixin)? If so, how? The docs aren’t exactly enlightening.

Here is my first go at it, get() doesn’t work and no doubt there are many other minor problems:

class arbitrary_dict(dict):
    """A dictionary that applies an arbitrary key-altering function
       before accessing the keys."""

    def __keytransform__(self, key):
        return key

    # Overridden methods. List from 
    # https://stackoverflow.com/questions/2390827/how-to-properly-subclass-dict

    def __init__(self, *args, **kwargs):
        self.update(*args, **kwargs)

    # Note: I'm using dict directly, since super(dict, self) doesn't work.
    # I'm not sure why, perhaps dict is not a new-style class.

    def __getitem__(self, key):
        return dict.__getitem__(self, self.__keytransform__(key))

    def __setitem__(self, key, value):
        return dict.__setitem__(self, self.__keytransform__(key), value)

    def __delitem__(self, key):
        return dict.__delitem__(self, self.__keytransform__(key))

    def __contains__(self, key):
        return dict.__contains__(self, self.__keytransform__(key))


class lcdict(arbitrary_dict):
    def __keytransform__(self, key):
        return str(key).lower()