<SYSTEM_TASK:>
Solve the following problem using Python, implementing the functions described below, one line at a time
<END_TASK>
<USER_TASK:>
Description:
 def flatten(self):
 """Flatten the scheme into a dictionary where the keys are compound 'dot' notation keys, and the values are the corresponding options. Returns: dict: The flattened `Scheme`. """

        if self._flat is None:
            flat = {}
            for arg in self.args:
                if isinstance(arg, Option):
                    flat[arg.name] = arg

                elif isinstance(arg, ListOption):
                    flat[arg.name] = arg

                elif isinstance(arg, DictOption):
                    flat[arg.name] = arg
                    if arg.scheme:
                        for k, v in arg.scheme.flatten().items():
                            flat[arg.name + '.' + k] = v

            self._flat = flat
        return self._flat