Python `max()` clone, with type hints in 6 overloads

mymax.py

from typing import Protocol, Any, TypeVar, overload, Callable, Iterable, Union


class Comparable(Protocol):
    def __lt__(self, other: Any) -> bool:
        ...


MISSING = object()
EMPTY_MSG = 'max() arg is an empty sequence'

T = TypeVar('T')
C = TypeVar('C', bound=Comparable)
DT = TypeVar('DT')


@overload
def max(__arg1: C, __arg2: C, *_args: C, key: None = ...) -> C:
    ...
@overload
def max(__arg1: T, __arg2: T, *_args: T, key: Callable[[T], C]) -> T:
    ...
@overload
def max(__iterable: Iterable[C], *, key: None = ...) -> C:
    ...
@overload
def max(__iterable: Iterable[T], *, key: Callable[[T], C]) -> T:
    ...
@overload
def max(__iterable: Iterable[C], *, key: None = ..., default: DT) -> Union[C, DT]:
    ...
@overload
def max(__iterable: Iterable[T], *, key: Callable[[T], C], default: DT) -> Union[T, DT]:
    ...
def max(first, *args, key=None, default=MISSING):
    if args:
        series = args
        candidate = first
    else:
        series = iter(first)
        try:
            candidate = next(series)
        except StopIteration as exc:
            if default is not MISSING:
                return default
            raise ValueError(EMPTY_MSG) from exc
    if key is None:
        for current in series:
            if candidate < current:
                candidate = current
    else:
        candidate_key = key(candidate)
        for current in series:
            current_key = key(current)
            if candidate_key < current_key:
                candidate = current
                candidate_key = current_key
    return candidate