Only use annotations of `@overload`ed functions to type check the implementation

Consider the following real-world example of a function. The type of one of the parameters of this function depends on another one:

from typing import *

K = TypeVar("K", bound=Hashable)
V = TypeVar("V")


@overload
def dict_get_factory(dictionary: Dict[K, V], 
                     key: K, 
                     factory: Callable[[], V], 
                     takes_key: Literal[False]) -> V: ...

@overload
def dict_get_factory(dictionary: Dict[K, V], 
                     key: K, 
                     factory: Callable[[K], V], 
                     takes_key: Literal[True]) -> V: ...

def dict_get_factory(dictionary: Dict[K, V], 
                     key: K, 
                     factory: Union[Callable[[], V], Callable[[K], V]], 
                     takes_key: bool=False) -> V:
    """Like dict.setdefault(), but uses a factory

    >>> dict_get_factory(d := {}, "foo", int)
    0
    >>> dict_get_factory(d, "bar", lambda key: len(key), takes_key=True)
    3
    >>> d
    {'foo': 0, 'bar': 3}
    """
    try:
        return dictionary[key]
    except KeyError:
        return dictionary.setdefault(key, 
                factory(key) if takes_key else factory())
                
i1: int = dict_get_factory({"foo": 1}, "bar", lambda: 1, False)   # ok
i2: int = dict_get_factory({"foo": 1}, "bar", lambda s: 1, True)  # ok

It doesn't typecheck as if you only look at the annotations of the implementation, the signature of the factory is ambiguous:

main.py:36: error: Too many arguments
main.py:36: error: Too few arguments

I don't think it's possible to annotate the implementation in such a way that it would typecheck. So it would be nice if mypy used the signatures of the @overloaded methods, and one wouldn't have to annotate the signature of the implementation at all. Even if you could annotate it in such a way that it would typecheck, wouldn't it be redundant?

Start with the overload example in the issue and inspect mypy's handling of implementation signatures and overload checking. Determine the intended behavior for calls inside the implementation, then add focused regression coverage and verify that the example no longer reports conflicting factory-call errors.