from typing import Iterable, Set, Union

models = None

DL_ATOMS = "abc"


def set_models(alphabet: Iterable[str], v: Iterable[Iterable[str]]):
    global models, DL_ATOMS
    DL_ATOMS = "".join(alphabet)
    models = tuple(set(model) for model in v)


def propagate(atoms: Set[str]) -> Set[str]:
    atoms = atoms.intersection(DL_ATOMS)
    sup_models = (model for model in models if atoms <= model)
    first: set = next(sup_models, None)
    if first is None:
        return set()
    return first.intersection(*sup_models)


def check(atoms: Set[str]) -> Union[Set[str], None]:
    atoms = atoms.intersection(DL_ATOMS)
    if atoms not in models:
        return None
    sub_models = (model for model in models if model < atoms)
    return atoms - atoms.intersection(*sub_models)