from collections import defaultdict
from collections.abc import Iterator
from typing import Any

def transitive_closure(implications) -> set[Any]: ...
def deduce_alpha_implications(implications) -> defaultdict[Any, set[Any]]: ...
def apply_beta_to_alpha_route(alpha_implications, beta_rules) -> dict[Any, Any]: ...
def rules_2prereq(rules) -> defaultdict[Any, set[Any]]: ...

class TautologyDetected(Exception): ...

class Prover:
    def __init__(self) -> None: ...
    def split_alpha_beta(self) -> tuple[list[Any], list[Any]]: ...
    @property
    def rules_alpha(self) -> list[Any]: ...
    @property
    def rules_beta(self) -> list[Any]: ...
    def process_rule(self, a, b) -> None: ...

class FactRules:
    def __init__(self, rules) -> None: ...
    def print_rules(self) -> Iterator[str]: ...

class InconsistentAssumptions(ValueError): ...

class FactKB(dict):
    def __init__(self, rules) -> None: ...
    def deduce_all_facts(self, facts) -> None: ...