from collections.abc import Generator, Iterator
from typing import Any, NoReturn
from typing_extensions import Self

from sympy.core.basic import Basic
from sympy.core.cache import cacheit
from sympy.core.decorators import call_highest_priority
from sympy.core.relational import Eq, Ne, Relational
from sympy.core.singleton import Singleton
from sympy.series.order import Order
from sympy.sets.sets import Complement, FiniteSet, Intersection, Interval, Union

class SeqBase(Basic):
    is_commutative = ...
    _op_priority = ...
    @property
    def gen(self): ...
    @property
    def interval(self): ...
    @property
    def start(self): ...
    @property
    def stop(self): ...
    @property
    def length(self): ...
    @property
    def variables(self) -> tuple[()]: ...
    @property
    def free_symbols(self) -> set[Basic]: ...
    @cacheit
    def coeff(self, pt): ...
    def coeff_mul(self, other) -> Order: ...
    def __add__(self, other) -> SeqAdd: ...
    @call_highest_priority("__add__")
    def __radd__(self, other): ...
    def __sub__(self, other) -> SeqAdd: ...
    @call_highest_priority("__sub__")
    def __rsub__(self, other): ...
    def __neg__(self) -> Order: ...
    def __mul__(self, other) -> SeqMul: ...
    @call_highest_priority("__mul__")
    def __rmul__(self, other): ...
    def __iter__(self) -> Iterator[Any]: ...
    def __getitem__(self, index) -> list[Any] | None: ...
    def find_linear_recurrence(self, n, d=..., gfvar=...) -> list[Any] | tuple[list[Any], None] | tuple[Any | list[Any], Any]: ...

class EmptySequence(SeqBase, metaclass=Singleton):
    @property
    def interval(self): ...
    @property
    def length(self): ...
    def coeff_mul(self, coeff) -> Self: ...
    def __iter__(self) -> Iterator[Any]: ...

class SeqExpr(SeqBase):
    @property
    def gen(self) -> Basic: ...
    @property
    def interval(self) -> FiniteSet | Interval: ...
    @property
    def start(self): ...
    @property
    def stop(self): ...
    @property
    def length(self): ...
    @property
    def variables(self) -> tuple[Any]: ...

class SeqPer(SeqExpr):
    def __new__(cls, periodical, limits=...) -> Self: ...
    @property
    def period(self) -> int: ...
    @property
    def periodical(self) -> Basic: ...
    def coeff_mul(self, coeff) -> SeqPer: ...

class SeqFormula(SeqExpr):
    def __new__(cls, formula, limits=...) -> Self: ...
    @property
    def formula(self) -> Basic: ...
    def coeff_mul(self, coeff) -> SeqFormula: ...
    def expand(self, *args, **kwargs) -> SeqFormula: ...

class RecursiveSeq(SeqBase):
    def __new__(cls, recurrence, yn, n, initial=..., start=...) -> Self: ...
    @property
    def recurrence(self) -> Eq | Relational | Ne: ...
    @property
    def yn(self) -> Basic: ...
    @property
    def y(self) -> type[Basic]: ...
    @property
    def n(self) -> Basic: ...
    @property
    def initial(self) -> Basic: ...
    @property
    def start(self) -> Basic: ...
    @property
    def stop(self): ...
    @property
    def interval(self) -> tuple[Basic, Any]: ...
    def __iter__(self) -> Generator[Any, Any, NoReturn]: ...

def sequence(seq, limits=...) -> SeqPer | SeqFormula: ...

class SeqExprOp(SeqBase):
    @property
    def gen(self) -> tuple[Any, ...]: ...
    @property
    def interval(self) -> FiniteSet | Intersection | Union | Complement: ...
    @property
    def start(self): ...
    @property
    def stop(self): ...
    @property
    def variables(self) -> tuple[Any, ...]: ...
    @property
    def length(self): ...

class SeqAdd(SeqExprOp):
    def __new__(cls, *args, **kwargs) -> SeqAdd | Self: ...
    @staticmethod
    def reduce(args) -> SeqAdd: ...

class SeqMul(SeqExprOp):
    def __new__(cls, *args, **kwargs) -> SeqMul | Self: ...
    @staticmethod
    def reduce(args) -> SeqMul: ...