venv/lib/python3.11/site-packages/sympy/assumptions/tests/test_query.py

from sympy.abc import t, w, x, y, z, n, k, m, p, i
from sympy.assumptions import (ask, AssumptionsContext, Q, register_handler,
        remove_handler)
from sympy.assumptions.assume import assuming, global_assumptions, Predicate
from sympy.assumptions.cnf import CNF, Literal
from sympy.assumptions.facts import (single_fact_lookup,
    get_known_facts, generate_known_facts_dict, get_known_facts_keys)
from sympy.assumptions.handlers import AskHandler
from sympy.assumptions.ask_generated import (get_all_known_facts,
    get_known_facts_dict)
from sympy.core.add import Add
from sympy.core.numbers import (I, Integer, Rational, oo, zoo, pi)
from sympy.core.singleton import S
from sympy.core.power import Pow
from sympy.core.symbol import Str, symbols, Symbol
from sympy.functions.combinatorial.factorials import factorial
from sympy.functions.elementary.complexes import (Abs, im, re, sign)
from sympy.functions.elementary.exponential import (exp, log)
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.trigonometric import (
    acos, acot, asin, atan, cos, cot, sin, tan)
from sympy.logic.boolalg import Equivalent, Implies, Xor, And, to_cnf
from sympy.matrices import Matrix, SparseMatrix
from sympy.testing.pytest import (XFAIL, slow, raises, warns_deprecated_sympy,
    _both_exp_pow)
import math


def test_int_1():
    z = 1
    assert ask(Q.commutative(z)) is True
    assert ask(Q.integer(z)) is True
    assert ask(Q.rational(z)) is True
    assert ask(Q.real(z)) is True
    assert ask(Q.complex(z)) is True
    assert ask(Q.irrational(z)) is False
    assert ask(Q.imaginary(z)) is False
    assert ask(Q.positive(z)) is True
    assert ask(Q.negative(z)) is False
    assert ask(Q.even(z)) is False
    assert ask(Q.odd(z)) is True
    assert ask(Q.finite(z)) is True
    assert ask(Q.prime(z)) is False
    assert ask(Q.composite(z)) is False
    assert ask(Q.hermitian(z)) is True
    assert ask(Q.antihermitian(z)) is False


def test_int_11():
    z = 11
    assert ask(Q.commutative(z)) is True
    assert ask(Q.integer(z)) is True
    assert ask(Q.rational(z)) is True
    assert ask(Q.real(z)) is True
    assert ask(Q.complex(z)) is True
    assert ask(Q.irrational(z)) is False
    assert ask(Q.imaginary(z)) is False
    assert ask(Q.positive(z)) is True
    assert ask(Q.negative(z)) is False
    assert ask(Q.even(z)) is False
    assert ask(Q.odd(z)) is True
    assert ask(Q.finite(z)) is True
    assert ask(Q.prime(z)) is True
    assert ask(Q.composite(z)) is False
    assert ask(Q.hermitian(z)) is True
    assert ask(Q.antihermitian(z)) is False


def test_int_12():
    z = 12
    assert ask(Q.commutative(z)) is True
    assert ask(Q.integer(z)) is True
    assert ask(Q.rational(z)) is True
    assert ask(Q.real(z)) is True
    assert ask(Q.complex(z)) is True
    assert ask(Q.irrational(z)) is False
    assert ask(Q.imaginary(z)) is False
    assert ask(Q.positive(z)) is True
    assert ask(Q.negative(z)) is False
    assert ask(Q.even(z)) is True
    assert ask(Q.odd(z)) is False
    assert ask(Q.finite(z)) is True
    assert ask(Q.prime(z)) is False
    assert ask(Q.composite(z)) is True
    assert ask(Q.hermitian(z)) is True
    assert ask(Q.antihermitian(z)) is False


def test_float_1():
    z = 1.0
    assert ask(Q.commutative(z)) is True
    assert ask(Q.integer(z)) is None
    assert ask(Q.rational(z)) is None
    assert ask(Q.real(z)) is True
    assert ask(Q.complex(z)) is True
    assert ask(Q.irrational(z)) is None
    assert ask(Q.imaginary(z)) is False
    assert ask(Q.positive(z)) is True
    assert ask(Q.negative(z)) is False
    assert ask(Q.even(z)) is None
    assert ask(Q.odd(z)) is None
    assert ask(Q.finite(z)) is True
    assert ask(Q.prime(z)) is None
    assert ask(Q.composite(z)) is None
    assert ask(Q.hermitian(z)) is True
    assert ask(Q.antihermitian(z)) is False

    z = 7.2123
    assert ask(Q.commutative(z)) is True
    assert ask(Q.integer(z)) is False
    assert ask(Q.rational(z)) is None
    assert ask(Q.real(z)) is True
    assert ask(Q.complex(z)) is True
    assert ask(Q.irrational(z)) is None
    assert ask(Q.imaginary(z)) is False
    assert ask(Q.positive(z)) is True
    assert ask(Q.negative(z)) is False
    assert ask(Q.even(z)) is False
    assert ask(Q.odd(z)) is False
    assert ask(Q.finite(z)) is True
    assert ask(Q.prime(z)) is False
    assert ask(Q.composite(z)) is False
    assert ask(Q.hermitian(z)) is True
    assert ask(Q.antihermitian(z)) is False

    # test for issue #12168
    assert ask(Q.rational(math.pi)) is None


def test_zero_0():
    z = Integer(0)
    assert ask(Q.nonzero(z)) is False
    assert ask(Q.zero(z)) is True
    assert ask(Q.commutative(z)) is True
    assert ask(Q.integer(z)) is True
    assert ask(Q.rational(z)) is True
    assert ask(Q.real(z)) is True
    assert ask(Q.complex(z)) is True
    assert ask(Q.imaginary(z)) is False
    assert ask(Q.positive(z)) is False
    assert ask(Q.negative(z)) is False
    assert ask(Q.even(z)) is True
    assert ask(Q.odd(z)) is False
    assert ask(Q.finite(z)) is True
    assert ask(Q.prime(z)) is False
    assert ask(Q.composite(z)) is False
    assert ask(Q.hermitian(z)) is True
    assert ask(Q.antihermitian(z)) is True


def test_negativeone():
    z = Integer(-1)
    assert ask(Q.nonzero(z)) is True
    assert ask(Q.zero(z)) is False
    assert ask(Q.commutative(z)) is True
    assert ask(Q.integer(z)) is True
    assert ask(Q.rational(z)) is True
    assert ask(Q.real(z)) is True
    assert ask(Q.complex(z)) is True
    assert ask(Q.irrational(z)) is False
    assert ask(Q.imaginary(z)) is False
    assert ask(Q.positive(z)) is False
    assert ask(Q.negative(z)) is True
    assert ask(Q.even(z)) is False
    assert ask(Q.odd(z)) is True
    assert ask(Q.finite(z)) is True
    assert ask(Q.prime(z)) is False
    assert ask(Q.composite(z)) is False
    assert ask(Q.hermitian(z)) is True
    assert ask(Q.antihermitian(z)) is False


def test_infinity():
    assert ask(Q.commutative(oo)) is True
    assert ask(Q.integer(oo)) is False
    assert ask(Q.rational(oo)) is False
    assert ask(Q.algebraic(oo)) is False
    assert ask(Q.real(oo)) is False
    assert ask(Q.extended_real(oo)) is True
    assert ask(Q.complex(oo)) is False
    assert ask(Q.irrational(oo)) is False
    assert ask(Q.imaginary(oo)) is False
    assert ask(Q.positive(oo)) is False
    assert ask(Q.extended_positive(oo)) is True
    assert ask(Q.negative(oo)) is False
    assert ask(Q.even(oo)) is False
    assert ask(Q.odd(oo)) is False
    assert ask(Q.finite(oo)) is False
    assert ask(Q.infinite(oo)) is True
    assert ask(Q.prime(oo)) is False
    assert ask(Q.composite(oo)) is False
    assert ask(Q.hermitian(oo)) is False
    assert ask(Q.antihermitian(oo)) is False
    assert ask(Q.positive_infinite(oo)) is True
    assert ask(Q.negative_infinite(oo)) is False


def test_neg_infinity():
    mm = S.NegativeInfinity
    assert ask(Q.commutative(mm)) is True
    assert ask(Q.integer(mm)) is False
    assert ask(Q.rational(mm)) is False
    assert ask(Q.algebraic(mm)) is False
    assert ask(Q.real(mm)) is False
    assert ask(Q.extended_real(mm)) is True
    assert ask(Q.complex(mm)) is False
    assert ask(Q.irrational(mm)) is False
    assert ask(Q.imaginary(mm)) is False
    assert ask(Q.positive(mm)) is False
    assert ask(Q.negative(mm)) is False
    assert ask(Q.extended_negative(mm)) is True
    assert ask(Q.even(mm)) is False
    assert ask(Q.odd(mm)) is False
    assert ask(Q.finite(mm)) is False
    assert ask(Q.infinite(oo)) is True
    assert ask(Q.prime(mm)) is False
    assert ask(Q.composite(mm)) is False
    assert ask(Q.hermitian(mm)) is False
    assert ask(Q.antihermitian(mm)) is False
    assert ask(Q.positive_infinite(-oo)) is False
    assert ask(Q.negative_infinite(-oo)) is True


def test_complex_infinity():
    assert ask(Q.commutative(zoo)) is True
    assert ask(Q.integer(zoo)) is False
    assert ask(Q.rational(zoo)) is False
    assert ask(Q.algebraic(zoo)) is False
    assert ask(Q.real(zoo)) is False
    assert ask(Q.extended_real(zoo)) is False
    assert ask(Q.complex(zoo)) is False
    assert ask(Q.irrational(zoo)) is False
    assert ask(Q.imaginary(zoo)) is False
    assert ask(Q.positive(zoo)) is False
    assert ask(Q.negative(zoo)) is False
    assert ask(Q.zero(zoo)) is False
    assert ask(Q.nonzero(zoo)) is False
    assert ask(Q.even(zoo)) is False
    assert ask(Q.odd(zoo)) is False
    assert ask(Q.finite(zoo)) is False
    assert ask(Q.infinite(zoo)) is True
    assert ask(Q.prime(zoo)) is False
    assert ask(Q.composite(zoo)) is False
    assert ask(Q.hermitian(zoo)) is False
    assert ask(Q.antihermitian(zoo)) is False
    assert ask(Q.positive_infinite(zoo)) is False
    assert ask(Q.negative_infinite(zoo)) is False


def test_nan():
    nan = S.NaN
    assert ask(Q.commutative(nan)) is True
    assert ask(Q.integer(nan)) is None
    assert ask(Q.rational(nan)) is None
    assert ask(Q.algebraic(nan)) is None
    assert ask(Q.real(nan)) is None
    assert ask(Q.extended_real(nan)) is None
    assert ask(Q.complex(nan)) is None
    assert ask(Q.irrational(nan)) is None
    assert ask(Q.imaginary(nan)) is None
    assert ask(Q.positive(nan)) is None
    assert ask(Q.nonzero(nan)) is None
    assert ask(Q.zero(nan)) is None
    assert ask(Q.even(nan)) is None
    assert ask(Q.odd(nan)) is None
    assert ask(Q.finite(nan)) is None
    assert ask(Q.infinite(nan)) is None
    assert ask(Q.prime(nan)) is None
    assert ask(Q.composite(nan)) is None
    assert ask(Q.hermitian(nan)) is None
    assert ask(Q.antihermitian(nan)) is None


def test_Rational_number():
    r = Rational(3, 4)
    assert ask(Q.commutative(r)) is True
    assert ask(Q.integer(r)) is False
    assert ask(Q.rational(r)) is True
    assert ask(Q.real(r)) is True
    assert ask(Q.complex(r)) is True
    assert ask(Q.irrational(r)) is False
    assert ask(Q.imaginary(r)) is False
    assert ask(Q.positive(r)) is True
    assert ask(Q.negative(r)) is False
    assert ask(Q.even(r)) is False
    assert ask(Q.odd(r)) is False
    assert ask(Q.finite(r)) is True
    assert ask(Q.prime(r)) is False
    assert ask(Q.composite(r)) is False
    assert ask(Q.hermitian(r)) is True
    assert ask(Q.antihermitian(r)) is False

    r = Rational(1, 4)
    assert ask(Q.positive(r)) is True
    assert ask(Q.negative(r)) is False

    r = Rational(5, 4)
    assert ask(Q.negative(r)) is False
    assert ask(Q.positive(r)) is True

    r = Rational(5, 3)
    assert ask(Q.positive(r)) is True
    assert ask(Q.negative(r)) is False

    r = Rational(-3, 4)
    assert ask(Q.positive(r)) is False
    assert ask(Q.negative(r)) is True

    r = Rational(-1, 4)
    assert ask(Q.positive(r)) is False
    assert ask(Q.negative(r)) is True

    r = Rational(-5, 4)
    assert ask(Q.negative(r)) is True
    assert ask(Q.positive(r)) is False

    r = Rational(-5, 3)
    assert ask(Q.positive(r)) is False
    assert ask(Q.negative(r)) is True


def test_sqrt_2():
    z = sqrt(2)
    assert ask(Q.commutative(z)) is True
    assert ask(Q.integer(z)) is False
    assert ask(Q.rational(z)) is False
    assert ask(Q.real(z)) is True
    assert ask(Q.complex(z)) is True
    assert ask(Q.irrational(z)) is True
    assert ask(Q.imaginary(z)) is False
    assert ask(Q.positive(z)) is True
    assert ask(Q.negative(z)) is False
    assert ask(Q.even(z)) is False
    assert ask(Q.odd(z)) is False
    assert ask(Q.finite(z)) is True
    assert ask(Q.prime(z)) is False
    assert ask(Q.composite(z)) is False
    assert ask(Q.hermitian(z)) is True
    assert ask(Q.antihermitian(z)) is False


def test_pi():
    z = S.Pi
    assert ask(Q.commutative(z)) is True
    assert ask(Q.integer(z)) is False
    assert ask(Q.rational(z)) is False
    assert ask(Q.algebraic(z)) is False
    assert ask(Q.real(z)) is True
    assert ask(Q.complex(z)) is True
    assert ask(Q.irrational(z)) is True
    assert ask(Q.imaginary(z)) is False
    assert ask(Q.positive(z)) is True
    assert ask(Q.negative(z)) is False
    assert ask(Q.even(z)) is False
    assert ask(Q.odd(z)) is False
    assert ask(Q.finite(z)) is True
    assert ask(Q.prime(z)) is False
    assert ask(Q.composite(z)) is False
    assert ask(Q.hermitian(z)) is True
    assert ask(Q.antihermitian(z)) is False

    z = S.Pi + 1
    assert ask(Q.commutative(z)) is True
    assert ask(Q.integer(z)) is False
    assert ask(Q.rational(z)) is False
    assert ask(Q.algebraic(z)) is False
    assert ask(Q.real(z)) is True
    assert ask(Q.complex(z)) is True
    assert ask(Q.irrational(z)) is True
    assert ask(Q.imaginary(z)) is False
    assert ask(Q.positive(z)) is True
    assert ask(Q.negative(z)) is False
    assert ask(Q.even(z)) is False
    assert ask(Q.odd(z)) is False
    assert ask(Q.finite(z)) is True
    assert ask(Q.prime(z)) is False
    assert ask(Q.composite(z)) is False
    assert ask(Q.hermitian(z)) is True
    assert ask(Q.antihermitian(z)) is False

    z = 2*S.Pi
    assert ask(Q.commutative(z)) is True
    assert ask(Q.integer(z)) is False
    assert ask(Q.rational(z)) is False
    assert ask(Q.algebraic(z)) is False
    assert ask(Q.real(z)) is True
    assert ask(Q.complex(z)) is True
    assert ask(Q.irrational(z)) is True
    assert ask(Q.imaginary(z)) is False
    assert ask(Q.positive(z)) is True
    assert ask(Q.negative(z)) is False
    assert ask(Q.even(z)) is False
    assert ask(Q.odd(z)) is False
    assert ask(Q.finite(z)) is True
    assert ask(Q.prime(z)) is False
    assert ask(Q.composite(z)) is False
    assert ask(Q.hermitian(z)) is True
    assert ask(Q.antihermitian(z)) is False

    z = S.Pi ** 2
    assert ask(Q.commutative(z)) is True
    assert ask(Q.integer(z)) is False
    assert ask(Q.rational(z)) is False
    assert ask(Q.algebraic(z)) is False
    assert ask(Q.real(z)) is True
    assert ask(Q.complex(z)) is True
    assert ask(Q.irrational(z)) is True
    assert ask(Q.imaginary(z)) is False
    assert ask(Q.positive(z)) is True
    assert ask(Q.negative(z)) is False
    assert ask(Q.even(z)) is False
    assert ask(Q.odd(z)) is False
    assert ask(Q.finite(z)) is True
    assert ask(Q.prime(z)) is False
    assert ask(Q.composite(z)) is False
    assert ask(Q.hermitian(z)) is True
    assert ask(Q.antihermitian(z)) is False

    z = (1 + S.Pi) ** 2
    assert ask(Q.commutative(z)) is True
    assert ask(Q.integer(z)) is False
    assert ask(Q.rational(z)) is False
    assert ask(Q.algebraic(z)) is None
    assert ask(Q.real(z)) is True
    assert ask(Q.complex(z)) is True
    assert ask(Q.irrational(z)) is True
    assert ask(Q.imaginary(z)) is False
    assert ask(Q.positive(z)) is True
    assert ask(Q.negative(z)) is False
    assert ask(Q.even(z)) is False
    assert ask(Q.odd(z)) is False
    assert ask(Q.finite(z)) is True
    assert ask(Q.prime(z)) is False
    assert ask(Q.composite(z)) is False
    assert ask(Q.hermitian(z)) is True
    assert ask(Q.antihermitian(z)) is False


def test_E():
    z = S.Exp1
    assert ask(Q.commutative(z)) is True
    assert ask(Q.integer(z)) is False
    assert ask(Q.rational(z)) is False
    assert ask(Q.algebraic(z)) is False
    assert ask(Q.real(z)) is True
    assert ask(Q.complex(z)) is True
    assert ask(Q.irrational(z)) is True
    assert ask(Q.imaginary(z)) is False
    assert ask(Q.positive(z)) is True
    assert ask(Q.negative(z)) is False
    assert ask(Q.even(z)) is False
    assert ask(Q.odd(z)) is False
    assert ask(Q.finite(z)) is True
    assert ask(Q.prime(z)) is False
    assert ask(Q.composite(z)) is False
    assert ask(Q.hermitian(z)) is True
    assert ask(Q.antihermitian(z)) is False


def test_GoldenRatio():
    z = S.GoldenRatio
    assert ask(Q.commutative(z)) is True
    assert ask(Q.integer(z)) is False
    assert ask(Q.rational(z)) is False
    assert ask(Q.algebraic(z)) is True
    assert ask(Q.real(z)) is True
    assert ask(Q.complex(z)) is True
    assert ask(Q.irrational(z)) is True
    assert ask(Q.imaginary(z)) is False
    assert ask(Q.positive(z)) is True
    assert ask(Q.negative(z)) is False
    assert ask(Q.even(z)) is False
    assert ask(Q.odd(z)) is False
    assert ask(Q.finite(z)) is True
    assert ask(Q.prime(z)) is False
    assert ask(Q.composite(z)) is False
    assert ask(Q.hermitian(z)) is True
    assert ask(Q.antihermitian(z)) is False


def test_TribonacciConstant():
    z = S.TribonacciConstant
    assert ask(Q.commutative(z)) is True
    assert ask(Q.integer(z)) is False
    assert ask(Q.rational(z)) is False
    assert ask(Q.algebraic(z)) is True
    assert ask(Q.real(z)) is True
    assert ask(Q.complex(z)) is True
    assert ask(Q.irrational(z)) is True
    assert ask(Q.imaginary(z)) is False
    assert ask(Q.positive(z)) is True
    assert ask(Q.negative(z)) is False
    assert ask(Q.even(z)) is False
    assert ask(Q.odd(z)) is False
    assert ask(Q.finite(z)) is True
    assert ask(Q.prime(z)) is False
    assert ask(Q.composite(z)) is False
    assert ask(Q.hermitian(z)) is True
    assert ask(Q.antihermitian(z)) is False


def test_I():
    z = I
    assert ask(Q.commutative(z)) is True
    assert ask(Q.integer(z)) is False
    assert ask(Q.rational(z)) is False
    assert ask(Q.algebraic(z)) is True
    assert ask(Q.real(z)) is False
    assert ask(Q.complex(z)) is True
    assert ask(Q.irrational(z)) is False
    assert ask(Q.imaginary(z)) is True
    assert ask(Q.positive(z)) is False
    assert ask(Q.negative(z)) is False
    assert ask(Q.even(z)) is False
    assert ask(Q.odd(z)) is False
    assert ask(Q.finite(z)) is True
    assert ask(Q.prime(z)) is False
    assert ask(Q.composite(z)) is False
    assert ask(Q.hermitian(z)) is False
    assert ask(Q.antihermitian(z)) is True

    z = 1 + I
    assert ask(Q.commutative(z)) is True
    assert ask(Q.integer(z)) is False
    assert ask(Q.rational(z)) is False
    assert ask(Q.algebraic(z)) is True
    assert ask(Q.real(z)) is False
    assert ask(Q.complex(z)) is True
    assert ask(Q.irrational(z)) is False
    assert ask(Q.imaginary(z)) is False
    assert ask(Q.positive(z)) is False
    assert ask(Q.negative(z)) is False
    assert ask(Q.even(z)) is False
    assert ask(Q.odd(z)) is False
    assert ask(Q.finite(z)) is True
    assert ask(Q.prime(z)) is False
    assert ask(Q.composite(z)) is False
    assert ask(Q.hermitian(z)) is False
    assert ask(Q.antihermitian(z)) is False

    z = I*(1 + I)
    assert ask(Q.commutative(z)) is True
    assert ask(Q.integer(z)) is False
    assert ask(Q.rational(z)) is False
    assert ask(Q.algebraic(z)) is True
    assert ask(Q.real(z)) is False
    assert ask(Q.complex(z)) is True
    assert ask(Q.irrational(z)) is False
    assert ask(Q.imaginary(z)) is False
    assert ask(Q.positive(z)) is False
    assert ask(Q.negative(z)) is False
    assert ask(Q.even(z)) is False
    assert ask(Q.odd(z)) is False
    assert ask(Q.finite(z)) is True
    assert ask(Q.prime(z)) is False
    assert ask(Q.composite(z)) is False
    assert ask(Q.hermitian(z)) is False
    assert ask(Q.antihermitian(z)) is False

    z = I**(I)
    assert ask(Q.imaginary(z)) is False
    assert ask(Q.real(z)) is True

    z = (-I)**(I)
    assert ask(Q.imaginary(z)) is False
    assert ask(Q.real(z)) is True

    z = (3*I)**(I)
    assert ask(Q.imaginary(z)) is False
    assert ask(Q.real(z)) is False

    z = (1)**(I)
    assert ask(Q.imaginary(z)) is False
    assert ask(Q.real(z)) is True

    z = (-1)**(I)
    assert ask(Q.imaginary(z)) is False
    assert ask(Q.real(z)) is True

    z = (1+I)**(I)
    assert ask(Q.imaginary(z)) is False
    assert ask(Q.real(z)) is False

    z = (I)**(I+3)
    assert ask(Q.imaginary(z)) is True
    assert ask(Q.real(z)) is False

    z = (I)**(I+2)
    assert ask(Q.imaginary(z)) is False
    assert ask(Q.real(z)) is True

    z = (I)**(2)
    assert ask(Q.imaginary(z)) is False
    assert ask(Q.real(z)) is True

    z = (I)**(3)
    assert ask(Q.imaginary(z)) is True
    assert ask(Q.real(z)) is False

    z = (3)**(I)
    assert ask(Q.imaginary(z)) is False
    assert ask(Q.real(z)) is False

    z = (I)**(0)
    assert ask(Q.imaginary(z)) is False
    assert ask(Q.real(z)) is True

def test_bounded():
    x, y, z = symbols('x,y,z')
    a = x + y
    x, y = a.args
    assert ask(Q.finite(a), Q.positive_infinite(y)) is None
    assert ask(Q.finite(x)) is None
    assert ask(Q.finite(x), Q.finite(x)) is True
    assert ask(Q.finite(x), Q.finite(y)) is None
    assert ask(Q.finite(x), Q.complex(x)) is True
    assert ask(Q.finite(x), Q.extended_real(x)) is None

    assert ask(Q.finite(x + 1)) is None
    assert ask(Q.finite(x + 1), Q.finite(x)) is True
    a = x + y
    x, y = a.args
    # B + B
    assert ask(Q.finite(a), Q.finite(x) & Q.finite(y)) is True
    assert ask(Q.finite(a), Q.positive(x) & Q.finite(y)) is True
    assert ask(Q.finite(a), Q.finite(x) & Q.positive(y)) is True
    assert ask(Q.finite(a), Q.positive(x) & Q.positive(y)) is True
    assert ask(Q.finite(a), Q.positive(x) & Q.finite(y)
        & ~Q.positive(y)) is True
    assert ask(Q.finite(a), Q.finite(x) & ~Q.positive(x)
        & Q.positive(y)) is True
    assert ask(Q.finite(a), Q.finite(x) & Q.finite(y) & ~Q.positive(x)
        & ~Q.positive(y)) is True
    # B + U
    assert ask(Q.finite(a), Q.finite(x) & ~Q.finite(y)) is False
    assert ask(Q.finite(a), Q.positive(x) & ~Q.finite(y)) is False
    assert ask(Q.finite(a), Q.finite(x)
        & Q.positive_infinite(y)) is False
    assert ask(Q.finite(a), Q.positive(x)
        & Q.positive_infinite(y)) is False
    assert ask(Q.finite(a), Q.positive(x) & ~Q.finite(y)
        & ~Q.positive(y)) is False
    assert ask(Q.finite(a), Q.finite(x) & ~Q.positive(x)
        & Q.positive_infinite(y)) is False
    assert ask(Q.finite(a), Q.finite(x) & ~Q.positive(x) & ~Q.finite(y)
        & ~Q.positive(y)) is False
    # B + ?
    assert ask(Q.finite(a), Q.finite(x)) is None
    assert ask(Q.finite(a), Q.positive(x)) is None
    assert ask(Q.finite(a), Q.finite(x)
        & Q.extended_positive(y)) is None
    assert ask(Q.finite(a), Q.positive(x)
        & Q.extended_positive(y)) is None
    assert ask(Q.finite(a), Q.positive(x) & ~Q.positive(y)) is None
    assert ask(Q.finite(a), Q.finite(x) & ~Q.positive(x)
        & Q.extended_positive(y)) is None
    assert ask(Q.finite(a), Q.finite(x) & ~Q.positive(x)
        & ~Q.positive(y)) is None
    # U + U
    assert ask(Q.finite(a), ~Q.finite(x) & ~Q.finite(y)) is None
    assert ask(Q.finite(a), Q.positive_infinite(x)
        & ~Q.finite(y)) is None
    assert ask(Q.finite(a), ~Q.finite(x)
        & Q.positive_infinite(y)) is None
    assert ask(Q.finite(a), Q.positive_infinite(x)
        & Q.positive_infinite(y)) is False
    assert ask(Q.finite(a), Q.positive_infinite(x) & ~Q.finite(y)
        & ~Q.extended_positive(y)) is None
    assert ask(Q.finite(a), ~Q.finite(x) & ~Q.extended_positive(x)
        & Q.positive_infinite(y)) is None
    assert ask(Q.finite(a), ~Q.finite(x) & ~Q.finite(y)
        & ~Q.extended_positive(x) & ~Q.extended_positive(y)) is False
    # U + ?
    assert ask(Q.finite(a), ~Q.finite(y)) is None
    assert ask(Q.finite(a), Q.extended_positive(x)
        & ~Q.finite(y)) is None
    assert ask(Q.finite(a), Q.positive_infinite(y)) is None
    assert ask(Q.finite(a), Q.extended_positive(x)
        & Q.positive_infinite(y)) is False
    assert ask(Q.finite(a), Q.extended_positive(x)
        & ~Q.finite(y) & ~Q.extended_positive(y)) is None
    assert ask(Q.finite(a), ~Q.extended_positive(x)
        & Q.positive_infinite(y)) is None
    assert ask(Q.finite(a), ~Q.extended_positive(x) & ~Q.finite(y)
        & ~Q.extended_positive(y)) is False
    # ? + ?
    assert ask(Q.finite(a)) is None
    assert ask(Q.finite(a), Q.extended_positive(x)) is None
    assert ask(Q.finite(a), Q.extended_positive(y)) is None
    assert ask(Q.finite(a), Q.extended_positive(x)
        & Q.extended_positive(y)) is None
    assert ask(Q.finite(a), Q.extended_positive(x)
        & ~Q.extended_positive(y)) is None
    assert ask(Q.finite(a), ~Q.extended_positive(x)
        & Q.extended_positive(y)) is None
    assert ask(Q.finite(a), ~Q.extended_positive(x)
        & ~Q.extended_positive(y)) is None

    x, y, z = symbols('x,y,z')
    a = x + y + z
    x, y, z = a.args
    assert ask(Q.finite(a), Q.negative(x) & Q.negative(y)
        & Q.negative(z)) is True
    assert ask(Q.finite(a), Q.negative(x) & Q.negative(y)
        & Q.finite(z)) is True
    assert ask(Q.finite(a), Q.negative(x) & Q.negative(y)
        & Q.positive(z)) is True
    assert ask(Q.finite(a), Q.negative(x) & Q.negative(y)
        & Q.negative_infinite(z)) is False
    assert ask(Q.finite(a), Q.negative(x) & Q.negative(y)
        & ~Q.finite(z)) is False
    assert ask(Q.finite(a), Q.negative(x) & Q.negative(y)
        & Q.positive_infinite(z)) is False
    assert ask(Q.finite(a), Q.negative(x) & Q.negative(y)
        & Q.extended_negative(z)) is None
    assert ask(Q.finite(a), Q.negative(x) & Q.negative(y)) is None
    assert ask(Q.finite(a), Q.negative(x) & Q.negative(y)
        & Q.extended_positive(z)) is None
    assert ask(Q.finite(a), Q.negative(x) & Q.finite(y)
        & Q.finite(z)) is True
    assert ask(Q.finite(a), Q.negative(x) & Q.finite(y)
        & Q.positive(z)) is True
    assert ask(Q.finite(a), Q.negative(x) & Q.finite(y)
        & Q.negative_infinite(z)) is False
    assert ask(Q.finite(a), Q.negative(x) & Q.finite(y)
        & ~Q.finite(z)) is False
    assert ask(Q.finite(a), Q.negative(x) & Q.finite(y)
        & Q.positive_infinite(z)) is False
    assert ask(Q.finite(a), Q.negative(x) & Q.finite(y)
        & Q.extended_negative(z)) is None
    assert ask(Q.finite(a), Q.negative(x) & Q.finite(y)) is None
    assert ask(Q.finite(a), Q.negative(x) & Q.finite(y)
        & Q.extended_positive(z)) is None
    assert ask(Q.finite(a), Q.negative(x) & Q.positive(y)
        & Q.positive(z)) is True
    assert ask(Q.finite(a), Q.negative(x) & Q.positive(y)
        & Q.negative_infinite(z)) is False
    assert ask(Q.finite(a), Q.negative(x) & Q.positive(y)
        & ~Q.finite(z)) is False
    assert ask(Q.finite(a), Q.negative(x) & Q.positive(y)
        & Q.positive_infinite(z)) is False
    assert ask(Q.finite(a), Q.negative(x) & Q.positive(y)
        & Q.extended_negative(z)) is None
    assert ask(Q.finite(a), Q.negative(x) & Q.extended_positive(y)
        & Q.finite(y)) is None
    assert ask(Q.finite(a), Q.negative(x) & Q.positive(y)
        & Q.extended_positive(z)) is None
    assert ask(Q.finite(a), Q.negative(x) & Q.negative_infinite(y)
        & Q.negative_infinite(z)) is False
    assert ask(Q.finite(a), Q.negative(x) & Q.negative_infinite(y)
        & ~Q.finite(z)) is None
    assert ask(Q.finite(a), Q.negative(x) & Q.negative_infinite(y)
        & Q.positive_infinite(z)) is None
    assert ask(Q.finite(a), Q.negative(x) & Q.negative_infinite(y)
        & Q.extended_negative(z)) is False
    assert ask(Q.finite(a), Q.negative(x)
        & Q.negative_infinite(y)) is None
    assert ask(Q.finite(a), Q.negative(x) & Q.negative_infinite(y)
        & Q.extended_positive(z)) is None
    assert ask(Q.finite(a), Q.negative(x) & ~Q.finite(y)
        & ~Q.finite(z)) is None
    assert ask(Q.finite(a), Q.negative(x) & ~Q.finite(y)
        & Q.positive_infinite(z)) is None
    assert ask(Q.finite(a), Q.negative(x) & ~Q.finite(y)
        & Q.extended_negative(z)) is None
    assert ask(Q.finite(a), Q.negative(x) & ~Q.finite(y)) is None
    assert ask(Q.finite(a), Q.negative(x) & ~Q.finite(y)
        & Q.extended_positive(z)) is None
    assert ask(Q.finite(a), Q.negative(x) & Q.positive_infinite(y)
        & Q.positive_infinite(z)) is False
    assert ask(Q.finite(a), Q.negative(x) & Q.positive_infinite(y)
        & Q.negative_infinite(z)) is None
    assert ask(Q.finite(a), Q.negative(x) &
         Q.positive_infinite(y)) is None
    assert ask(Q.finite(a), Q.negative(x) & Q.positive_infinite(y)
        & Q.extended_positive(z)) is False
    assert ask(Q.finite(a), Q.negative(x) & Q.extended_negative(y)
        & Q.extended_negative(z)) is None
    assert ask(Q.finite(a), Q.negative(x)
        & Q.extended_negative(y)) is None
    assert ask(Q.finite(a), Q.negative(x) & Q.extended_negative(y)
        & Q.extended_positive(z)) is None
    assert ask(Q.finite(a), Q.negative(x)) is None
    assert ask(Q.finite(a), Q.negative(x)
        & Q.extended_positive(z)) is None
    assert ask(Q.finite(a), Q.negative(x) & Q.extended_positive(y)
        & Q.extended_positive(z)) is None
    assert ask(Q.finite(a), Q.finite(x) & Q.finite(y)
        & Q.finite(z)) is True
    assert ask(Q.finite(a), Q.finite(x) & Q.finite(y)
        & Q.positive(z)) is True
    assert ask(Q.finite(a), Q.finite(x) & Q.finite(y)
        & Q.negative_infinite(z)) is False
    assert ask(Q.finite(a), Q.finite(x) & Q.finite(y)
        & ~Q.finite(z)) is False
    assert ask(Q.finite(a), Q.finite(x) & Q.finite(y)
        & Q.positive_infinite(z)) is False
    assert ask(Q.finite(a), Q.finite(x) & Q.finite(y)
        & Q.extended_negative(z)) is None
    assert ask(Q.finite(a), Q.finite(x) & Q.finite(y)) is None
    assert ask(Q.finite(a), Q.finite(x) & Q.finite(y)
        & Q.extended_positive(z)) is None
    assert ask(Q.finite(a), Q.finite(x) & Q.positive(y)
        & Q.positive(z)) is True
    assert ask(Q.finite(a), Q.finite(x) & Q.positive(y)
        & Q.negative_infinite(z)) is False
    assert ask(Q.finite(a), Q.finite(x) & Q.positive(y)
        & ~Q.finite(z)) is False
    assert ask(Q.finite(a), Q.finite(x) & Q.positive(y)
        & Q.positive_infinite(z)) is False
    assert ask(Q.finite(a), Q.finite(x) & Q.positive(y)
        & Q.extended_negative(z)) is None
    assert ask(Q.finite(a), Q.finite(x) & Q.positive(y)) is None
    assert ask(Q.finite(a), Q.finite(x) & Q.positive(y)
        & Q.extended_positive(z)) is None
    assert ask(Q.finite(a), Q.finite(x) & Q.negative_infinite(y)
        & Q.negative_infinite(z)) is False
    assert ask(Q.finite(a), Q.finite(x) & Q.negative_infinite(y)
        & ~Q.finite(z)) is None
    assert ask(Q.finite(a), Q.finite(x) & Q.negative_infinite(y)
        & Q.positive_infinite(z)) is None
    assert ask(Q.finite(a), Q.finite(x) & Q.negative_infinite(y)
        & Q.extended_negative(z)) is False
    assert ask(Q.finite(a), Q.finite(x)
        & Q.negative_infinite(y)) is None
    assert ask(Q.finite(a), Q.finite(x) & Q.negative_infinite(y)
        & Q.extended_positive(z)) is None
    assert ask(Q.finite(a), Q.finite(x) & ~Q.finite(y)
        & ~Q.finite(z)) is None
    assert ask(Q.finite(a), Q.finite(x) & ~Q.finite(y)
        & Q.positive_infinite(z)) is None
    assert ask(Q.finite(a), Q.finite(x) & ~Q.finite(y)
        & Q.extended_negative(z)) is None
    assert ask(Q.finite(a), Q.finite(x) & ~Q.finite(y)) is None
    assert ask(Q.finite(a), Q.finite(x) & ~Q.finite(y)
        & Q.extended_positive(z)) is None
    assert ask(Q.finite(a), Q.finite(x) & Q.positive_infinite(y)
        & Q.positive_infinite(z)) is False
    assert ask(Q.finite(a), Q.finite(x) & Q.positive_infinite(y)
        & Q.extended_negative(z)) is None
    assert ask(Q.finite(a), Q.finite(x)
        & Q.positive_infinite(y)) is None
    assert ask(Q.finite(a), Q.finite(x) & Q.positive_infinite(y)
        & Q.extended_positive(z)) is False
    assert ask(Q.finite(a), Q.finite(x) & Q.extended_negative(y)
        & Q.extended_negative(z)) is None
    assert ask(Q.finite(a), Q.finite(x)
        & Q.extended_negative(y)) is None
    assert ask(Q.finite(a), Q.finite(x) & Q.extended_negative(y)
        & Q.extended_positive(z)) is None
    assert ask(Q.finite(a), Q.finite(x)) is None
    assert ask(Q.finite(a), Q.finite(x)
        & Q.extended_positive(z)) is None
    assert ask(Q.finite(a), Q.finite(x) & Q.extended_positive(y)
        & Q.extended_positive(z)) is None
    assert ask(Q.finite(a), Q.positive(x) & Q.positive(y)
        & Q.positive(z)) is True
    assert ask(Q.finite(a), Q.positive(x) & Q.positive(y)
        & Q.negative_infinite(z)) is False
    assert ask(Q.finite(a), Q.positive(x) & Q.positive(y)
        & ~Q.finite(z)) is False
    assert ask(Q.finite(a), Q.positive(x) & Q.positive(y)
        & Q.positive_infinite(z)) is False
    assert ask(Q.finite(a), Q.positive(x) & Q.positive(y)
        & Q.extended_negative(z)) is None
    assert ask(Q.finite(a), Q.positive(x) & Q.positive(y)) is None
    assert ask(Q.finite(a), Q.positive(x) & Q.positive(y)
        & Q.extended_positive(z)) is None
    assert ask(Q.finite(a), Q.positive(x) & Q.negative_infinite(y)
        & Q.negative_infinite(z)) is False
    assert ask(Q.finite(a), Q.positive(x) & Q.negative_infinite(y)
        & ~Q.finite(z)) is None
    assert ask(Q.finite(a), Q.positive(x) & Q.negative_infinite(y)
        & Q.positive_infinite(z)) is None
    assert ask(Q.finite(a), Q.positive(x) & Q.negative_infinite(y)
        & Q.extended_negative(z)) is False
    assert ask(Q.finite(a), Q.positive(x)
        & Q.negative_infinite(y)) is None
    assert ask(Q.finite(a), Q.positive(x) & Q.negative_infinite(y)
        & Q.extended_positive(z)) is None
    assert ask(Q.finite(a), Q.positive(x) & ~Q.finite(y)
        & ~Q.finite(z)) is None
    assert ask(Q.finite(a), Q.positive(x) & ~Q.finite(y)
        & Q.positive_infinite(z)) is None
    assert ask(Q.finite(a), Q.positive(x) & ~Q.finite(y)
        & Q.extended_negative(z)) is None
    assert ask(Q.finite(a), Q.positive(x) & ~Q.finite(y)) is None
    assert ask(Q.finite(a), Q.positive(x) & ~Q.finite(y)
        & Q.extended_positive(z)) is None
    assert ask(Q.finite(a), Q.positive(x) & Q.positive_infinite(y)
        & Q.positive_infinite(z)) is False
    assert ask(Q.finite(a), Q.positive(x) & Q.positive_infinite(y)
        & Q.extended_negative(z)) is None
    assert ask(Q.finite(a), Q.positive(x)
        & Q.positive_infinite(y)) is None
    assert ask(Q.finite(a), Q.positive(x) & Q.positive_infinite(y)
        & Q.extended_positive(z)) is False
    assert ask(Q.finite(a), Q.positive(x) & Q.extended_negative(y)
        & Q.extended_negative(z)) is None
    assert ask(Q.finite(a), Q.positive(x)
        & Q.extended_negative(y)) is None
    assert ask(Q.finite(a), Q.positive(x) & Q.extended_negative(y)
        & Q.extended_positive(z)) is None
    assert ask(Q.finite(a), Q.positive(x)) is None
    assert ask(Q.finite(a), Q.positive(x)
        & Q.extended_positive(z)) is None
    assert ask(Q.finite(a), Q.positive(x) & Q.extended_positive(y)
        & Q.extended_positive(z)) is None
    assert ask(Q.finite(a), Q.negative_infinite(x)
        & Q.negative_infinite(y) & Q.negative_infinite(z)) is False
    assert ask(Q.finite(a), Q.negative_infinite(x)
        & Q.negative_infinite(y) & ~Q.finite(z)) is None
    assert ask(Q.finite(a), Q.negative_infinite(x)
        & Q.negative_infinite(y)& Q.positive_infinite(z)) is None
    assert ask(Q.finite(a), Q.negative_infinite(x)
        & Q.negative_infinite(y) & Q.extended_negative(z)) is False
    assert ask(Q.finite(a), Q.negative_infinite(x)
        & Q.negative_infinite(y)) is None
    assert ask(Q.finite(a), Q.negative_infinite(x)
        & Q.negative_infinite(y) & Q.extended_positive(z)) is None
    assert ask(Q.finite(a), Q.negative_infinite(x)
        & ~Q.finite(y) & ~Q.finite(z)) is None
    assert ask(Q.finite(a), Q.negative_infinite(x)
        & ~Q.finite(y) & Q.positive_infinite(z)) is None
    assert ask(Q.finite(a), Q.negative_infinite(x)
        & ~Q.finite(y) & Q.extended_negative(z)) is None
    assert ask(Q.finite(a), Q.negative_infinite(x)
        & ~Q.finite(y)) is None
    assert ask(Q.finite(a), Q.negative_infinite(x)
        & ~Q.finite(y) & Q.extended_positive(z)) is None
    assert ask(Q.finite(a), Q.negative_infinite(x)
        & Q.positive_infinite(y) & Q.positive_infinite(z)) is None
    assert ask(Q.finite(a), Q.negative_infinite(x)
        & Q.positive_infinite(y) & Q.extended_negative(z)) is None
    assert ask(Q.finite(a), Q.negative_infinite(x)
        & Q.positive_infinite(y)) is None
    assert ask(Q.finite(a), Q.negative_infinite(x)
        & Q.positive_infinite(y) & Q.extended_positive(z)) is None
    assert ask(Q.finite(a), Q.negative_infinite(x)
        & Q.extended_negative(y) & Q.extended_negative(z)) is False
    assert ask(Q.finite(a), Q.negative_infinite(x)
        & Q.extended_negative(y)) is None
    assert ask(Q.finite(a), Q.negative_infinite(x)
        & Q.extended_negative(y) & Q.extended_positive(z)) is None
    assert ask(Q.finite(a), Q.negative_infinite(x)) is None
    assert ask(Q.finite(a), Q.negative_infinite(x)
        & Q.extended_positive(z)) is None
    assert ask(Q.finite(a), Q.negative_infinite(x)
        & Q.extended_positive(y) & Q.extended_positive(z)) is None
    assert ask(Q.finite(a), ~Q.finite(x) & ~Q.finite(y)
        & ~Q.finite(z)) is None
    assert ask(Q.finite(a), ~Q.finite(x) & Q.positive_infinite(z)
        & ~Q.finite(z)) is None
    assert ask(Q.finite(a), ~Q.finite(x) & ~Q.finite(y)
        & Q.extended_negative(z)) is None
    assert ask(Q.finite(a), ~Q.finite(x) & ~Q.finite(y)) is None
    assert ask(Q.finite(a), ~Q.finite(x) & ~Q.finite(y)
        & Q.extended_positive(z)) is None
    assert ask(Q.finite(a), ~Q.finite(x) & Q.positive_infinite(y)
        & Q.positive_infinite(z)) is None
    assert ask(Q.finite(a), ~Q.finite(x) & Q.positive_infinite(y)
        & Q.extended_negative(z)) is None
    assert ask(Q.finite(a), ~Q.finite(x)
        & Q.positive_infinite(y)) is None
    assert ask(Q.finite(a), ~Q.finite(x) & Q.positive_infinite(y)
        & Q.extended_positive(z)) is None
    assert ask(Q.finite(a), ~Q.finite(x) & Q.extended_negative(y)
        & Q.extended_negative(z)) is None
    assert ask(Q.finite(a), ~Q.finite(x)
        & Q.extended_negative(y)) is None
    assert ask(Q.finite(a), ~Q.finite(x) & Q.extended_negative(y)
        & Q.extended_positive(z)) is None
    assert ask(Q.finite(a), ~Q.finite(x)) is None
    assert ask(Q.finite(a), ~Q.finite(x)
        & Q.extended_positive(z)) is None
    assert ask(Q.finite(a), ~Q.finite(x) & Q.extended_positive(y)
        & Q.extended_positive(z)) is None
    assert ask(Q.finite(a), Q.positive_infinite(x)
        & Q.positive_infinite(y) & Q.positive_infinite(z)) is False
    assert ask(Q.finite(a), Q.positive_infinite(x)
        & Q.positive_infinite(y) & Q.extended_negative(z)) is None
    assert ask(Q.finite(a), Q.positive_infinite(x)
        & Q.positive_infinite(y)) is None
    assert ask(Q.finite(a), Q.positive_infinite(x)
        & Q.positive_infinite(y) & Q.extended_positive(z)) is False
    assert ask(Q.finite(a), Q.positive_infinite(x)
        & Q.extended_negative(y) & Q.extended_negative(z)) is None
    assert ask(Q.finite(a), Q.positive_infinite(x)
        & Q.extended_negative(y)) is None
    assert ask(Q.finite(a), Q.positive_infinite(x)
        & Q.extended_negative(y) & Q.extended_positive(z)) is None
    assert ask(Q.finite(a), Q.positive_infinite(x)) is None
    assert ask(Q.finite(a), Q.positive_infinite(x)
        & Q.extended_positive(z)) is None
    assert ask(Q.finite(a), Q.positive_infinite(x)
        & Q.extended_positive(y) & Q.extended_positive(z)) is False
    assert ask(Q.finite(a), Q.extended_negative(x)
        & Q.extended_negative(y) & Q.extended_negative(z)) is None
    assert ask(Q.finite(a), Q.extended_negative(x)
        & Q.extended_negative(y)) is None
    assert ask(Q.finite(a), Q.extended_negative(x)
        & Q.extended_negative(y) & Q.extended_positive(z)) is None
    assert ask(Q.finite(a), Q.extended_negative(x)) is None
    assert ask(Q.finite(a), Q.extended_negative(x)
        & Q.extended_positive(z)) is None
    assert ask(Q.finite(a), Q.extended_negative(x)
        & Q.extended_positive(y) & Q.extended_positive(z)) is None
    assert ask(Q.finite(a)) is None
    assert ask(Q.finite(a), Q.extended_positive(z)) is None
    assert ask(Q.finite(a), Q.extended_positive(y)
        & Q.extended_positive(z)) is None
    assert ask(Q.finite(a), Q.extended_positive(x)
        & Q.extended_positive(y) & Q.extended_positive(z)) is None

    assert ask(Q.finite(2*x)) is None
    assert ask(Q.finite(2*x), Q.finite(x)) is True

    x, y, z = symbols('x,y,z')
    a = x*y
    x, y = a.args
    assert ask(Q.finite(a), Q.finite(x) & Q.finite(y)) is True
    assert ask(Q.finite(a), Q.finite(x) & ~Q.zero(x) & ~Q.finite(y)) is False
    assert ask(Q.finite(a), Q.finite(x)) is None
    assert ask(Q.finite(a), ~Q.finite(x) & Q.finite(y) &~Q.zero(y)) is False
    assert ask(Q.finite(a), ~Q.finite(x) & ~Q.finite(y)) is False
    assert ask(Q.finite(a), ~Q.finite(x)) is None
    assert ask(Q.finite(a), Q.finite(y)) is None
    assert ask(Q.finite(a), ~Q.finite(y)) is None
    assert ask(Q.finite(a)) is None
    a = x*y*z
    x, y, z = a.args
    assert ask(Q.finite(a), Q.finite(x) & Q.finite(y)
        & Q.finite(z)) is True
    assert ask(Q.finite(a), Q.finite(x) & ~Q.zero(x) & Q.finite(y)
        & ~Q.zero(y) & ~Q.finite(z)) is False
    assert ask(Q.finite(a), Q.finite(x) & Q.finite(y)) is None
    assert ask(Q.finite(a), Q.finite(x) & ~Q.zero(x) & ~Q.finite(y)
        & Q.finite(z) & ~Q.zero(z)) is False
    assert ask(Q.finite(a), Q.finite(x) & ~Q.zero(x) & ~Q.finite(y)
        & ~Q.finite(z)) is False
    assert ask(Q.finite(a), Q.finite(x) & ~Q.finite(y)) is None
    assert ask(Q.finite(a), Q.finite(x) & Q.finite(z)) is None
    assert ask(Q.finite(a), Q.finite(x) & ~Q.finite(z)) is None
    assert ask(Q.finite(a), Q.finite(x)) is None
    assert ask(Q.finite(a), ~Q.finite(x) & Q.finite(y) & ~Q.zero(y)
        & Q.finite(z) & ~Q.zero(z)) is False
    assert ask(Q.finite(a), ~Q.finite(x) & ~Q.zero(x) & Q.finite(y)
        & ~Q.zero(y) & ~Q.finite(z)) is False
    assert ask(Q.finite(a), ~Q.finite(x) & Q.finite(y)) is None
    assert ask(Q.finite(a), ~Q.finite(x) & ~Q.finite(y)
        & Q.finite(z) & ~Q.zero(z)) is False
    assert ask(Q.finite(a), ~Q.finite(x) & ~Q.finite(y)
        & ~Q.finite(z)) is False
    assert ask(Q.finite(a), ~Q.finite(x) & ~Q.finite(y)) is None
    assert ask(Q.finite(a), ~Q.finite(x) & Q.finite(z)) is None
    assert ask(Q.finite(a), ~Q.finite(x) & ~Q.finite(z)) is None
    assert ask(Q.finite(a), ~Q.finite(x)) is None
    assert ask(Q.finite(a), Q.finite(y) & Q.finite(z)) is None
    assert ask(Q.finite(a), Q.finite(y) & ~Q.finite(z)) is None
    assert ask(Q.finite(a), Q.finite(y)) is None
    assert ask(Q.finite(a), ~Q.finite(y) & Q.finite(z)) is None
    assert ask(Q.finite(a), ~Q.finite(y) & ~Q.finite(z)) is None
    assert ask(Q.finite(a), ~Q.finite(y)) is None
    assert ask(Q.finite(a), Q.finite(z)) is None
    assert ask(Q.finite(a), ~Q.finite(z)) is None
    assert ask(Q.finite(a), ~Q.finite(z) & Q.extended_nonzero(x)
        & Q.extended_nonzero(y) & Q.extended_nonzero(z)) is None
    assert ask(Q.finite(a), Q.extended_nonzero(x) & ~Q.finite(y)
        & Q.extended_nonzero(y) & ~Q.finite(z)
        & Q.extended_nonzero(z)) is False

    x, y, z = symbols('x,y,z')
    assert ask(Q.finite(x**2)) is None
    assert ask(Q.finite(2**x)) is None
    assert ask(Q.finite(2**x), Q.finite(x)) is True
    assert ask(Q.finite(x**x)) is None
    assert ask(Q.finite(S.Half ** x)) is None
    assert ask(Q.finite(S.Half ** x), Q.extended_positive(x)) is True
    assert ask(Q.finite(S.Half ** x), Q.extended_negative(x)) is None
    assert ask(Q.finite(2**x), Q.extended_negative(x)) is True
    assert ask(Q.finite(sqrt(x))) is None
    assert ask(Q.finite(2**x), ~Q.finite(x)) is False
    assert ask(Q.finite(x**2), ~Q.finite(x)) is False

    # https://github.com/sympy/sympy/issues/27707
    assert ask(Q.finite(x**y), Q.real(x) & Q.real(y)) is None
    assert ask(Q.finite(x**y), Q.real(x) & Q.negative(y)) is None
    assert ask(Q.finite(x**y), Q.zero(x) & Q.negative(y)) is False
    assert ask(Q.finite(x**y), Q.real(x) & Q.positive(y)) is True
    assert ask(Q.finite(x**y), Q.nonzero(x) & Q.real(y)) is True
    assert ask(Q.finite(x**y), Q.nonzero(x) & Q.negative(y)) is True
    assert ask(Q.finite(x**y), Q.zero(x) & Q.positive(y)) is True

    # sign function
    assert ask(Q.finite(sign(x))) is True
    assert ask(Q.finite(sign(x)), ~Q.finite(x)) is True

    # exponential functions
    assert ask(Q.finite(log(x))) is None
    assert ask(Q.finite(log(x)), Q.finite(x)) is None
    assert ask(Q.finite(log(x)), ~Q.zero(x)) is True
    assert ask(Q.finite(log(x)), Q.infinite(x)) is False
    assert ask(Q.finite(log(x)), Q.zero(x)) is False
    assert ask(Q.finite(exp(x))) is None
    assert ask(Q.finite(exp(x)), Q.finite(x)) is True
    assert ask(Q.finite(exp(2))) is True

    # trigonometric functions
    assert ask(Q.finite(sin(x))) is True
    assert ask(Q.finite(sin(x)), ~Q.finite(x)) is True
    assert ask(Q.finite(cos(x))) is True
    assert ask(Q.finite(cos(x)), ~Q.finite(x)) is True
    assert ask(Q.finite(2*sin(x))) is True
    assert ask(Q.finite(sin(x)**2)) is True
    assert ask(Q.finite(cos(x)**2)) is True
    assert ask(Q.finite(cos(x) + sin(x))) is True


def test_unbounded():
    assert ask(Q.infinite(I * oo)) is True
    assert ask(Q.infinite(1 + I*oo)) is True
    assert ask(Q.infinite(3 * (I * oo))) is True
    assert ask(Q.infinite(-I * oo)) is True
    assert ask(Q.infinite(1 + zoo)) is True
    assert ask(Q.infinite(I * zoo)) is True
    assert ask(Q.infinite(x / y), Q.infinite(x) & Q.finite(y) & ~Q.zero(y)) is True
    assert ask(Q.infinite(I * oo - I * oo)) is None
    assert ask(Q.infinite(x * I * oo)) is None
    assert ask(Q.infinite(1 / x), Q.finite(x) & ~Q.zero(x)) is False
    assert ask(Q.infinite(1 / (I * oo))) is False


def test_issue_27441():
    # https://github.com/sympy/sympy/issues/27441
    assert ask(Q.composite(y), Q.integer(y) & Q.positive(y) & ~Q.prime(y)) is None


def test_issue_27447():
    x,y,z = symbols('x y z')
    a = x*y
    assert ask(Q.finite(a), Q.finite(x)  & ~Q.finite(y)) is None
    assert ask(Q.finite(a), ~Q.finite(x)  & Q.finite(y)) is None

    a = x*y*z
    assert ask(Q.finite(a), Q.finite(x) & Q.finite(y)
        & ~Q.finite(z)) is None
    assert ask(Q.finite(a), Q.finite(x) & ~Q.finite(y)
        & Q.finite(z) ) is None
    assert ask(Q.finite(a), Q.finite(x) & ~Q.finite(y)
        & ~Q.finite(z)) is None
    assert ask(Q.finite(a), ~Q.finite(x) & Q.finite(y)
        & Q.finite(z)) is None
    assert ask(Q.finite(a), ~Q.finite(x) & Q.finite(y)
        & ~Q.finite(z)) is None
    assert ask(Q.finite(a), ~Q.finite(x) & ~Q.finite(y)
        & Q.finite(z)) is None


@XFAIL
def test_issue_27662_xfail():
    assert ask(Q.finite(x*y), ~Q.finite(x)
        & Q.zero(y)) is None


@XFAIL
def test_bounded_xfail():
    """We need to support relations in ask for this to work"""
    assert ask(Q.finite(sin(x)**x)) is True
    assert ask(Q.finite(cos(x)**x)) is True


def test_commutative():
    """By default objects are Q.commutative that is why it returns True
    for both key=True and key=False"""
    assert ask(Q.commutative(x)) is True
    assert ask(Q.commutative(x), ~Q.commutative(x)) is False
    assert ask(Q.commutative(x), Q.complex(x)) is True
    assert ask(Q.commutative(x), Q.imaginary(x)) is True
    assert ask(Q.commutative(x), Q.real(x)) is True
    assert ask(Q.commutative(x), Q.positive(x)) is True
    assert ask(Q.commutative(x), ~Q.commutative(y)) is True

    assert ask(Q.commutative(2*x)) is True
    assert ask(Q.commutative(2*x), ~Q.commutative(x)) is False

    assert ask(Q.commutative(x + 1)) is True
    assert ask(Q.commutative(x + 1), ~Q.commutative(x)) is False

    assert ask(Q.commutative(x**2)) is True
    assert ask(Q.commutative(x**2), ~Q.commutative(x)) is False

    assert ask(Q.commutative(log(x))) is True


@_both_exp_pow
def test_complex():
    assert ask(Q.complex(x)) is None
    assert ask(Q.complex(x), Q.complex(x)) is True
    assert ask(Q.complex(x), Q.complex(y)) is None
    assert ask(Q.complex(x), ~Q.complex(x)) is False
    assert ask(Q.complex(x), Q.real(x)) is True
    assert ask(Q.complex(x), ~Q.real(x)) is None
    assert ask(Q.complex(x), Q.rational(x)) is True
    assert ask(Q.complex(x), Q.irrational(x)) is True
    assert ask(Q.complex(x), Q.positive(x)) is True
    assert ask(Q.complex(x), Q.imaginary(x)) is True
    assert ask(Q.complex(x), Q.algebraic(x)) is True

    # a+b
    assert ask(Q.complex(x + 1), Q.complex(x)) is True
    assert ask(Q.complex(x + 1), Q.real(x)) is True
    assert ask(Q.complex(x + 1), Q.rational(x)) is True
    assert ask(Q.complex(x + 1), Q.irrational(x)) is True
    assert ask(Q.complex(x + 1), Q.imaginary(x)) is True
    assert ask(Q.complex(x + 1), Q.integer(x)) is True
    assert ask(Q.complex(x + 1), Q.even(x)) is True
    assert ask(Q.complex(x + 1), Q.odd(x)) is True
    assert ask(Q.complex(x + y), Q.complex(x) & Q.complex(y)) is True
    assert ask(Q.complex(x + y), Q.real(x) & Q.imaginary(y)) is True

    # a*x +b
    assert ask(Q.complex(2*x + 1), Q.complex(x)) is True
    assert ask(Q.complex(2*x + 1), Q.real(x)) is True
    assert ask(Q.complex(2*x + 1), Q.positive(x)) is True
    assert ask(Q.complex(2*x + 1), Q.rational(x)) is True
    assert ask(Q.complex(2*x + 1), Q.irrational(x)) is True
    assert ask(Q.complex(2*x + 1), Q.imaginary(x)) is True
    assert ask(Q.complex(2*x + 1), Q.integer(x)) is True
    assert ask(Q.complex(2*x + 1), Q.even(x)) is True
    assert ask(Q.complex(2*x + 1), Q.odd(x)) is True

    # x**2
    assert ask(Q.complex(x**2), Q.complex(x)) is True
    assert ask(Q.complex(x**2), Q.real(x)) is True
    assert ask(Q.complex(x**2), Q.positive(x)) is True
    assert ask(Q.complex(x**2), Q.rational(x)) is True
    assert ask(Q.complex(x**2), Q.irrational(x)) is True
    assert ask(Q.complex(x**2), Q.imaginary(x)) is True
    assert ask(Q.complex(x**2), Q.integer(x)) is True
    assert ask(Q.complex(x**2), Q.even(x)) is True
    assert ask(Q.complex(x**2), Q.odd(x)) is True

    # 2**x
    assert ask(Q.complex(2**x), Q.complex(x)) is True
    assert ask(Q.complex(2**x), Q.real(x)) is True
    assert ask(Q.complex(2**x), Q.positive(x)) is True
    assert ask(Q.complex(2**x), Q.rational(x)) is True
    assert ask(Q.complex(2**x), Q.irrational(x)) is True
    assert ask(Q.complex(2**x), Q.imaginary(x)) is True
    assert ask(Q.complex(2**x), Q.integer(x)) is True
    assert ask(Q.complex(2**x), Q.even(x)) is True
    assert ask(Q.complex(2**x), Q.odd(x)) is True
    assert ask(Q.complex(x**y), Q.complex(x) & Q.complex(y)) is True

    # trigonometric expressions
    assert ask(Q.complex(sin(x))) is True
    assert ask(Q.complex(sin(2*x + 1))) is True
    assert ask(Q.complex(cos(x))) is True
    assert ask(Q.complex(cos(2*x + 1))) is True

    # exponential
    assert ask(Q.complex(exp(x))) is True
    assert ask(Q.complex(exp(x))) is True

    # Q.complexes
    assert ask(Q.complex(Abs(x))) is True
    assert ask(Q.complex(re(x))) is True
    assert ask(Q.complex(im(x))) is True


def test_even_query():
    assert ask(Q.even(x)) is None
    assert ask(Q.even(x), Q.integer(x)) is None
    assert ask(Q.even(x), ~Q.integer(x)) is False
    assert ask(Q.even(x), Q.rational(x)) is None
    assert ask(Q.even(x), Q.positive(x)) is None

    assert ask(Q.even(2*x)) is None
    assert ask(Q.even(2*x), Q.integer(x)) is True
    assert ask(Q.even(2*x), Q.even(x)) is True
    assert ask(Q.even(2*x), Q.irrational(x)) is False
    assert ask(Q.even(2*x), Q.odd(x)) is True
    assert ask(Q.even(2*x), ~Q.integer(x)) is None
    assert ask(Q.even(3*x), Q.integer(x)) is None
    assert ask(Q.even(3*x), Q.even(x)) is True
    assert ask(Q.even(3*x), Q.odd(x)) is False

    assert ask(Q.even(x + 1), Q.odd(x)) is True
    assert ask(Q.even(x + 1), Q.even(x)) is False
    assert ask(Q.even(x + 2), Q.odd(x)) is False
    assert ask(Q.even(x + 2), Q.even(x)) is True
    assert ask(Q.even(7 - x), Q.odd(x)) is True
    assert ask(Q.even(7 + x), Q.odd(x)) is True
    assert ask(Q.even(x + y), Q.odd(x) & Q.odd(y)) is True
    assert ask(Q.even(x + y), Q.odd(x) & Q.even(y)) is False
    assert ask(Q.even(x + y), Q.even(x) & Q.even(y)) is True

    assert ask(Q.even(2*x + 1), Q.integer(x)) is False
    assert ask(Q.even(2*x*y), Q.rational(x) & Q.rational(x)) is None
    assert ask(Q.even(2*x*y), Q.irrational(x) & Q.irrational(x)) is None

    assert ask(Q.even(x + y + z), Q.odd(x) & Q.odd(y) & Q.even(z)) is True
    assert ask(Q.even(x + y + z + t),
        Q.odd(x) & Q.odd(y) & Q.even(z) & Q.integer(t)) is None

    assert ask(Q.even(Abs(x)), Q.even(x)) is True
    assert ask(Q.even(Abs(x)), ~Q.even(x)) is None
    assert ask(Q.even(re(x)), Q.even(x)) is True
    assert ask(Q.even(re(x)), ~Q.even(x)) is None
    assert ask(Q.even(im(x)), Q.even(x)) is True
    assert ask(Q.even(im(x)), Q.real(x)) is True

    assert ask(Q.even((-1)**n), Q.integer(n)) is False

    assert ask(Q.even(k**2), Q.even(k)) is True
    assert ask(Q.even(n**2), Q.odd(n)) is False
    assert ask(Q.even(2**k), Q.even(k)) is None
    assert ask(Q.even(x**2)) is None

    assert ask(Q.even(k**m), Q.even(k) & Q.integer(m) & ~Q.negative(m)) is None
    assert ask(Q.even(n**m), Q.odd(n) & Q.integer(m) & ~Q.negative(m)) is False

    assert ask(Q.even(k**p), Q.even(k) & Q.integer(p) & Q.positive(p)) is True
    assert ask(Q.even(n**p), Q.odd(n) & Q.integer(p) & Q.positive(p)) is False

    assert ask(Q.even(m**k), Q.even(k) & Q.integer(m) & ~Q.negative(m)) is None
    assert ask(Q.even(p**k), Q.even(k) & Q.integer(p) & Q.positive(p)) is None

    assert ask(Q.even(m**n), Q.odd(n) & Q.integer(m) & ~Q.negative(m)) is None
    assert ask(Q.even(p**n), Q.odd(n) & Q.integer(p) & Q.positive(p)) is None

    assert ask(Q.even(k**x), Q.even(k)) is None
    assert ask(Q.even(n**x), Q.odd(n)) is None

    assert ask(Q.even(x*y), Q.integer(x) & Q.integer(y)) is None
    assert ask(Q.even(x*x), Q.integer(x)) is None
    assert ask(Q.even(x*(x + y)), Q.integer(x) & Q.odd(y)) is True
    assert ask(Q.even(x*(x + y)), Q.integer(x) & Q.even(y)) is None


@XFAIL
def test_evenness_in_ternary_integer_product_with_odd():
    # Tests that oddness inference is independent of term ordering.
    # Term ordering at the point of testing depends on SymPy's symbol order, so
    # we try to force a different order by modifying symbol names.
    assert ask(Q.even(x*y*(y + z)), Q.integer(x) & Q.integer(y) & Q.odd(z)) is True
    assert ask(Q.even(y*x*(x + z)), Q.integer(x) & Q.integer(y) & Q.odd(z)) is True


def test_evenness_in_ternary_integer_product_with_even():
    assert ask(Q.even(x*y*(y + z)), Q.integer(x) & Q.integer(y) & Q.even(z)) is None


def test_extended_real():
    assert ask(Q.extended_real(x), Q.positive_infinite(x)) is True
    assert ask(Q.extended_real(x), Q.positive(x)) is True
    assert ask(Q.extended_real(x), Q.zero(x)) is True
    assert ask(Q.extended_real(x), Q.negative(x)) is True
    assert ask(Q.extended_real(x), Q.negative_infinite(x)) is True

    assert ask(Q.extended_real(-x), Q.positive(x)) is True
    assert ask(Q.extended_real(-x), Q.negative(x)) is True

    assert ask(Q.extended_real(x + S.Infinity), Q.real(x)) is True

    assert ask(Q.extended_real(x), Q.infinite(x)) is None


@_both_exp_pow
def test_rational():
    assert ask(Q.rational(x), Q.integer(x)) is True
    assert ask(Q.rational(x), Q.irrational(x)) is False
    assert ask(Q.rational(x), Q.real(x)) is None
    assert ask(Q.rational(x), Q.positive(x)) is None
    assert ask(Q.rational(x), Q.negative(x)) is None
    assert ask(Q.rational(x), Q.nonzero(x)) is None
    assert ask(Q.rational(x), ~Q.algebraic(x)) is False

    assert ask(Q.rational(2*x), Q.rational(x)) is True
    assert ask(Q.rational(2*x), Q.integer(x)) is True
    assert ask(Q.rational(2*x), Q.even(x)) is True
    assert ask(Q.rational(2*x), Q.odd(x)) is True
    assert ask(Q.rational(2*x), Q.irrational(x)) is False

    assert ask(Q.rational(x/2), Q.rational(x)) is True
    assert ask(Q.rational(x/2), Q.integer(x)) is True
    assert ask(Q.rational(x/2), Q.even(x)) is True
    assert ask(Q.rational(x/2), Q.odd(x)) is True
    assert ask(Q.rational(x/2), Q.irrational(x)) is False

    assert ask(Q.rational(1/x), Q.rational(x) & Q.nonzero(x)) is True
    assert ask(Q.rational(1/x), Q.integer(x) & Q.nonzero(x)) is True
    assert ask(Q.rational(1/x), Q.even(x) & Q.nonzero(x)) is True
    assert ask(Q.rational(1/x), Q.odd(x)) is True
    assert ask(Q.rational(1/x), Q.irrational(x)) is False

    assert ask(Q.rational(2/x), Q.rational(x) & Q.nonzero(x)) is True
    assert ask(Q.rational(2/x), Q.integer(x) & Q.nonzero(x)) is True
    assert ask(Q.rational(2/x), Q.even(x) & Q.nonzero(x)) is True
    assert ask(Q.rational(2/x), Q.odd(x)) is True
    assert ask(Q.rational(2/x), Q.irrational(x)) is False

    assert ask(Q.rational(x), ~Q.algebraic(x)) is False

    # with multiple symbols
    assert ask(Q.rational(x*y), Q.irrational(x) & Q.irrational(y)) is None
    assert ask(Q.rational(y/x), Q.rational(x) & Q.rational(y) & Q.nonzero(x)) is True
    assert ask(Q.rational(y/x), Q.integer(x) & Q.rational(y) & Q.nonzero(x)) is True
    assert ask(Q.rational(y/x), Q.even(x) & Q.rational(y) & Q.nonzero(x)) is True
    assert ask(Q.rational(y/x), Q.odd(x) & Q.rational(y)) is True
    assert ask(Q.rational(y/x), Q.irrational(x) & Q.rational(y) & Q.nonzero(y)) is False

    for f in [exp, sin, tan, asin, atan, cos]:
        assert ask(Q.rational(f(7))) is False
        assert ask(Q.rational(f(7, evaluate=False))) is False
        assert ask(Q.rational(f(0, evaluate=False))) is True
        assert ask(Q.rational(f(x)), Q.rational(x)) is None
        assert ask(Q.rational(f(x)), Q.rational(x) & Q.nonzero(x)) is False

    for g in [log, acos]:
        assert ask(Q.rational(g(7))) is False
        assert ask(Q.rational(g(7, evaluate=False))) is False
        assert ask(Q.rational(g(1, evaluate=False))) is True
        assert ask(Q.rational(g(x)), Q.rational(x)) is None
        assert ask(Q.rational(g(x)), Q.rational(x) & Q.nonzero(x - 1)) is False

    for h in [cot, acot]:
        assert ask(Q.rational(h(7))) is False
        assert ask(Q.rational(h(7, evaluate=False))) is False
        assert ask(Q.rational(h(x)), Q.rational(x)) is False

    # https://github.com/sympy/sympy/issues/27442
    assert ask(Q.rational(x**y),Q.irrational(x) & Q.rational(y)) is None
    assert ask(Q.rational(x**y),Q.integer(x) & Q.prime(x) & Q.rational(y)) is None
    assert ask(Q.rational(x**y),Q.integer(x) & Q.integer(y)) is None
    assert ask(Q.rational(x**y),Q.integer(x) & Q.eq(x,0) & Q.integer(y)) is None
    assert ask(Q.rational(x**y),Q.eq(x,1) & Q.rational(y)) is None
    assert ask(Q.rational(x**y),Q.eq(x,-1) & Q.rational(y)) is None
    assert ask(Q.rational(x**y), Q.prime(x) & Q.rational(y)) is None
    assert ask(Q.rational(x**y), ~Q.rational(x) & Q.integer(y) ) is None
    assert ask(Q.rational(Pow(-1, x, evaluate=False), Q.rational(x))) is None
    assert ask(Q.rational(x**y), Q.integer(y) & ~Q. algebraic(x)) is None
    assert ask(Q.rational(x**y), Q.integer(y) & ~Q. algebraic(x) & ~Q.zero(x)) is None
    assert ask(Q.rational(x**y), Q.integer(y) & ~Q.algebraic(x) & Q.complex(x) & ~Q.real(x)) is None
    assert ask(Q.rational(x**y), Q.integer(y) & ~Q.algebraic(x) & Q.complex(x)) is None


def test_hermitian():
    assert ask(Q.hermitian(x)) is None
    assert ask(Q.hermitian(x), Q.antihermitian(x)) is None
    assert ask(Q.hermitian(x), Q.imaginary(x)) is False
    assert ask(Q.hermitian(x), Q.prime(x)) is True
    assert ask(Q.hermitian(x), Q.real(x)) is True
    assert ask(Q.hermitian(x), Q.zero(x)) is True

    assert ask(Q.hermitian(x + 1), Q.antihermitian(x)) is None
    assert ask(Q.hermitian(x + 1), Q.complex(x)) is None
    assert ask(Q.hermitian(x + 1), Q.hermitian(x)) is True
    assert ask(Q.hermitian(x + 1), Q.imaginary(x)) is False
    assert ask(Q.hermitian(x + 1), Q.real(x)) is True
    assert ask(Q.hermitian(x + I), Q.antihermitian(x)) is None
    assert ask(Q.hermitian(x + I), Q.complex(x)) is None
    assert ask(Q.hermitian(x + I), Q.hermitian(x)) is False
    assert ask(Q.hermitian(x + I), Q.imaginary(x)) is None
    assert ask(Q.hermitian(x + I), Q.real(x)) is False
    assert ask(
        Q.hermitian(x + y), Q.antihermitian(x) & Q.antihermitian(y)) is None
    assert ask(Q.hermitian(x + y), Q.antihermitian(x) & Q.complex(y)) is None
    assert ask(
        Q.hermitian(x + y), Q.antihermitian(x) & Q.hermitian(y)) is None
    assert ask(Q.hermitian(x + y), Q.antihermitian(x) & Q.imaginary(y)) is None
    assert ask(Q.hermitian(x + y), Q.antihermitian(x) & Q.real(y)) is None
    assert ask(Q.hermitian(x + y), Q.hermitian(x) & Q.complex(y)) is None
    assert ask(Q.hermitian(x + y), Q.hermitian(x) & Q.hermitian(y)) is True
    assert ask(Q.hermitian(x + y), Q.hermitian(x) & Q.imaginary(y)) is False
    assert ask(Q.hermitian(x + y), Q.hermitian(x) & Q.real(y)) is True
    assert ask(Q.hermitian(x + y), Q.imaginary(x) & Q.complex(y)) is None
    assert ask(Q.hermitian(x + y), Q.imaginary(x) & Q.imaginary(y)) is None
    assert ask(Q.hermitian(x + y), Q.imaginary(x) & Q.real(y)) is False
    assert ask(Q.hermitian(x + y), Q.real(x) & Q.complex(y)) is None
    assert ask(Q.hermitian(x + y), Q.real(x) & Q.real(y)) is True

    assert ask(Q.hermitian(I*x), Q.antihermitian(x)) is True
    assert ask(Q.hermitian(I*x), Q.complex(x)) is None
    assert ask(Q.hermitian(I*x), Q.hermitian(x)) is False
    assert ask(Q.hermitian(I*x), Q.imaginary(x)) is True
    assert ask(Q.hermitian(I*x), Q.real(x)) is False
    assert ask(Q.hermitian(x*y), Q.hermitian(x) & Q.real(y)) is True

    assert ask(
        Q.hermitian(x + y + z), Q.real(x) & Q.real(y) & Q.real(z)) is True
    assert ask(Q.hermitian(x + y + z),
        Q.real(x) & Q.real(y) & Q.imaginary(z)) is False
    assert ask(Q.hermitian(x + y + z),
        Q.real(x) & Q.imaginary(y) & Q.imaginary(z)) is None
    assert ask(Q.hermitian(x + y + z),
        Q.imaginary(x) & Q.imaginary(y) & Q.imaginary(z)) is None

    assert ask(Q.antihermitian(x)) is None
    assert ask(Q.antihermitian(x), Q.real(x)) is False
    assert ask(Q.antihermitian(x), Q.prime(x)) is False

    assert ask(Q.antihermitian(x + 1), Q.antihermitian(x)) is False
    assert ask(Q.antihermitian(x + 1), Q.complex(x)) is None
    assert ask(Q.antihermitian(x + 1), Q.hermitian(x)) is None
    assert ask(Q.antihermitian(x + 1), Q.imaginary(x)) is False
    assert ask(Q.antihermitian(x + 1), Q.real(x)) is None
    assert ask(Q.antihermitian(x + I), Q.antihermitian(x)) is True
    assert ask(Q.antihermitian(x + I), Q.complex(x)) is None
    assert ask(Q.antihermitian(x + I), Q.hermitian(x)) is None
    assert ask(Q.antihermitian(x + I), Q.imaginary(x)) is True
    assert ask(Q.antihermitian(x + I), Q.real(x)) is False
    assert ask(Q.antihermitian(x), Q.zero(x)) is True

    assert ask(
        Q.antihermitian(x + y), Q.antihermitian(x) & Q.antihermitian(y)
    ) is True
    assert ask(
        Q.antihermitian(x + y), Q.antihermitian(x) & Q.complex(y)) is None
    assert ask(
        Q.antihermitian(x + y), Q.antihermitian(x) & Q.hermitian(y)) is None
    assert ask(
        Q.antihermitian(x + y), Q.antihermitian(x) & Q.imaginary(y)) is True
    assert ask(Q.antihermitian(x + y), Q.antihermitian(x) & Q.real(y)
        ) is False
    assert ask(Q.antihermitian(x + y), Q.hermitian(x) & Q.complex(y)) is None
    assert ask(Q.antihermitian(x + y), Q.hermitian(x) & Q.hermitian(y)
        ) is None
    assert ask(
        Q.antihermitian(x + y), Q.hermitian(x) & Q.imaginary(y)) is None
    assert ask(Q.antihermitian(x + y), Q.hermitian(x) & Q.real(y)) is None
    assert ask(Q.antihermitian(x + y), Q.imaginary(x) & Q.complex(y)) is None
    assert ask(Q.antihermitian(x + y), Q.imaginary(x) & Q.imaginary(y)) is True
    assert ask(Q.antihermitian(x + y), Q.imaginary(x) & Q.real(y)) is False
    assert ask(Q.antihermitian(x + y), Q.real(x) & Q.complex(y)) is None
    assert ask(Q.antihermitian(x + y), Q.real(x) & Q.real(y)) is None

    assert ask(Q.antihermitian(I*x), Q.real(x)) is True
    assert ask(Q.antihermitian(I*x), Q.antihermitian(x)) is False
    assert ask(Q.antihermitian(I*x), Q.complex(x)) is None
    assert ask(Q.antihermitian(x*y), Q.antihermitian(x) & Q.real(y)) is True

    assert ask(Q.antihermitian(x + y + z),
        Q.real(x) & Q.real(y) & Q.real(z)) is None
    assert ask(Q.antihermitian(x + y + z),
        Q.real(x) & Q.real(y) & Q.imaginary(z)) is None
    assert ask(Q.antihermitian(x + y + z),
        Q.real(x) & Q.imaginary(y) & Q.imaginary(z)) is False
    assert ask(Q.antihermitian(x + y + z),
        Q.imaginary(x) & Q.imaginary(y) & Q.imaginary(z)) is True


@_both_exp_pow
def test_imaginary():
    assert ask(Q.imaginary(x)) is None
    assert ask(Q.imaginary(x), Q.real(x)) is False
    assert ask(Q.imaginary(x), Q.prime(x)) is False

    assert ask(Q.imaginary(x + 1), Q.real(x)) is False
    assert ask(Q.imaginary(x + 1), Q.imaginary(x)) is False
    assert ask(Q.imaginary(x + I), Q.real(x)) is False
    assert ask(Q.imaginary(x + I), Q.imaginary(x)) is True
    assert ask(Q.imaginary(x + y), Q.imaginary(x) & Q.imaginary(y)) is True
    assert ask(Q.imaginary(x + y), Q.real(x) & Q.real(y)) is False
    assert ask(Q.imaginary(x + y), Q.imaginary(x) & Q.real(y)) is False
    assert ask(Q.imaginary(x + y), Q.complex(x) & Q.real(y)) is None
    assert ask(
        Q.imaginary(x + y + z), Q.real(x) & Q.real(y) & Q.real(z)) is False
    assert ask(Q.imaginary(x + y + z),
        Q.real(x) & Q.real(y) & Q.imaginary(z)) is None
    assert ask(Q.imaginary(x + y + z),
        Q.real(x) & Q.imaginary(y) & Q.imaginary(z)) is False

    assert ask(Q.imaginary(I*x), Q.real(x)) is True
    assert ask(Q.imaginary(I*x), Q.imaginary(x)) is False
    assert ask(Q.imaginary(I*x), Q.complex(x)) is None
    assert ask(Q.imaginary(x*y), Q.imaginary(x) & Q.real(y)) is True
    assert ask(Q.imaginary(x*y), Q.real(x) & Q.real(y)) is False

    assert ask(Q.imaginary(I**x), Q.negative(x)) is None
    assert ask(Q.imaginary(I**x), Q.positive(x)) is None
    assert ask(Q.imaginary(I**x), Q.even(x)) is False
    assert ask(Q.imaginary(I**x), Q.odd(x)) is True
    assert ask(Q.imaginary(I**x), Q.imaginary(x)) is False
    assert ask(Q.imaginary((2*I)**x), Q.imaginary(x)) is False
    assert ask(Q.imaginary(x**0), Q.imaginary(x)) is False
    assert ask(Q.imaginary(x**y), Q.imaginary(x) & Q.imaginary(y)) is None
    assert ask(Q.imaginary(x**y), Q.imaginary(x) & Q.real(y)) is None
    assert ask(Q.imaginary(x**y), Q.real(x) & Q.imaginary(y)) is None
    assert ask(Q.imaginary(x**y), Q.real(x) & Q.real(y)) is None
    assert ask(Q.imaginary(x**y), Q.imaginary(x) & Q.integer(y)) is None
    assert ask(Q.imaginary(x**y), Q.imaginary(y) & Q.integer(x)) is None
    assert ask(Q.imaginary(x**y), Q.imaginary(x) & Q.odd(y)) is True
    assert ask(Q.imaginary(x**y), Q.imaginary(x) & Q.rational(y)) is None
    assert ask(Q.imaginary(x**y), Q.imaginary(x) & Q.even(y)) is False

    assert ask(Q.imaginary(x**y), Q.real(x) & Q.integer(y)) is False
    assert ask(Q.imaginary(x**y), Q.positive(x) & Q.real(y)) is False
    assert ask(Q.imaginary(x**y), Q.negative(x) & Q.real(y)) is None
    assert ask(Q.imaginary(x**y), Q.negative(x) & Q.real(y) & ~Q.rational(y)) is False
    assert ask(Q.imaginary(x**y), Q.integer(x) & Q.imaginary(y)) is None
    assert ask(Q.imaginary(x**y), Q.negative(x) & Q.rational(y) & Q.integer(2*y)) is True
    assert ask(Q.imaginary(x**y), Q.negative(x) & Q.rational(y) & ~Q.integer(2*y)) is False
    assert ask(Q.imaginary(x**y), Q.negative(x) & Q.rational(y)) is None
    assert ask(Q.imaginary(x**y), Q.real(x) & Q.rational(y) & ~Q.integer(2*y)) is False
    assert ask(Q.imaginary(x**y), Q.real(x) & Q.rational(y) & Q.integer(2*y)) is None

    # logarithm
    assert ask(Q.imaginary(log(I))) is True
    assert ask(Q.imaginary(log(2*I))) is False
    assert ask(Q.imaginary(log(I + 1))) is False
    assert ask(Q.imaginary(log(x)), Q.complex(x)) is None
    assert ask(Q.imaginary(log(x)), Q.imaginary(x)) is None
    assert ask(Q.imaginary(log(x)), Q.positive(x)) is False
    assert ask(Q.imaginary(log(exp(x))), Q.complex(x)) is None
    assert ask(Q.imaginary(log(exp(x))), Q.imaginary(x)) is None  # zoo/I/a+I*b
    assert ask(Q.imaginary(log(exp(I)))) is True

    # exponential
    assert ask(Q.imaginary(exp(x)**x), Q.imaginary(x)) is False
    eq = Pow(exp(pi*I*x, evaluate=False), x, evaluate=False)
    assert ask(Q.imaginary(eq), Q.even(x)) is False
    eq = Pow(exp(pi*I*x/2, evaluate=False), x, evaluate=False)
    assert ask(Q.imaginary(eq), Q.odd(x)) is True
    assert ask(Q.imaginary(exp(3*I*pi*x)**x), Q.integer(x)) is False
    assert ask(Q.imaginary(exp(2*pi*I, evaluate=False))) is False
    assert ask(Q.imaginary(exp(pi*I/2, evaluate=False))) is True

    # issue 7886
    assert ask(Q.imaginary(Pow(x, Rational(1, 4))), Q.real(x) & Q.negative(x)) is False


def test_integer():
    assert ask(Q.integer(x)) is None
    assert ask(Q.integer(x), Q.integer(x)) is True
    assert ask(Q.integer(x), ~Q.integer(x)) is False
    assert ask(Q.integer(x), ~Q.real(x)) is False
    assert ask(Q.integer(x), ~Q.positive(x)) is None
    assert ask(Q.integer(x), Q.even(x) | Q.odd(x)) is True

    assert ask(Q.integer(2*x), Q.integer(x)) is True
    assert ask(Q.integer(2*x), Q.even(x)) is True
    assert ask(Q.integer(2*x), Q.prime(x)) is True
    assert ask(Q.integer(2*x), Q.rational(x)) is None
    assert ask(Q.integer(2*x), Q.real(x)) is None
    assert ask(Q.integer(sqrt(2)*x), Q.integer(x)) is False
    assert ask(Q.integer(sqrt(2)*x), Q.irrational(x)) is None

    assert ask(Q.integer(x/2), Q.odd(x)) is False
    assert ask(Q.integer(x/2), Q.even(x)) is True
    assert ask(Q.integer(x/3), Q.odd(x)) is None
    assert ask(Q.integer(x/3), Q.even(x)) is None

    # https://github.com/sympy/sympy/issues/7286
    assert ask(Q.integer(Abs(x)),Q.integer(x)) is True
    assert ask(Q.integer(Abs(-x)),Q.integer(x)) is True
    assert ask(Q.integer(Abs(x)), ~Q.integer(x)) is None
    assert ask(Q.integer(Abs(x)),Q.complex(x)) is None
    assert ask(Q.integer(Abs(x+I*y)),Q.real(x) & Q.real(y)) is None

    # https://github.com/sympy/sympy/issues/27739
    assert ask(Q.integer(x/y), Q.integer(x) & Q.integer(y)) is None
    assert ask(Q.integer(1/x), Q.integer(x)) is None
    assert ask(Q.integer(x**y), Q.integer(x) & Q.integer(y)) is None
    assert ask(Q.integer(sqrt(5))) is False
    assert ask(Q.integer(x**y), Q.nonzero(x) & Q.zero(y)) is True
    assert ask(Q.integer(x**y), Q.integer(x) & Q.integer(y) & Q.positive(y)) is True
    assert ask(Q.integer(-1**x), Q.integer(x)) is True
    assert ask(Q.integer(x**y), Q.integer(x) & Q.integer(y) & Q.positive(y)) is True
    assert ask(Q.integer(x**y), Q.zero(x) & Q.integer(y) & Q.positive(y)) is True
    assert ask(Q.integer(pi**x), Q.zero(x)) is True
    assert ask(Q.integer(x**y), Q.imaginary(x) & Q.zero(y)) is True


def test_negative():
    assert ask(Q.negative(x), Q.negative(x)) is True
    assert ask(Q.negative(x), Q.positive(x)) is False
    assert ask(Q.negative(x), ~Q.real(x)) is False
    assert ask(Q.negative(x), Q.prime(x)) is False
    assert ask(Q.negative(x), ~Q.prime(x)) is None

    assert ask(Q.negative(-x), Q.positive(x)) is True
    assert ask(Q.negative(-x), ~Q.positive(x)) is None
    assert ask(Q.negative(-x), Q.negative(x)) is False
    assert ask(Q.negative(-x), Q.positive(x)) is True

    assert ask(Q.negative(x - 1), Q.negative(x)) is True
    assert ask(Q.negative(x + y)) is None
    assert ask(Q.negative(x + y), Q.negative(x)) is None
    assert ask(Q.negative(x + y), Q.negative(x) & Q.negative(y)) is True
    assert ask(Q.negative(x + y), Q.negative(x) & Q.nonpositive(y)) is True
    assert ask(Q.negative(2 + I)) is False
    # although this could be False, it is representative of expressions
    # that don't evaluate to a zero with precision
    assert ask(Q.negative(cos(I)**2 + sin(I)**2 - 1)) is None
    assert ask(Q.negative(-I + I*(cos(2)**2 + sin(2)**2))) is None

    assert ask(Q.negative(x**2)) is None
    assert ask(Q.negative(x**2), Q.real(x)) is False
    assert ask(Q.negative(x**1.4), Q.real(x)) is None

    assert ask(Q.negative(x**I), Q.positive(x)) is None

    assert ask(Q.negative(x*y)) is None
    assert ask(Q.negative(x*y), Q.positive(x) & Q.positive(y)) is False
    assert ask(Q.negative(x*y), Q.positive(x) & Q.negative(y)) is True
    assert ask(Q.negative(x*y), Q.complex(x) & Q.complex(y)) is None

    assert ask(Q.negative(x**y)) is None
    assert ask(Q.negative(x**y), Q.negative(x) & Q.even(y)) is False
    assert ask(Q.negative(x**y), Q.negative(x) & Q.odd(y)) is True
    assert ask(Q.negative(x**y), Q.positive(x) & Q.integer(y)) is False

    assert ask(Q.negative(Abs(x))) is False


def test_nonzero():
    assert ask(Q.nonzero(x)) is None
    assert ask(Q.nonzero(x), Q.real(x)) is None
    assert ask(Q.nonzero(x), Q.positive(x)) is True
    assert ask(Q.nonzero(x), Q.negative(x)) is True
    assert ask(Q.nonzero(x), Q.negative(x) | Q.positive(x)) is True

    assert ask(Q.nonzero(x + y)) is None
    assert ask(Q.nonzero(x + y), Q.positive(x) & Q.positive(y)) is True
    assert ask(Q.nonzero(x + y), Q.positive(x) & Q.negative(y)) is None
    assert ask(Q.nonzero(x + y), Q.negative(x) & Q.negative(y)) is True

    assert ask(Q.nonzero(2*x)) is None
    assert ask(Q.nonzero(2*x), Q.positive(x)) is True
    assert ask(Q.nonzero(2*x), Q.negative(x)) is True
    assert ask(Q.nonzero(x*y), Q.nonzero(x)) is None
    assert ask(Q.nonzero(x*y), Q.nonzero(x) & Q.nonzero(y)) is True

    assert ask(Q.nonzero(x**y), Q.nonzero(x)) is True

    assert ask(Q.nonzero(Abs(x))) is None
    assert ask(Q.nonzero(Abs(x)), Q.nonzero(x)) is True

    assert ask(Q.nonzero(log(exp(2*I)))) is False
    # although this could be False, it is representative of expressions
    # that don't evaluate to a zero with precision
    assert ask(Q.nonzero(cos(1)**2 + sin(1)**2 - 1)) is None


def test_zero():
    assert ask(Q.zero(x)) is None
    assert ask(Q.zero(x), Q.real(x)) is None
    assert ask(Q.zero(x), Q.positive(x)) is False
    assert ask(Q.zero(x), Q.negative(x)) is False
    assert ask(Q.zero(x), Q.negative(x) | Q.positive(x)) is False

    assert ask(Q.zero(x), Q.nonnegative(x) & Q.nonpositive(x)) is True

    assert ask(Q.zero(x + y)) is None
    assert ask(Q.zero(x + y), Q.positive(x) & Q.positive(y)) is False
    assert ask(Q.zero(x + y), Q.positive(x) & Q.negative(y)) is None
    assert ask(Q.zero(x + y), Q.negative(x) & Q.negative(y)) is False

    assert ask(Q.zero(2*x)) is None
    assert ask(Q.zero(2*x), Q.positive(x)) is False
    assert ask(Q.zero(2*x), Q.negative(x)) is False
    assert ask(Q.zero(x*y), Q.nonzero(x)) is None

    assert ask(Q.zero(Abs(x))) is None
    assert ask(Q.zero(Abs(x)), Q.zero(x)) is True

    assert ask(Q.integer(x), Q.zero(x)) is True
    assert ask(Q.even(x), Q.zero(x)) is True
    assert ask(Q.odd(x), Q.zero(x)) is False
    assert ask(Q.zero(x), Q.even(x)) is None
    assert ask(Q.zero(x), Q.odd(x)) is False
    assert ask(Q.zero(x) | Q.zero(y), Q.zero(x*y)) is True


def test_odd_query():
    assert ask(Q.odd(x)) is None
    assert ask(Q.odd(x), Q.odd(x)) is True
    assert ask(Q.odd(x), Q.integer(x)) is None
    assert ask(Q.odd(x), ~Q.integer(x)) is False
    assert ask(Q.odd(x), Q.rational(x)) is None
    assert ask(Q.odd(x), Q.positive(x)) is None

    assert ask(Q.odd(-x), Q.odd(x)) is True

    assert ask(Q.odd(2*x)) is None
    assert ask(Q.odd(2*x), Q.integer(x)) is False
    assert ask(Q.odd(2*x), Q.odd(x)) is False
    assert ask(Q.odd(2*x), Q.irrational(x)) is False
    assert ask(Q.odd(2*x), ~Q.integer(x)) is None
    assert ask(Q.odd(3*x), Q.integer(x)) is None

    assert ask(Q.odd(x/3), Q.odd(x)) is None
    assert ask(Q.odd(x/3), Q.even(x)) is None

    assert ask(Q.odd(x + 1), Q.even(x)) is True
    assert ask(Q.odd(x + 2), Q.even(x)) is False
    assert ask(Q.odd(x + 2), Q.odd(x)) is True
    assert ask(Q.odd(3 - x), Q.odd(x)) is False
    assert ask(Q.odd(3 - x), Q.even(x)) is True
    assert ask(Q.odd(3 + x), Q.odd(x)) is False
    assert ask(Q.odd(3 + x), Q.even(x)) is True
    assert ask(Q.odd(x + y), Q.odd(x) & Q.odd(y)) is False
    assert ask(Q.odd(x + y), Q.odd(x) & Q.even(y)) is True
    assert ask(Q.odd(x - y), Q.even(x) & Q.odd(y)) is True
    assert ask(Q.odd(x - y), Q.odd(x) & Q.odd(y)) is False

    assert ask(Q.odd(x + y + z), Q.odd(x) & Q.odd(y) & Q.even(z)) is False
    assert ask(Q.odd(x + y + z + t),
        Q.odd(x) & Q.odd(y) & Q.even(z) & Q.integer(t)) is None

    assert ask(Q.odd(2*x + 1), Q.integer(x)) is True
    assert ask(Q.odd(2*x + y), Q.integer(x) & Q.odd(y)) is True
    assert ask(Q.odd(2*x + y), Q.integer(x) & Q.even(y)) is False
    assert ask(Q.odd(2*x + y), Q.integer(x) & Q.integer(y)) is None
    assert ask(Q.odd(x*y), Q.odd(x) & Q.even(y)) is False
    assert ask(Q.odd(x*y), Q.odd(x) & Q.odd(y)) is True
    assert ask(Q.odd(2*x*y), Q.rational(x) & Q.rational(x)) is None
    assert ask(Q.odd(2*x*y), Q.irrational(x) & Q.irrational(x)) is None

    assert ask(Q.odd(Abs(x)), Q.odd(x)) is True

    assert ask(Q.odd((-1)**n), Q.integer(n)) is True

    assert ask(Q.odd(k**2), Q.even(k)) is False
    assert ask(Q.odd(n**2), Q.odd(n)) is True
    assert ask(Q.odd(3**k), Q.even(k)) is None

    assert ask(Q.odd(k**m), Q.even(k) & Q.integer(m) & ~Q.negative(m)) is None
    assert ask(Q.odd(n**m), Q.odd(n) & Q.integer(m) & ~Q.negative(m)) is True

    assert ask(Q.odd(k**p), Q.even(k) & Q.integer(p) & Q.positive(p)) is False
    assert ask(Q.odd(n**p), Q.odd(n) & Q.integer(p) & Q.positive(p)) is True

    assert ask(Q.odd(m**k), Q.even(k) & Q.integer(m) & ~Q.negative(m)) is None
    assert ask(Q.odd(p**k), Q.even(k) & Q.integer(p) & Q.positive(p)) is None

    assert ask(Q.odd(m**n), Q.odd(n) & Q.integer(m) & ~Q.negative(m)) is None
    assert ask(Q.odd(p**n), Q.odd(n) & Q.integer(p) & Q.positive(p)) is None

    assert ask(Q.odd(k**x), Q.even(k)) is None
    assert ask(Q.odd(n**x), Q.odd(n)) is None

    assert ask(Q.odd(x*y), Q.integer(x) & Q.integer(y)) is None
    assert ask(Q.odd(x*x), Q.integer(x)) is None
    assert ask(Q.odd(x*(x + y)), Q.integer(x) & Q.odd(y)) is False
    assert ask(Q.odd(x*(x + y)), Q.integer(x) & Q.even(y)) is None


@XFAIL
def test_oddness_in_ternary_integer_product_with_odd():
    # Tests that oddness inference is independent of term ordering.
    # Term ordering at the point of testing depends on SymPy's symbol order, so
    # we try to force a different order by modifying symbol names.
    assert ask(Q.odd(x*y*(y + z)), Q.integer(x) & Q.integer(y) & Q.odd(z)) is False
    assert ask(Q.odd(y*x*(x + z)), Q.integer(x) & Q.integer(y) & Q.odd(z)) is False


def test_oddness_in_ternary_integer_product_with_even():
    assert ask(Q.odd(x*y*(y + z)), Q.integer(x) & Q.integer(y) & Q.even(z)) is None


def test_prime():
    assert ask(Q.prime(x), Q.prime(x)) is True
    assert ask(Q.prime(x), ~Q.prime(x)) is False
    assert ask(Q.prime(x), Q.integer(x)) is None
    assert ask(Q.prime(x), ~Q.integer(x)) is False

    assert ask(Q.prime(2*x), Q.integer(x)) is None
    assert ask(Q.prime(x*y)) is None
    assert ask(Q.prime(x*y), Q.prime(x)) is None
    assert ask(Q.prime(x*y), Q.integer(x) & Q.integer(y)) is None
    assert ask(Q.prime(4*x), Q.integer(x)) is False
    assert ask(Q.prime(4*x)) is None

    assert ask(Q.prime(x**2), Q.integer(x)) is False
    assert ask(Q.prime(x**2), Q.prime(x)) is False

    # https://github.com/sympy/sympy/issues/27446
    assert ask(Q.prime(4**x), Q.integer(x)) is False
    assert ask(Q.prime(p**x), Q.prime(p) & Q.integer(x) & Q.ne(x, 1)) is False
    assert ask(Q.prime(n**x), Q.integer(x) & Q.composite(n)) is False
    assert ask(Q.prime(x**y), Q.integer(x) & Q.integer(y)) is None
    assert ask(Q.prime(2**x), Q.integer(x)) is None
    assert ask(Q.prime(p**x), Q.prime(p) & Q.integer(x)) is None

    # Ideally, these should return True since the base is prime and the exponent is one,
    # but currently, they return None.
    assert ask(Q.prime(x**y), Q.prime(x) & Q.eq(y,1)) is None
    assert ask(Q.prime(x**y), Q.prime(x) & Q.integer(y) & Q.gt(y,0) & Q.lt(y,2)) is None

    assert ask(Q.prime(Pow(x,1, evaluate=False)), Q.prime(x)) is True


@_both_exp_pow
def test_positive():
    assert ask(Q.positive(cos(I) ** 2 + sin(I) ** 2 - 1)) is None
    assert ask(Q.positive(x), Q.positive(x)) is True
    assert ask(Q.positive(x), Q.negative(x)) is False
    assert ask(Q.positive(x), Q.nonzero(x)) is None

    assert ask(Q.positive(-x), Q.positive(x)) is False
    assert ask(Q.positive(-x), Q.negative(x)) is True

    assert ask(Q.positive(x + y), Q.positive(x) & Q.positive(y)) is True
    assert ask(Q.positive(x + y), Q.positive(x) & Q.nonnegative(y)) is True
    assert ask(Q.positive(x + y), Q.positive(x) & Q.negative(y)) is None
    assert ask(Q.positive(x + y), Q.positive(x) & Q.imaginary(y)) is False

    assert ask(Q.positive(2*x), Q.positive(x)) is True
    assumptions = Q.positive(x) & Q.negative(y) & Q.negative(z) & Q.positive(w)
    assert ask(Q.positive(x*y*z)) is None
    assert ask(Q.positive(x*y*z), assumptions) is True
    assert ask(Q.positive(-x*y*z), assumptions) is False

    assert ask(Q.positive(x**I), Q.positive(x)) is None

    assert ask(Q.positive(x**2), Q.positive(x)) is True
    assert ask(Q.positive(x**2), Q.negative(x)) is True
    assert ask(Q.positive(x**3), Q.negative(x)) is False
    assert ask(Q.positive(1/(1 + x**2)), Q.real(x)) is True
    assert ask(Q.positive(2**I)) is False
    assert ask(Q.positive(2 + I)) is False
    # although this could be False, it is representative of expressions
    # that don't evaluate to a zero with precision
    assert ask(Q.positive(cos(I)**2 + sin(I)**2 - 1)) is None
    assert ask(Q.positive(-I + I*(cos(2)**2 + sin(2)**2))) is None

    #exponential
    assert ask(Q.positive(exp(x)), Q.real(x)) is True
    assert ask(~Q.negative(exp(x)), Q.real(x)) is True
    assert ask(Q.positive(x + exp(x)), Q.real(x)) is None
    assert ask(Q.positive(exp(x)), Q.imaginary(x)) is None
    assert ask(Q.positive(exp(2*pi*I, evaluate=False)), Q.imaginary(x)) is True
    assert ask(Q.negative(exp(pi*I, evaluate=False)), Q.imaginary(x)) is True
    assert ask(Q.positive(exp(x*pi*I)), Q.even(x)) is True
    assert ask(Q.positive(exp(x*pi*I)), Q.odd(x)) is False
    assert ask(Q.positive(exp(x*pi*I)), Q.real(x)) is None

    # logarithm
    assert ask(Q.positive(log(x)), Q.imaginary(x)) is False
    assert ask(Q.positive(log(x)), Q.negative(x)) is False
    assert ask(Q.positive(log(x)), Q.positive(x)) is None
    assert ask(Q.positive(log(x + 2)), Q.positive(x)) is True

    # factorial
    assert ask(Q.positive(factorial(x)), Q.integer(x) & Q.positive(x))
    assert ask(Q.positive(factorial(x)), Q.integer(x)) is None

    #absolute value
    assert ask(Q.positive(Abs(x))) is None  # Abs(0) = 0
    assert ask(Q.positive(Abs(x)), Q.positive(x)) is True


def test_nonpositive():
    assert ask(Q.nonpositive(-1))
    assert ask(Q.nonpositive(0))
    assert ask(Q.nonpositive(1)) is False
    assert ask(~Q.positive(x), Q.nonpositive(x))
    assert ask(Q.nonpositive(x), Q.positive(x)) is False
    assert ask(Q.nonpositive(sqrt(-1))) is False
    assert ask(Q.nonpositive(x), Q.imaginary(x)) is False


def test_nonnegative():
    assert ask(Q.nonnegative(-1)) is False
    assert ask(Q.nonnegative(0))
    assert ask(Q.nonnegative(1))
    assert ask(~Q.negative(x), Q.nonnegative(x))
    assert ask(Q.nonnegative(x), Q.negative(x)) is False
    assert ask(Q.nonnegative(sqrt(-1))) is False
    assert ask(Q.nonnegative(x), Q.imaginary(x)) is False

def test_real_basic():
    assert ask(Q.real(x)) is None
    assert ask(Q.real(x), Q.real(x)) is True
    assert ask(Q.real(x), Q.nonzero(x)) is True
    assert ask(Q.real(x), Q.positive(x)) is True
    assert ask(Q.real(x), Q.negative(x)) is True
    assert ask(Q.real(x), Q.integer(x)) is True
    assert ask(Q.real(x), Q.even(x)) is True
    assert ask(Q.real(x), Q.prime(x)) is True

    assert ask(Q.real(x/sqrt(2)), Q.real(x)) is True
    assert ask(Q.real(x/sqrt(-2)), Q.real(x)) is False

    assert ask(Q.real(x + 1), Q.real(x)) is True
    assert ask(Q.real(x + I), Q.real(x)) is False
    assert ask(Q.real(x + I), Q.complex(x)) is None

    assert ask(Q.real(2*x), Q.real(x)) is True
    assert ask(Q.real(I*x), Q.real(x)) is False
    assert ask(Q.real(I*x), Q.imaginary(x)) is True
    assert ask(Q.real(I*x), Q.complex(x)) is None


def test_real_pow():
    assert ask(Q.real(x**2), Q.real(x)) is True
    assert ask(Q.real(sqrt(x)), Q.negative(x)) is False
    assert ask(Q.real(x**y), Q.real(x) & Q.integer(y)) is None
    assert ask(Q.real(x**y), Q.real(x) & Q.real(y)) is None
    assert ask(Q.real(x**y), Q.positive(x) & Q.real(y)) is True
    assert ask(Q.real(x**y), Q.imaginary(x) & Q.imaginary(y)) is None  # I**I or (2*I)**I
    assert ask(Q.real(x**y), Q.imaginary(x) & Q.real(y)) is None  # I**1 or I**0
    assert ask(Q.real(x**y), Q.real(x) & Q.imaginary(y)) is None  # could be exp(2*pi*I) or 2**I
    assert ask(Q.real(x**0), Q.imaginary(x)) is True
    assert ask(Q.real(x**y), Q.positive(x) & Q.real(y)) is True
    assert ask(Q.real(x**y), Q.real(x) & Q.rational(y)) is None
    assert ask(Q.real(x**y), Q.imaginary(x) & Q.integer(y)) is None
    assert ask(Q.real(x**y), Q.imaginary(x) & Q.odd(y)) is False
    assert ask(Q.real(x**y), Q.imaginary(x) & Q.even(y)) is True
    assert ask(Q.real(x**(y/z)), Q.real(x) & Q.real(y/z) & Q.rational(y/z) & Q.even(z) & Q.positive(x)) is True
    assert ask(Q.real(x**(y/z)), Q.real(x) & Q.rational(y/z) & Q.even(z) & Q.negative(x)) is None
    assert ask(Q.real(x**(y/z)), Q.real(x) & Q.integer(y/z)) is None
    assert ask(Q.real(x**(y/z)), Q.real(x) & Q.real(y/z) & Q.positive(x)) is True
    assert ask(Q.real(x**(y/z)), Q.real(x) & Q.real(y/z) & Q.negative(x)) is None
    assert ask(Q.real((-I)**i), Q.imaginary(i)) is True
    assert ask(Q.real(I**i), Q.imaginary(i)) is True
    assert ask(Q.real(i**i), Q.imaginary(i)) is None  # i might be 2*I
    assert ask(Q.real(x**i), Q.imaginary(i)) is None  # x could be 0
    assert ask(Q.real(x**(I*pi/log(x))), Q.real(x)) is True

    # https://github.com/sympy/sympy/issues/27485
    assert ask(Q.real(n**p), Q.negative(n) & Q.positive(p)) is None

    # https://github.com/sympy/sympy/issues/16530
    assert ask(Q.real(1/Abs(x))) is None
    assert ask(Q.real(x**y), Q.zero(x) & Q.real(y)) is None
    assert ask(Q.real(x**y), Q.zero(x) & Q.positive(y)) is True


@_both_exp_pow
def test_real_functions():
    # trigonometric functions
    assert ask(Q.real(sin(x))) is None
    assert ask(Q.real(cos(x))) is None
    assert ask(Q.real(sin(x)), Q.real(x)) is True
    assert ask(Q.real(cos(x)), Q.real(x)) is True

    # exponential function
    assert ask(Q.real(exp(x))) is None
    assert ask(Q.real(exp(x)), Q.real(x)) is True
    assert ask(Q.real(x + exp(x)), Q.real(x)) is True
    assert ask(Q.real(exp(2*pi*I, evaluate=False))) is True
    assert ask(Q.real(exp(pi*I, evaluate=False))) is True
    assert ask(Q.real(exp(pi*I/2, evaluate=False))) is False

    # logarithm
    assert ask(Q.real(log(I))) is False
    assert ask(Q.real(log(2*I))) is False
    assert ask(Q.real(log(I + 1))) is False
    assert ask(Q.real(log(x)), Q.complex(x)) is None
    assert ask(Q.real(log(x)), Q.imaginary(x)) is False
    assert ask(Q.real(log(exp(x))), Q.imaginary(x)) is None  # exp(2*pi*I) is 1, log(exp(pi*I)) is pi*I (disregarding periodicity)
    assert ask(Q.real(log(exp(x))), Q.complex(x)) is None
    eq = Pow(exp(2*pi*I*x, evaluate=False), x, evaluate=False)
    assert ask(Q.real(eq), Q.integer(x)) is True
    assert ask(Q.real(exp(x)**x), Q.imaginary(x)) is True
    assert ask(Q.real(exp(x)**x), Q.complex(x)) is None

    # Q.complexes
    assert ask(Q.real(re(x))) is True
    assert ask(Q.real(im(x))) is True


def test_matrix():

    # hermitian
    assert ask(Q.hermitian(Matrix([[2, 2 + I, 4], [2 - I, 3, I], [4, -I, 1]]))) == True
    assert ask(Q.hermitian(Matrix([[2, 2 + I, 4], [2 + I, 3, I], [4, -I, 1]]))) == False
    z = symbols('z', complex=True)
    assert ask(Q.hermitian(Matrix([[2, 2 + I, z], [2 - I, 3, I], [4, -I, 1]]))) == None
    assert ask(Q.hermitian(SparseMatrix(((25, 15, -5), (15, 18, 0), (-5, 0, 11))))) == True
    assert ask(Q.hermitian(SparseMatrix(((25, 15, -5), (15, I, 0), (-5, 0, 11))))) == False
    assert ask(Q.hermitian(SparseMatrix(((25, 15, -5), (15, z, 0), (-5, 0, 11))))) == None

    # antihermitian
    A = Matrix([[0, -2 - I, 0], [2 - I, 0, -I], [0, -I, 0]])
    B = Matrix([[-I, 2 + I, 0], [-2 + I, 0, 2 + I], [0, -2 + I, -I]])
    assert ask(Q.antihermitian(A)) is True
    assert ask(Q.antihermitian(B)) is True
    assert ask(Q.antihermitian(A**2)) is False
    C = (B**3)
    C.simplify()
    assert ask(Q.antihermitian(C)) is True
    _A = Matrix([[0, -2 - I, 0], [z, 0, -I], [0, -I, 0]])
    assert ask(Q.antihermitian(_A)) is None


@_both_exp_pow
def test_algebraic():
    assert ask(Q.algebraic(x)) is None

    assert ask(Q.algebraic(I)) is True
    assert ask(Q.algebraic(2*I)) is True
    assert ask(Q.algebraic(I/3)) is True

    assert ask(Q.algebraic(sqrt(7))) is True
    assert ask(Q.algebraic(2*sqrt(7))) is True
    assert ask(Q.algebraic(sqrt(7)/3)) is True

    assert ask(Q.algebraic(I*sqrt(3))) is True
    assert ask(Q.algebraic(sqrt(1 + I*sqrt(3)))) is True

    assert ask(Q.algebraic(1 + I*sqrt(3)**Rational(17, 31))) is True
    assert ask(Q.algebraic(1 + I*sqrt(3)**(17/pi))) is None

    for f in [exp, sin, tan, asin, atan, cos]:
        assert ask(Q.algebraic(f(7))) is False
        assert ask(Q.algebraic(f(7, evaluate=False))) is False
        assert ask(Q.algebraic(f(0, evaluate=False))) is True
        assert ask(Q.algebraic(f(x)), Q.algebraic(x)) is None
        assert ask(Q.algebraic(f(x)), Q.algebraic(x) & Q.nonzero(x)) is False

    for g in [log, acos]:
        assert ask(Q.algebraic(g(7))) is False
        assert ask(Q.algebraic(g(7, evaluate=False))) is False
        assert ask(Q.algebraic(g(1, evaluate=False))) is True
        assert ask(Q.algebraic(g(x)), Q.algebraic(x)) is None
        assert ask(Q.algebraic(g(x)), Q.algebraic(x) & Q.nonzero(x - 1)) is False

    for h in [cot, acot]:
        assert ask(Q.algebraic(h(7))) is False
        assert ask(Q.algebraic(h(7, evaluate=False))) is False
        assert ask(Q.algebraic(h(x)), Q.algebraic(x)) is False

    assert ask(Q.algebraic(sqrt(sin(7)))) is None
    assert ask(Q.algebraic(sqrt(y + I*sqrt(7)))) is None

    assert ask(Q.algebraic(2.47)) is True

    assert ask(Q.algebraic(x), Q.transcendental(x)) is False
    assert ask(Q.transcendental(x), Q.algebraic(x)) is False

    #https://github.com/sympy/sympy/issues/27445
    assert ask(Q.algebraic(Pow(1, x, evaluate=False)), Q.algebraic(x)) is None
    assert ask(Q.algebraic(Pow(x, y))) is None
    assert ask(Q.algebraic(Pow(1, x, evaluate=False))) is None
    assert ask(Q.algebraic(x**(pi*I))) is None
    assert ask(Q.algebraic(pi**n),Q.integer(n) & Q.positive(n)) is False
    assert ask(Q.algebraic(x**y),Q.algebraic(x) & Q.rational(y)) is True


def test_global():
    """Test ask with global assumptions"""
    assert ask(Q.integer(x)) is None
    global_assumptions.add(Q.integer(x))
    assert ask(Q.integer(x)) is True
    global_assumptions.clear()
    assert ask(Q.integer(x)) is None


def test_custom_context():
    """Test ask with custom assumptions context"""
    assert ask(Q.integer(x)) is None
    local_context = AssumptionsContext()
    local_context.add(Q.integer(x))
    assert ask(Q.integer(x), context=local_context) is True
    assert ask(Q.integer(x)) is None


def test_functions_in_assumptions():
    assert ask(Q.negative(x), Q.real(x) >> Q.positive(x)) is False
    assert ask(Q.negative(x), Equivalent(Q.real(x), Q.positive(x))) is False
    assert ask(Q.negative(x), Xor(Q.real(x), Q.negative(x))) is False


def test_composite_ask():
    assert ask(Q.negative(x) & Q.integer(x),
        assumptions=Q.real(x) >> Q.positive(x)) is False


def test_composite_proposition():
    assert ask(True) is True
    assert ask(False) is False
    assert ask(~Q.negative(x), Q.positive(x)) is True
    assert ask(~Q.real(x), Q.commutative(x)) is None
    assert ask(Q.negative(x) & Q.integer(x), Q.positive(x)) is False
    assert ask(Q.negative(x) & Q.integer(x)) is None
    assert ask(Q.real(x) | Q.integer(x), Q.positive(x)) is True
    assert ask(Q.real(x) | Q.integer(x)) is None
    assert ask(Q.real(x) >> Q.positive(x), Q.negative(x)) is False
    assert ask(Implies(
        Q.real(x), Q.positive(x), evaluate=False), Q.negative(x)) is False
    assert ask(Implies(Q.real(x), Q.positive(x), evaluate=False)) is None
    assert ask(Equivalent(Q.integer(x), Q.even(x)), Q.even(x)) is True
    assert ask(Equivalent(Q.integer(x), Q.even(x))) is None
    assert ask(Equivalent(Q.positive(x), Q.integer(x)), Q.integer(x)) is None
    assert ask(Q.real(x) | Q.integer(x), Q.real(x) | Q.integer(x)) is True

def test_tautology():
    assert ask(Q.real(x) | ~Q.real(x)) is True
    assert ask(Q.real(x) & ~Q.real(x)) is False

def test_composite_assumptions():
    assert ask(Q.real(x), Q.real(x) & Q.real(y)) is True
    assert ask(Q.positive(x), Q.positive(x) | Q.positive(y)) is None
    assert ask(Q.positive(x), Q.real(x) >> Q.positive(y)) is None
    assert ask(Q.real(x), ~(Q.real(x) >> Q.real(y))) is True

def test_key_extensibility():
    """test that you can add keys to the ask system at runtime"""
    # make sure the key is not defined
    raises(AttributeError, lambda: ask(Q.my_key(x)))

    # Old handler system
    class MyAskHandler(AskHandler):
        @staticmethod
        def Symbol(expr, assumptions):
            return True
    try:
        with warns_deprecated_sympy():
            register_handler('my_key', MyAskHandler)
        with warns_deprecated_sympy():
            assert ask(Q.my_key(x)) is True
        with warns_deprecated_sympy():
            assert ask(Q.my_key(x + 1)) is None
    finally:
        # We have to disable the stacklevel testing here because this raises
        # the warning twice from two different places
        with warns_deprecated_sympy():
            remove_handler('my_key', MyAskHandler)
        del Q.my_key
    raises(AttributeError, lambda: ask(Q.my_key(x)))

    # New handler system
    class MyPredicate(Predicate):
        pass
    try:
        Q.my_key = MyPredicate()
        @Q.my_key.register(Symbol)
        def _(expr, assumptions):
            return True
        assert ask(Q.my_key(x)) is True
        assert ask(Q.my_key(x+1)) is None
    finally:
        del Q.my_key
    raises(AttributeError, lambda: ask(Q.my_key(x)))


def test_type_extensibility():
    """test that new types can be added to the ask system at runtime
    """
    from sympy.core import Basic

    class MyType(Basic):
        pass

    @Q.prime.register(MyType)
    def _(expr, assumptions):
        return True

    assert ask(Q.prime(MyType())) is True


def test_single_fact_lookup():
    known_facts = And(Implies(Q.integer, Q.rational),
                      Implies(Q.rational, Q.real),
                      Implies(Q.real, Q.complex))
    known_facts_keys = {Q.integer, Q.rational, Q.real, Q.complex}

    known_facts_cnf = to_cnf(known_facts)
    mapping = single_fact_lookup(known_facts_keys, known_facts_cnf)

    assert mapping[Q.rational] == {Q.real, Q.rational, Q.complex}


def test_generate_known_facts_dict():
    known_facts = And(Implies(Q.integer(x), Q.rational(x)),
                      Implies(Q.rational(x), Q.real(x)),
                      Implies(Q.real(x), Q.complex(x)))
    known_facts_keys = {Q.integer(x), Q.rational(x), Q.real(x), Q.complex(x)}

    assert generate_known_facts_dict(known_facts_keys, known_facts) == \
        {Q.complex: ({Q.complex}, set()),
         Q.integer: ({Q.complex, Q.integer, Q.rational, Q.real}, set()),
         Q.rational: ({Q.complex, Q.rational, Q.real}, set()),
         Q.real: ({Q.complex, Q.real}, set())}


@slow
def test_known_facts_consistent():
    """"Test that ask_generated.py is up-to-date"""
    x = Symbol('x')
    fact = get_known_facts(x)
    # test cnf clauses of fact between unary predicates
    cnf = CNF.to_CNF(fact)
    clauses = set()
    clauses.update(frozenset(Literal(lit.arg.function, lit.is_Not) for lit in sorted(cl, key=str)) for cl in cnf.clauses)
    assert get_all_known_facts() == clauses
    # test dictionary of fact between unary predicates
    keys = [pred(x) for pred in get_known_facts_keys()]
    mapping = generate_known_facts_dict(keys, fact)
    assert get_known_facts_dict() == mapping


def test_Add_queries():
    assert ask(Q.prime(12345678901234567890 + (cos(1)**2 + sin(1)**2))) is True
    assert ask(Q.even(Add(S(2), S(2), evaluate=False))) is True
    assert ask(Q.prime(Add(S(2), S(2), evaluate=False))) is False
    assert ask(Q.integer(Add(S(2), S(2), evaluate=False))) is True


def test_positive_assuming():
    with assuming(Q.positive(x + 1)):
        assert not ask(Q.positive(x))


def test_issue_5421():
    raises(TypeError, lambda: ask(pi/log(x), Q.real))


def test_issue_3906():
    raises(TypeError, lambda: ask(Q.positive))


def test_issue_5833():
    assert ask(Q.positive(log(x)**2), Q.positive(x)) is None
    assert ask(~Q.negative(log(x)**2), Q.positive(x)) is True


def test_issue_6732():
    raises(ValueError, lambda: ask(Q.positive(x), Q.positive(x) & Q.negative(x)))
    raises(ValueError, lambda: ask(Q.negative(x), Q.positive(x) & Q.negative(x)))


def test_issue_7246():
    assert ask(Q.positive(atan(p)), Q.positive(p)) is True
    assert ask(Q.positive(atan(p)), Q.negative(p)) is False
    assert ask(Q.positive(atan(p)), Q.zero(p)) is False
    assert ask(Q.positive(atan(x))) is None

    assert ask(Q.positive(asin(p)), Q.positive(p)) is None
    assert ask(Q.positive(asin(p)), Q.zero(p)) is None
    assert ask(Q.positive(asin(Rational(1, 7)))) is True
    assert ask(Q.positive(asin(x)), Q.positive(x) & Q.nonpositive(x - 1)) is True
    assert ask(Q.positive(asin(x)), Q.negative(x) & Q.nonnegative(x + 1)) is False

    assert ask(Q.positive(acos(p)), Q.positive(p)) is None
    assert ask(Q.positive(acos(Rational(1, 7)))) is True
    assert ask(Q.positive(acos(x)), Q.nonnegative(x + 1) & Q.nonpositive(x - 1)) is True
    assert ask(Q.positive(acos(x)), Q.nonnegative(x - 1)) is None

    assert ask(Q.positive(acot(x)), Q.positive(x)) is True
    assert ask(Q.positive(acot(x)), Q.real(x)) is True
    assert ask(Q.positive(acot(x)), Q.imaginary(x)) is False
    assert ask(Q.positive(acot(x))) is None


@XFAIL
def test_issue_7246_failing():
    #Move this test to test_issue_7246 once
    #the new assumptions module is improved.
    assert ask(Q.positive(acos(x)), Q.zero(x)) is True


def test_check_old_assumption():
    x = symbols('x', real=True)
    assert ask(Q.real(x)) is True
    assert ask(Q.imaginary(x)) is False
    assert ask(Q.complex(x)) is True

    x = symbols('x', imaginary=True)
    assert ask(Q.real(x)) is False
    assert ask(Q.imaginary(x)) is True
    assert ask(Q.complex(x)) is True

    x = symbols('x', complex=True)
    assert ask(Q.real(x)) is None
    assert ask(Q.complex(x)) is True

    x = symbols('x', positive=True)
    assert ask(Q.positive(x)) is True
    assert ask(Q.negative(x)) is False
    assert ask(Q.real(x)) is True

    x = symbols('x', commutative=False)
    assert ask(Q.commutative(x)) is False

    x = symbols('x', negative=True)
    assert ask(Q.positive(x)) is False
    assert ask(Q.negative(x)) is True

    x = symbols('x', nonnegative=True)
    assert ask(Q.negative(x)) is False
    assert ask(Q.positive(x)) is None
    assert ask(Q.zero(x)) is None

    x = symbols('x', finite=True)
    assert ask(Q.finite(x)) is True

    x = symbols('x', prime=True)
    assert ask(Q.prime(x)) is True
    assert ask(Q.composite(x)) is False

    x = symbols('x', composite=True)
    assert ask(Q.prime(x)) is False
    assert ask(Q.composite(x)) is True

    x = symbols('x', even=True)
    assert ask(Q.even(x)) is True
    assert ask(Q.odd(x)) is False

    x = symbols('x', odd=True)
    assert ask(Q.even(x)) is False
    assert ask(Q.odd(x)) is True

    x = symbols('x', nonzero=True)
    assert ask(Q.nonzero(x)) is True
    assert ask(Q.zero(x)) is False

    x = symbols('x', zero=True)
    assert ask(Q.zero(x)) is True

    x = symbols('x', integer=True)
    assert ask(Q.integer(x)) is True

    x = symbols('x', rational=True)
    assert ask(Q.rational(x)) is True
    assert ask(Q.irrational(x)) is False

    x = symbols('x', irrational=True)
    assert ask(Q.irrational(x)) is True
    assert ask(Q.rational(x)) is False


def test_issue_9636():
    assert ask(Q.integer(1.0)) is None
    assert ask(Q.prime(3.0)) is None
    assert ask(Q.composite(4.0)) is None
    assert ask(Q.even(2.0)) is None
    assert ask(Q.odd(3.0)) is None


def test_autosimp_used_to_fail():
    # See issue #9807
    assert ask(Q.imaginary(0**I)) is None
    assert ask(Q.imaginary(0**(-I))) is None
    assert ask(Q.real(0**I)) is None
    assert ask(Q.real(0**(-I))) is None


def test_custom_AskHandler():
    from sympy.logic.boolalg import conjuncts

    # Old handler system
    class MersenneHandler(AskHandler):
        @staticmethod
        def Integer(expr, assumptions):
            if ask(Q.integer(log(expr + 1, 2))):
                return True
        @staticmethod
        def Symbol(expr, assumptions):
            if expr in conjuncts(assumptions):
                return True
    try:
        with warns_deprecated_sympy():
            register_handler('mersenne', MersenneHandler)
        n = Symbol('n', integer=True)
        with warns_deprecated_sympy():
            assert ask(Q.mersenne(7))
        with warns_deprecated_sympy():
            assert ask(Q.mersenne(n), Q.mersenne(n))
    finally:
        del Q.mersenne

    # New handler system
    class MersennePredicate(Predicate):
        pass
    try:
        Q.mersenne = MersennePredicate()
        @Q.mersenne.register(Integer)
        def _(expr, assumptions):
            if ask(Q.integer(log(expr + 1, 2))):
                return True
        @Q.mersenne.register(Symbol)
        def _(expr, assumptions):
            if expr in conjuncts(assumptions):
                return True
        assert ask(Q.mersenne(7))
        assert ask(Q.mersenne(n), Q.mersenne(n))
    finally:
        del Q.mersenne


def test_polyadic_predicate():

    class SexyPredicate(Predicate):
        pass
    try:
        Q.sexyprime = SexyPredicate()

        @Q.sexyprime.register(Integer, Integer)
        def _(int1, int2, assumptions):
            args = sorted([int1, int2])
            if not all(ask(Q.prime(a), assumptions) for a in args):
                return False
            return args[1] - args[0] == 6

        @Q.sexyprime.register(Integer, Integer, Integer)
        def _(int1, int2, int3, assumptions):
            args = sorted([int1, int2, int3])
            if not all(ask(Q.prime(a), assumptions) for a in args):
                return False
            return args[2] - args[1] == 6 and args[1] - args[0] == 6

        assert ask(Q.sexyprime(5, 11))
        assert ask(Q.sexyprime(7, 13, 19))
    finally:
        del Q.sexyprime


def test_Predicate_handler_is_unique():

    # Undefined predicate does not have a handler
    assert Predicate('mypredicate').handler is None

    # Handler of defined predicate is unique to the class
    class MyPredicate(Predicate):
        pass
    mp1 = MyPredicate(Str('mp1'))
    mp2 = MyPredicate(Str('mp2'))
    assert mp1.handler is mp2.handler


def test_relational():
    assert ask(Q.eq(x, 0), Q.zero(x))
    assert not ask(Q.eq(x, 0), Q.nonzero(x))
    assert not ask(Q.ne(x, 0), Q.zero(x))
    assert ask(Q.ne(x, 0), Q.nonzero(x))


def test_issue_25221():
    assert ask(Q.transcendental(x), Q.algebraic(x) | Q.positive(y,y)) is None
    assert ask(Q.transcendental(x), Q.algebraic(x) | (0 > y)) is None
    assert ask(Q.transcendental(x), Q.algebraic(x) | Q.gt(0,y)) is None


def test_issue_27440():
    nan = S.NaN
    assert ask(Q.negative(nan)) is None