text: 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 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
  
      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 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_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')
      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.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)) 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.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(y)
          & Q.finite(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(x) & Q.finite(y)
          & Q.finite(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(y)
          & Q.finite(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
  
      # 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
  
  
  @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)) is True
      assert ask(Q.rational(1/x), Q.integer(x)) is True
      assert ask(Q.rational(1/x), Q.even(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)) 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), ~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)) is True
      assert ask(Q.rational(y/x), Q.integer(x) & Q.rational(y)) is True
      assert ask(Q.rational(y/x), Q.even(x) & Q.rational(y)) 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)) 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
  
  
  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
  
  
  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
      assert ask(Q.prime(x**y), Q.integer(x) & Q.integer(y)) is False
  
  
  @_both_exp_pow
  def test_positive():
      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 True
      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.real(x) & Q.integer(y)) 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 False
      assert ask(Q.real(x**(y/z)), Q.real(x) & Q.integer(y/z)) is True
      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 False
      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
  
  
  @_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 False
  
      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 False
      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
  
  
  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()
      for cl in cnf.clauses:
          clauses.add(frozenset(Literal(lit.arg.function, lit.is_Not) for lit in sorted(cl, key=str)))
      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=0))) is True
      assert ask(Q.prime(Add(S(2), S(2), evaluate=0))) is False
      assert ask(Q.integer(Add(S(2), S(2), evaluate=0))) 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 False
      assert ask(Q.prime(3.0)) is False
      assert ask(Q.composite(4.0)) is False
      assert ask(Q.even(2.0)) is False
      assert ask(Q.odd(3.0)) is False
  
  
  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))
  