test_assumptions.py

from sympy.core import Symbol, S, Rational, Integer
from sympy.utilities.pytest import raises, XFAIL
from sympy import I, sqrt, log, exp, sin, asin

from sympy.core.facts import InconsistentAssumptions


def test_symbol_unset():
    x = Symbol('x', real=True, integer=True)
    assert x.is_real is True
    assert x.is_integer is True
    assert x.is_imaginary is False
    assert x.is_noninteger is False
    assert x.is_number is False


def test_zero():
    z = Integer(0)
    assert z.is_commutative is True
    assert z.is_integer is True
    assert z.is_rational is True
    assert z.is_real is True
    assert z.is_complex is True
    assert z.is_noninteger is False
    assert z.is_irrational is False
    assert z.is_imaginary is False
    assert z.is_positive is False
    assert z.is_negative is False
    assert z.is_nonpositive is True
    assert z.is_nonnegative is True
    assert z.is_even is True
    assert z.is_odd is False
    assert z.is_bounded is True
    assert z.is_unbounded is False
    assert z.is_finite is False
    assert z.is_infinitesimal is True
    assert z.is_comparable is True
    assert z.is_prime is False
    assert z.is_composite is False
    assert z.is_number is True


def test_one():
    z = Integer(1)
    assert z.is_commutative is True
    assert z.is_integer is True
    assert z.is_rational is True
    assert z.is_real is True
    assert z.is_complex is True
    assert z.is_noninteger is False
    assert z.is_irrational is False
    assert z.is_imaginary is False
    assert z.is_positive is True
    assert z.is_negative is False
    assert z.is_nonpositive is False
    assert z.is_nonnegative is True
    assert z.is_even is False
    assert z.is_odd is True
    assert z.is_bounded is True
    assert z.is_unbounded is False
    assert z.is_finite is True
    assert z.is_infinitesimal is False
    assert z.is_comparable is True
    assert z.is_prime is False
    assert z.is_number is True


@XFAIL
def test_one_is_composite():
    assert S(1).is_composite is False


def test_negativeone():
    z = Integer(-1)
    assert z.is_commutative is True
    assert z.is_integer is True
    assert z.is_rational is True
    assert z.is_real is True
    assert z.is_complex is True
    assert z.is_noninteger is False
    assert z.is_irrational is False
    assert z.is_imaginary is False
    assert z.is_positive is False
    assert z.is_negative is True
    assert z.is_nonpositive is True
    assert z.is_nonnegative is False
    assert z.is_even is False
    assert z.is_odd is True
    assert z.is_bounded is True
    assert z.is_unbounded is False
    assert z.is_finite is True
    assert z.is_infinitesimal is False
    assert z.is_comparable is True
    assert z.is_prime is False
    assert z.is_composite is False
    assert z.is_number is True


def test_infinity():
    oo = S.Infinity

    assert oo.is_commutative is True
    assert oo.is_integer is None
    assert oo.is_rational is None
    assert oo.is_real is True
    assert oo.is_complex is True
    assert oo.is_noninteger is None
    assert oo.is_irrational is None
    assert oo.is_imaginary is False
    assert oo.is_positive is True
    assert oo.is_negative is False
    assert oo.is_nonpositive is False
    assert oo.is_nonnegative is True
    assert oo.is_even is None
    assert oo.is_odd is None
    assert oo.is_bounded is False
    assert oo.is_unbounded is True
    assert oo.is_finite is False
    assert oo.is_infinitesimal is False
    assert oo.is_comparable is True
    assert oo.is_prime is None
    assert oo.is_composite is None
    assert oo.is_number is True


def test_neg_infinity():
    mm = S.NegativeInfinity

    assert mm.is_commutative is True
    assert mm.is_integer is None
    assert mm.is_rational is None
    assert mm.is_real is True
    assert mm.is_complex is True
    assert mm.is_noninteger is None
    assert mm.is_irrational is None
    assert mm.is_imaginary is False
    assert mm.is_positive is False
    assert mm.is_negative is True
    assert mm.is_nonpositive is True
    assert mm.is_nonnegative is False
    assert mm.is_even is None
    assert mm.is_odd is None
    assert mm.is_bounded is False
    assert mm.is_unbounded is True
    assert mm.is_finite is False
    assert mm.is_infinitesimal is False
    assert mm.is_comparable is True
    assert mm.is_prime is False
    assert mm.is_composite is False
    assert mm.is_number is True


def test_nan():
    nan = S.NaN

    assert nan.is_commutative is True
    assert nan.is_integer is None
    assert nan.is_rational is None
    assert nan.is_real is None
    assert nan.is_complex is None
    assert nan.is_noninteger is None
    assert nan.is_irrational is None
    assert nan.is_imaginary is None
    assert nan.is_positive is None
    assert nan.is_negative is None
    assert nan.is_nonpositive is None
    assert nan.is_nonnegative is None
    assert nan.is_even is None
    assert nan.is_odd is None
    assert nan.is_bounded is None
    assert nan.is_unbounded is None
    assert nan.is_finite is None
    assert nan.is_infinitesimal is None
    assert nan.is_comparable is False
    assert nan.is_prime is None
    assert nan.is_composite is None
    assert nan.is_number is True


def test_pos_rational():
    r = Rational(3, 4)
    assert r.is_commutative is True
    assert r.is_integer is False
    assert r.is_rational is True
    assert r.is_real is True
    assert r.is_complex is True
    assert r.is_noninteger is True
    assert r.is_irrational is False
    assert r.is_imaginary is False
    assert r.is_positive is True
    assert r.is_negative is False
    assert r.is_nonpositive is False
    assert r.is_nonnegative is True
    assert r.is_even is False
    assert r.is_odd is False
    assert r.is_bounded is True
    assert r.is_unbounded is False
    assert r.is_finite is True
    assert r.is_infinitesimal is False
    assert r.is_comparable is True
    assert r.is_prime is False
    assert r.is_composite is False

    r = Rational(1, 4)
    assert r.is_nonpositive is False
    assert r.is_positive is True
    assert r.is_negative is False
    assert r.is_nonnegative is True
    r = Rational(5, 4)
    assert r.is_negative is False
    assert r.is_positive is True
    assert r.is_nonpositive is False
    assert r.is_nonnegative is True
    r = Rational(5, 3)
    assert r.is_nonnegative is True
    assert r.is_positive is True
    assert r.is_negative is False
    assert r.is_nonpositive is False


def test_neg_rational():
    r = Rational(-3, 4)
    assert r.is_positive is False
    assert r.is_nonpositive is True
    assert r.is_negative is True
    assert r.is_nonnegative is False
    r = Rational(-1, 4)
    assert r.is_nonpositive is True
    assert r.is_positive is False
    assert r.is_negative is True
    assert r.is_nonnegative is False
    r = Rational(-5, 4)
    assert r.is_negative is True
    assert r.is_positive is False
    assert r.is_nonpositive is True
    assert r.is_nonnegative is False
    r = Rational(-5, 3)
    assert r.is_nonnegative is False
    assert r.is_positive is False
    assert r.is_negative is True
    assert r.is_nonpositive is True


def test_pi():
    z = S.Pi
    assert z.is_commutative is True
    assert z.is_integer is False
    assert z.is_rational is False
    assert z.is_real is True
    assert z.is_complex is True
    assert z.is_noninteger is True
    assert z.is_irrational is True
    assert z.is_imaginary is False
    assert z.is_positive is True
    assert z.is_negative is False
    assert z.is_nonpositive is False
    assert z.is_nonnegative is True
    assert z.is_even is False
    assert z.is_odd is False
    assert z.is_bounded is True
    assert z.is_unbounded is False
    assert z.is_finite is True
    assert z.is_infinitesimal is False
    assert z.is_comparable is True
    assert z.is_prime is False
    assert z.is_composite is False


def test_E():
    z = S.Exp1
    assert z.is_commutative is True
    assert z.is_integer is False
    assert z.is_rational is False
    assert z.is_real is True
    assert z.is_complex is True
    assert z.is_noninteger is True
    assert z.is_irrational is True
    assert z.is_imaginary is False
    assert z.is_positive is True
    assert z.is_negative is False
    assert z.is_nonpositive is False
    assert z.is_nonnegative is True
    assert z.is_even is False
    assert z.is_odd is False
    assert z.is_bounded is True
    assert z.is_unbounded is False
    assert z.is_finite is True
    assert z.is_infinitesimal is False
    assert z.is_comparable is True
    assert z.is_prime is False
    assert z.is_composite is False


def test_I():
    z = S.ImaginaryUnit
    assert z.is_commutative is True
    assert z.is_integer is False
    assert z.is_rational is False
    assert z.is_real is False
    assert z.is_complex is True
    assert z.is_noninteger is False
    assert z.is_irrational is False
    assert z.is_imaginary is True
    assert z.is_positive is False
    assert z.is_negative is False
    assert z.is_nonpositive is False
    assert z.is_nonnegative is False
    assert z.is_even is False
    assert z.is_odd is False
    assert z.is_bounded is True
    assert z.is_unbounded is False
    assert z.is_finite is True
    assert z.is_infinitesimal is False
    assert z.is_comparable is False
    assert z.is_prime is False
    assert z.is_composite is False


def test_symbol_real():
    # issue 749
    a = Symbol('a', real=False)

    assert a.is_real is False
    assert a.is_integer is False
    assert a.is_negative is False
    assert a.is_positive is False
    assert a.is_nonnegative is False
    assert a.is_nonpositive is False
    assert a.is_zero is False


def test_symbol_zero():
    x = Symbol('x', zero=True)
    assert x.is_positive is False
    assert x.is_nonpositive is True
    assert x.is_negative is False
    assert x.is_nonnegative is True
    assert x.is_zero is True
    assert x.is_nonzero is False


def test_symbol_positive():
    x = Symbol('x', positive=True)
    assert x.is_positive is True
    assert x.is_nonpositive is False
    assert x.is_negative is False
    assert x.is_nonnegative is True
    assert x.is_zero is False
    assert x.is_nonzero is True


def test_neg_symbol_positive():
    x = -Symbol('x', positive=True)
    assert x.is_positive is False
    assert x.is_nonpositive is True
    assert x.is_negative is True
    assert x.is_nonnegative is False
    assert x.is_zero is False
    assert x.is_nonzero is True


def test_symbol_nonpositive():
    x = Symbol('x', nonpositive=True)
    assert x.is_positive is False
    assert x.is_nonpositive is True
    assert x.is_negative is None
    assert x.is_nonnegative is None
    assert x.is_zero is None
    assert x.is_nonzero is None


def test_neg_symbol_nonpositive():
    x = -Symbol('x', nonpositive=True)
    assert x.is_positive is None
    assert x.is_nonpositive is None
    assert x.is_negative is False
    assert x.is_nonnegative is True
    assert x.is_zero is None
    assert x.is_nonzero is None


def test_symbol_falsepositive():
    x = Symbol('x', positive=False)
    assert x.is_positive is False
    assert x.is_nonpositive is None
    assert x.is_negative is None
    assert x.is_nonnegative is None
    assert x.is_zero is None
    assert x.is_nonzero is None


@XFAIL
def test_neg_symbol_falsepositive():
    x = -Symbol('x', positive=False)
    assert x.is_positive is None
    assert x.is_nonpositive is None
    assert x.is_negative is False  # this currently returns None
    assert x.is_nonnegative is None
    assert x.is_zero is None
    assert x.is_nonzero is None


def test_symbol_falsepositive_real():
    x = Symbol('x', positive=False, real=True)
    assert x.is_positive is False
    assert x.is_nonpositive is True
    assert x.is_negative is None
    assert x.is_nonnegative is None
    assert x.is_zero is None
    assert x.is_nonzero is None


def test_neg_symbol_falsepositive_real():
    x = -Symbol('x', positive=False, real=True)
    assert x.is_positive is None
    assert x.is_nonpositive is None
    assert x.is_negative is False
    assert x.is_nonnegative is True
    assert x.is_zero is None
    assert x.is_nonzero is None


def test_symbol_falsenonnegative():
    x = Symbol('x', nonnegative=False)
    assert x.is_positive is False
    assert x.is_nonpositive is None
    assert x.is_negative is None
    assert x.is_nonnegative is False
    assert x.is_zero is False
    assert x.is_nonzero is True


@XFAIL
def test_neg_symbol_falsenonnegative():
    x = -Symbol('x', nonnegative=False)
    assert x.is_positive is None
    assert x.is_nonpositive is False  # this currently returns None
    assert x.is_negative is False  # this currently returns None
    assert x.is_nonnegative is None
    assert x.is_zero is False  # this currently returns None
    assert x.is_nonzero is True  # this currently returns None


def test_symbol_falsenonnegative_real():
    x = Symbol('x', nonnegative=False, real=True)
    assert x.is_positive is False
    assert x.is_nonpositive is True
    assert x.is_negative is True
    assert x.is_nonnegative is False
    assert x.is_zero is False
    assert x.is_nonzero is True


def test_neg_symbol_falsenonnegative_real():
    x = -Symbol('x', nonnegative=False, real=True)
    assert x.is_positive is True
    assert x.is_nonpositive is False
    assert x.is_negative is False
    assert x.is_nonnegative is True
    assert x.is_zero is False
    assert x.is_nonzero is True


def test_prime():
    assert S(-1).is_prime is False
    assert S(-2).is_prime is False
    assert S(-4).is_prime is False
    assert S(0).is_prime is False
    assert S(1).is_prime is False
    assert S(2).is_prime is True
    assert S(17).is_prime is True
    assert S(4).is_prime is False


def test_composite():
    assert S(-1).is_composite is False
    assert S(-2).is_composite is False
    assert S(-4).is_composite is False
    assert S(0).is_composite is False
    assert S(2).is_composite is False
    assert S(17).is_composite is False
    assert S(4).is_composite is True


def test_prime_symbol():
    x = Symbol('x', prime=True)
    assert x.is_prime is True
    assert x.is_integer is True
    assert x.is_positive is True
    assert x.is_negative is False
    assert x.is_nonpositive is False
    assert x.is_nonnegative is True

    x = Symbol('x', prime=False)
    assert x.is_prime is False
    assert x.is_integer is None
    assert x.is_positive is None
    assert x.is_negative is None
    assert x.is_nonpositive is None
    assert x.is_nonnegative is None


def test_symbol_noncommutative():
    x = Symbol('x', commutative=True)
    assert x.is_complex is None

    x = Symbol('x', commutative=False)
    assert x.is_integer is False
    assert x.is_rational is False
    assert x.is_irrational is False
    assert x.is_real is False
    assert x.is_complex is False


def test_other_symbol():
    x = Symbol('x', integer=True)
    assert x.is_integer is True
    assert x.is_real is True

    x = Symbol('x', integer=True, nonnegative=True)
    assert x.is_integer is True
    assert x.is_nonnegative is True
    assert x.is_negative is False
    assert x.is_positive is None

    x = Symbol('x', integer=True, nonpositive=True)
    assert x.is_integer is True
    assert x.is_nonpositive is True
    assert x.is_positive is False
    assert x.is_negative is None

    x = Symbol('x', odd=True)
    assert x.is_odd is True
    assert x.is_even is False
    assert x.is_integer is True

    x = Symbol('x', odd=False)
    assert x.is_odd is False
    assert x.is_even is None
    assert x.is_integer is None

    x = Symbol('x', even=True)
    assert x.is_even is True
    assert x.is_odd is False
    assert x.is_integer is True

    x = Symbol('x', even=False)
    assert x.is_even is False
    assert x.is_odd is None
    assert x.is_integer is None

    x = Symbol('x', integer=True, nonnegative=True)
    assert x.is_integer is True
    assert x.is_nonnegative is True

    x = Symbol('x', integer=True, nonpositive=True)
    assert x.is_integer is True
    assert x.is_nonpositive is True

    with raises(AttributeError):
        x.is_real = False


def test_issue726():
    """catch: hash instability"""
    x = Symbol("x")
    y = Symbol("y")
    a1 = x + y
    a2 = y + x
    a2.is_comparable

    h1 = hash(a1)
    h2 = hash(a2)
    assert h1 == h2


def test_issue1723():
    z = (-1)**Rational(1, 3)*(1 - I*sqrt(3))
    assert z.is_real in [True, None]


def test_hash_vs_typeinfo():
    """seemingly different typeinfo, but in fact equal"""

    # the following two are semantically equal
    x1 = Symbol('x', even=True)
    x2 = Symbol('x', integer=True, odd=False)

    assert hash(x1) == hash(x2)
    assert x1 == x2


def test_hash_vs_typeinfo_2():
    """different typeinfo should mean !eq"""
    # the following two are semantically different
    x = Symbol('x')
    x1 = Symbol('x', even=True)

    assert x != x1
    assert hash(x) != hash(x1)  # This might fail with very low probability


def test_hash_vs_eq():
    """catch: different hash for equal objects"""
    a = 1 + S.Pi    # important: do not fold it into a Number instance
    ha = hash(a)  # it should be Add/Mul/... to trigger the bug

    a.is_positive   # this uses .evalf() and deduces it is positive
    assert a.is_positive is True

    # be sure that hash stayed the same
    assert ha == hash(a)

    # now b should be the same expression
    b = a.expand(trig=True)
    hb = hash(b)

    assert a == b
    assert ha == hb


def test_Add_is_pos_neg():
    # these cover lines not covered by the rest of tests in core
    n = Symbol('n', negative=True, bounded=False)
    p = Symbol('p', positive=True, bounded=False)
    x = Symbol('x')
    assert (n + p).is_positive is None
    assert (n + x).is_positive is False
    assert (p + x).is_positive is True
    assert (n + p).is_negative is None
    assert (n + x).is_negative is True
    assert (p + x).is_negative is False


def test_special_is_rational():
    i = Symbol('i', integer=True)
    r = Symbol('r', rational=True)
    x = Symbol('x')
    assert sqrt(3).is_rational is False
    assert (3 + sqrt(3)).is_rational is False
    assert (3*sqrt(3)).is_rational is False
    assert exp(3).is_rational is False
    assert exp(i).is_rational is False
    assert exp(r).is_rational is False
    assert exp(x).is_rational is None
    assert exp(log(3), evaluate=False).is_rational is True
    assert log(exp(3), evaluate=False).is_rational is True
    assert log(3).is_rational is False
    assert log(i).is_rational is False
    assert log(r).is_rational is False
    assert log(x).is_rational is None
    assert (sqrt(3) + sqrt(5)).is_rational is None
    assert (sqrt(3) + S.Pi).is_rational is None
    assert (x**i).is_rational is None
    assert (i**i).is_rational is True
    assert (r**i).is_rational is True
    assert (r**r).is_rational is None
    assert (r**x).is_rational is None
    assert sin(1).is_rational is False
    assert sin(i).is_rational is False
    assert sin(r).is_rational is False
    assert sin(x).is_rational is None
    assert asin(r).is_rational is False
    assert sin(asin(3), evaluate=False).is_rational is True


@XFAIL
def test_issue_3176():
    x = Symbol('x')
    # both zero or both Muls...but neither "change would be very appreciated.
    # This is similar to x/x => 1 even though if x = 0, it is really nan.
    assert isinstance(x*0, type(0*S.Infinity))
    if 0*S.Infinity is S.NaN:
        b = Symbol('b', bounded=None)
        assert (b*0).is_zero is None


def test_sanitize_assumptions():
    # issue 3567
    x = Symbol('x', real=1, positive=0)
    assert x.is_real is True
    assert x.is_positive is False


def test_special_assumptions():
    x = Symbol('x')
    z2 = z = Symbol('z', zero=True)
    assert z2 == z == S.Zero
    assert (2*z).is_positive is False
    assert (2*z).is_negative is False
    assert (2*z).is_zero is True
    assert (z2*z).is_positive is False
    assert (z2*z).is_negative is False
    assert (z2*z).is_zero is True

    e = -3 - sqrt(5) + (-sqrt(10)/2 - sqrt(2)/2)**2
    assert (e < 0) is S.false
    assert (e > 0) is S.false
    assert (e == 0) is False  # it's not a literal 0
    assert e.equals(0) is True


def test_inconsistent():
    # cf. issues 2696 and 2446
    raises(InconsistentAssumptions, lambda: Symbol('x', real=True,
           commutative=False))


def test_issue_3532():
    assert ((-1)**(I)).is_real is True
    assert ((-1)**(I*2)).is_real is True
    assert ((-1)**(I/2)).is_real is True
    assert ((-1)**(I*S.Pi)).is_real is True
    assert (I**(I + 2)).is_real is True


def test_gh2730():
    assert (1/(1 + I)).is_real is False