def test_float_1(): z = 1.0 assert ask(Q.commutative(z)) == True assert ask(Q.integer(z)) == True assert ask(Q.rational(z)) == True assert ask(Q.real(z)) == True assert ask(Q.complex(z)) == True assert ask(Q.irrational(z)) == False assert ask(Q.imaginary(z)) == False assert ask(Q.positive(z)) == True assert ask(Q.negative(z)) == False assert ask(Q.even(z)) == False assert ask(Q.odd(z)) == True assert ask(Q.bounded(z)) == True assert ask(Q.infinitesimal(z)) == False assert ask(Q.prime(z)) == False assert ask(Q.composite(z)) == True z = 7.2123 assert ask(Q.commutative(z)) == True assert ask(Q.integer(z)) == False assert ask(Q.rational(z)) == True assert ask(Q.real(z)) == True assert ask(Q.complex(z)) == True assert ask(Q.irrational(z)) == False assert ask(Q.imaginary(z)) == False assert ask(Q.positive(z)) == True assert ask(Q.negative(z)) == False assert ask(Q.even(z)) == False assert ask(Q.odd(z)) == False assert ask(Q.bounded(z)) == True assert ask(Q.infinitesimal(z)) == False assert ask(Q.prime(z)) == False assert ask(Q.composite(z)) == False
def test_negative(): x, y = symbols('x,y') assert ask(Q.negative(x), Q.negative(x)) == True assert ask(Q.negative(x), Q.positive(x)) == False assert ask(Q.negative(x), ~Q.real(x)) == False assert ask(Q.negative(x), Q.prime(x)) == False assert ask(Q.negative(x), ~Q.prime(x)) == None assert ask(Q.negative(-x), Q.positive(x)) == True assert ask(Q.negative(-x), ~Q.positive(x)) == None assert ask(Q.negative(-x), Q.negative(x)) == False assert ask(Q.negative(-x), Q.positive(x)) == True assert ask(Q.negative(x - 1), Q.negative(x)) == True assert ask(Q.negative(x + y)) == None assert ask(Q.negative(x + y), Q.negative(x)) == None assert ask(Q.negative(x + y), Q.negative(x) & Q.negative(y)) == True assert ask(Q.negative(x**2)) == None assert ask(Q.negative(x**2), Q.real(x)) == False assert ask(Q.negative(x**1.4), Q.real(x)) == None assert ask(Q.negative(x * y)) == None assert ask(Q.negative(x * y), Q.positive(x) & Q.positive(y)) == False assert ask(Q.negative(x * y), Q.positive(x) & Q.negative(y)) == True assert ask(Q.negative(x * y), Q.complex(x) & Q.complex(y)) == None assert ask(Q.negative(x**y)) == None assert ask(Q.negative(x**y), Q.negative(x) & Q.even(y)) == False assert ask(Q.negative(x**y), Q.negative(x) & Q.odd(y)) == True assert ask(Q.negative(x**y), Q.positive(x) & Q.integer(y)) == False assert ask(Q.negative(Abs(x))) == False
def test_negative(): x, y = symbols('x,y') assert ask(Q.negative(x), Q.negative(x)) == True assert ask(Q.negative(x), Q.positive(x)) == False assert ask(Q.negative(x), ~Q.real(x)) == False assert ask(Q.negative(x), Q.prime(x)) == False assert ask(Q.negative(x), ~Q.prime(x)) == None assert ask(Q.negative(-x), Q.positive(x)) == True assert ask(Q.negative(-x), ~Q.positive(x)) == None assert ask(Q.negative(-x), Q.negative(x)) == False assert ask(Q.negative(-x), Q.positive(x)) == True assert ask(Q.negative(x-1), Q.negative(x)) == True assert ask(Q.negative(x+y)) == None assert ask(Q.negative(x+y), Q.negative(x)) == None assert ask(Q.negative(x+y), Q.negative(x) & Q.negative(y)) == True assert ask(Q.negative(x**2)) == None assert ask(Q.negative(x**2), Q.real(x)) == False assert ask(Q.negative(x**1.4), Q.real(x)) == None assert ask(Q.negative(x*y)) == None assert ask(Q.negative(x*y), Q.positive(x) & Q.positive(y)) == False assert ask(Q.negative(x*y), Q.positive(x) & Q.negative(y)) == True assert ask(Q.negative(x*y), Q.complex(x) & Q.complex(y)) == None assert ask(Q.negative(x**y)) == None assert ask(Q.negative(x**y), Q.negative(x) & Q.even(y)) == False assert ask(Q.negative(x**y), Q.negative(x) & Q.odd(y)) == True assert ask(Q.negative(x**y), Q.positive(x) & Q.integer(y)) == False assert ask(Q.negative(Abs(x))) == False
def test_I(): I = S.ImaginaryUnit z = I assert ask(Q.commutative(z)) == True assert ask(Q.integer(z)) == False assert ask(Q.rational(z)) == False assert ask(Q.real(z)) == False assert ask(Q.complex(z)) == True assert ask(Q.irrational(z)) == False assert ask(Q.imaginary(z)) == True assert ask(Q.positive(z)) == False assert ask(Q.negative(z)) == False assert ask(Q.even(z)) == False assert ask(Q.odd(z)) == False assert ask(Q.bounded(z)) == True assert ask(Q.infinitesimal(z)) == False assert ask(Q.prime(z)) == False assert ask(Q.composite(z)) == False z = 1 + I assert ask(Q.commutative(z)) == True assert ask(Q.integer(z)) == False assert ask(Q.rational(z)) == False assert ask(Q.real(z)) == False assert ask(Q.complex(z)) == True assert ask(Q.irrational(z)) == False assert ask(Q.imaginary(z)) == False assert ask(Q.positive(z)) == False assert ask(Q.negative(z)) == False assert ask(Q.even(z)) == False assert ask(Q.odd(z)) == False assert ask(Q.bounded(z)) == True assert ask(Q.infinitesimal(z)) == False assert ask(Q.prime(z)) == False assert ask(Q.composite(z)) == False z = I * (1 + I) assert ask(Q.commutative(z)) == True assert ask(Q.integer(z)) == False assert ask(Q.rational(z)) == False assert ask(Q.real(z)) == False assert ask(Q.complex(z)) == True assert ask(Q.irrational(z)) == False assert ask(Q.imaginary(z)) == False assert ask(Q.positive(z)) == False assert ask(Q.negative(z)) == False assert ask(Q.even(z)) == False assert ask(Q.odd(z)) == False assert ask(Q.bounded(z)) == True assert ask(Q.infinitesimal(z)) == False assert ask(Q.prime(z)) == False assert ask(Q.composite(z)) == False
def test_I(): I = S.ImaginaryUnit z = I assert ask(Q.commutative(z)) == True assert ask(Q.integer(z)) == False assert ask(Q.rational(z)) == False assert ask(Q.real(z)) == False assert ask(Q.complex(z)) == True assert ask(Q.irrational(z)) == False assert ask(Q.imaginary(z)) == True assert ask(Q.positive(z)) == False assert ask(Q.negative(z)) == False assert ask(Q.even(z)) == False assert ask(Q.odd(z)) == False assert ask(Q.bounded(z)) == True assert ask(Q.infinitesimal(z)) == False assert ask(Q.prime(z)) == False assert ask(Q.composite(z)) == False z = 1 + I assert ask(Q.commutative(z)) == True assert ask(Q.integer(z)) == False assert ask(Q.rational(z)) == False assert ask(Q.real(z)) == False assert ask(Q.complex(z)) == True assert ask(Q.irrational(z)) == False assert ask(Q.imaginary(z)) == False assert ask(Q.positive(z)) == False assert ask(Q.negative(z)) == False assert ask(Q.even(z)) == False assert ask(Q.odd(z)) == False assert ask(Q.bounded(z)) == True assert ask(Q.infinitesimal(z)) == False assert ask(Q.prime(z)) == False assert ask(Q.composite(z)) == False z = I*(1+I) assert ask(Q.commutative(z)) == True assert ask(Q.integer(z)) == False assert ask(Q.rational(z)) == False assert ask(Q.real(z)) == False assert ask(Q.complex(z)) == True assert ask(Q.irrational(z)) == False assert ask(Q.imaginary(z)) == False assert ask(Q.positive(z)) == False assert ask(Q.negative(z)) == False assert ask(Q.even(z)) == False assert ask(Q.odd(z)) == False assert ask(Q.bounded(z)) == True assert ask(Q.infinitesimal(z)) == False assert ask(Q.prime(z)) == False assert ask(Q.composite(z)) == False
def test_imaginary(): x, y, z = symbols('x,y,z') I = S.ImaginaryUnit assert ask(Q.imaginary(x)) == None assert ask(Q.imaginary(x), Q.real(x)) == False assert ask(Q.imaginary(x), Q.prime(x)) == False assert ask(Q.imaginary(x + 1), Q.real(x)) == False assert ask(Q.imaginary(x + 1), Q.imaginary(x)) == False assert ask(Q.imaginary(x + I), Q.real(x)) == False assert ask(Q.imaginary(x + I), Q.imaginary(x)) == True assert ask(Q.imaginary(x + y), Q.imaginary(x) & Q.imaginary(y)) == True assert ask(Q.imaginary(x + y), Q.real(x) & Q.real(y)) == False assert ask(Q.imaginary(x + y), Q.imaginary(x) & Q.real(y)) == False assert ask(Q.imaginary(x + y), Q.complex(x) & Q.real(y)) == None assert ask(Q.imaginary(I * x), Q.real(x)) == True assert ask(Q.imaginary(I * x), Q.imaginary(x)) == False assert ask(Q.imaginary(I * x), Q.complex(x)) == None assert ask(Q.imaginary(x * y), Q.imaginary(x) & Q.real(y)) == True assert ask(Q.imaginary(x + y + z), Q.real(x) & Q.real(y) & Q.real(z)) == False assert ask(Q.imaginary(x + y + z), Q.real(x) & Q.real(y) & Q.imaginary(z)) == None assert ask(Q.imaginary(x + y + z), Q.real(x) & Q.imaginary(y) & Q.imaginary(z)) == False
def Basic(expr, assumptions): _positive = ask(Q.positive(expr), assumptions) if _positive: _integer = ask(Q.integer(expr), assumptions) if _integer: _prime = ask(Q.prime(expr), assumptions) if _prime is None: return return not _prime else: return _integer else: return _positive
def Pow(expr, assumptions): """ Rational ** Integer -> Rational Irrational ** Rational -> Irrational Rational ** Irrational -> ? """ if ask(Q.integer(expr.exp), assumptions): return ask(Q.rational(expr.base), assumptions) elif ask(Q.rational(expr.exp), assumptions): if ask(Q.prime(expr.base), assumptions): return False
def test_incompatible_resolutors(): x = symbols('x') class Prime2AskHandler(AskHandler): @staticmethod def Number(expr, assumptions): return True register_handler('prime', Prime2AskHandler) raises(ValueError, 'ask(Q.prime(4))') remove_handler('prime', Prime2AskHandler) class InconclusiveHandler(AskHandler): @staticmethod def Number(expr, assumptions): return None register_handler('prime', InconclusiveHandler) assert ask(Q.prime(3)) == True
def test_E(): z = S.Exp1 assert ask(Q.commutative(z)) == True assert ask(Q.integer(z)) == False assert ask(Q.rational(z)) == False assert ask(Q.real(z)) == True assert ask(Q.complex(z)) == True assert ask(Q.irrational(z)) == True assert ask(Q.imaginary(z)) == False assert ask(Q.positive(z)) == True assert ask(Q.negative(z)) == False assert ask(Q.even(z)) == False assert ask(Q.odd(z)) == False assert ask(Q.bounded(z)) == True assert ask(Q.infinitesimal(z)) == False assert ask(Q.prime(z)) == False assert ask(Q.composite(z)) == False
def test_zero_0(): z = Integer(0) assert ask(Q.nonzero(z)) == False assert ask(Q.commutative(z)) == True assert ask(Q.integer(z)) == True assert ask(Q.rational(z)) == True assert ask(Q.real(z)) == True assert ask(Q.complex(z)) == True assert ask(Q.imaginary(z)) == False assert ask(Q.positive(z)) == False assert ask(Q.negative(z)) == False assert ask(Q.even(z)) == True assert ask(Q.odd(z)) == False assert ask(Q.bounded(z)) == True assert ask(Q.infinitesimal(z)) == True assert ask(Q.prime(z)) == False assert ask(Q.composite(z)) == False
def _(expr, assumptions): _positive = ask(Q.positive(expr), assumptions) if _positive: _integer = ask(Q.integer(expr), assumptions) if _integer: _prime = ask(Q.prime(expr), assumptions) if _prime is None: return # Positive integer which is not prime is not # necessarily composite if expr.equals(1): return False return not _prime else: return _integer else: return _positive
def Basic(expr, assumptions): _positive = ask(Q.positive(expr), assumptions) if _positive: _integer = ask(Q.integer(expr), assumptions) if _integer: _prime = ask(Q.prime(expr), assumptions) if _prime is None: return # Positive integer which is not prime is not # necessarily composite if expr.equals(1): return False return not _prime else: return _integer else: return _positive
def _(expr, assumptions): """ * Rational ** Integer -> Rational * Irrational ** Rational -> Irrational * Rational ** Irrational -> ? """ if expr.base == E: x = expr.exp if ask(Q.rational(x), assumptions): return ask(~Q.nonzero(x), assumptions) return if ask(Q.integer(expr.exp), assumptions): return ask(Q.rational(expr.base), assumptions) elif ask(Q.rational(expr.exp), assumptions): if ask(Q.prime(expr.base), assumptions): return False
def test_Rational_number(): r = Rational(3, 4) assert ask(Q.commutative(r)) == True assert ask(Q.integer(r)) == False assert ask(Q.rational(r)) == True assert ask(Q.real(r)) == True assert ask(Q.complex(r)) == True assert ask(Q.irrational(r)) == False assert ask(Q.imaginary(r)) == False assert ask(Q.positive(r)) == True assert ask(Q.negative(r)) == False assert ask(Q.even(r)) == False assert ask(Q.odd(r)) == False assert ask(Q.bounded(r)) == True assert ask(Q.infinitesimal(r)) == False assert ask(Q.prime(r)) == False assert ask(Q.composite(r)) == False r = Rational(1, 4) assert ask(Q.positive(r)) == True assert ask(Q.negative(r)) == False r = Rational(5, 4) assert ask(Q.negative(r)) == False assert ask(Q.positive(r)) == True r = Rational(5, 3) assert ask(Q.positive(r)) == True assert ask(Q.negative(r)) == False r = Rational(-3, 4) assert ask(Q.positive(r)) == False assert ask(Q.negative(r)) == True r = Rational(-1, 4) assert ask(Q.positive(r)) == False assert ask(Q.negative(r)) == True r = Rational(-5, 4) assert ask(Q.negative(r)) == True assert ask(Q.positive(r)) == False r = Rational(-5, 3) assert ask(Q.positive(r)) == False assert ask(Q.negative(r)) == True
def test_Rational_number(): r = Rational(3,4) assert ask(Q.commutative(r)) == True assert ask(Q.integer(r)) == False assert ask(Q.rational(r)) == True assert ask(Q.real(r)) == True assert ask(Q.complex(r)) == True assert ask(Q.irrational(r)) == False assert ask(Q.imaginary(r)) == False assert ask(Q.positive(r)) == True assert ask(Q.negative(r)) == False assert ask(Q.even(r)) == False assert ask(Q.odd(r)) == False assert ask(Q.bounded(r)) == True assert ask(Q.infinitesimal(r)) == False assert ask(Q.prime(r)) == False assert ask(Q.composite(r)) == False r = Rational(1,4) assert ask(Q.positive(r)) == True assert ask(Q.negative(r)) == False r = Rational(5,4) assert ask(Q.negative(r)) == False assert ask(Q.positive(r)) == True r = Rational(5,3) assert ask(Q.positive(r)) == True assert ask(Q.negative(r)) == False r = Rational(-3,4) assert ask(Q.positive(r)) == False assert ask(Q.negative(r)) == True r = Rational(-1,4) assert ask(Q.positive(r)) == False assert ask(Q.negative(r)) == True r = Rational(-5,4) assert ask(Q.negative(r)) == True assert ask(Q.positive(r)) == False r = Rational(-5,3) assert ask(Q.positive(r)) == False assert ask(Q.negative(r)) == True
def test_nan(): nan = S.NaN assert ask(Q.commutative(nan)) == True assert ask(Q.integer(nan)) == False assert ask(Q.rational(nan)) == False assert ask(Q.real(nan)) == False assert ask(Q.extended_real(nan)) == False assert ask(Q.complex(nan)) == False assert ask(Q.irrational(nan)) == False assert ask(Q.imaginary(nan)) == False assert ask(Q.positive(nan)) == False assert ask(Q.nonzero(nan)) == True assert ask(Q.even(nan)) == False assert ask(Q.odd(nan)) == False assert ask(Q.bounded(nan)) == False assert ask(Q.infinitesimal(nan)) == False assert ask(Q.prime(nan)) == False assert ask(Q.composite(nan)) == False
def test_neg_infinity(): mm = S.NegativeInfinity assert ask(Q.commutative(mm)) == True assert ask(Q.integer(mm)) == False assert ask(Q.rational(mm)) == False assert ask(Q.real(mm)) == False assert ask(Q.extended_real(mm)) == True assert ask(Q.complex(mm)) == False assert ask(Q.irrational(mm)) == False assert ask(Q.imaginary(mm)) == False assert ask(Q.positive(mm)) == False assert ask(Q.negative(mm)) == True assert ask(Q.even(mm)) == False assert ask(Q.odd(mm)) == False assert ask(Q.bounded(mm)) == False assert ask(Q.infinitesimal(mm)) == False assert ask(Q.prime(mm)) == False assert ask(Q.composite(mm)) == False
def test_infinity(): oo = S.Infinity assert ask(Q.commutative(oo)) == True assert ask(Q.integer(oo)) == False assert ask(Q.rational(oo)) == False assert ask(Q.real(oo)) == False assert ask(Q.extended_real(oo)) == True assert ask(Q.complex(oo)) == False assert ask(Q.irrational(oo)) == False assert ask(Q.imaginary(oo)) == False assert ask(Q.positive(oo)) == True assert ask(Q.negative(oo)) == False assert ask(Q.even(oo)) == False assert ask(Q.odd(oo)) == False assert ask(Q.bounded(oo)) == False assert ask(Q.infinitesimal(oo)) == False assert ask(Q.prime(oo)) == False assert ask(Q.composite(oo)) == False
def test_real(): x, y = symbols('x,y') assert ask(Q.real(x)) == None assert ask(Q.real(x), Q.real(x)) == True assert ask(Q.real(x), Q.nonzero(x)) == True assert ask(Q.real(x), Q.positive(x)) == True assert ask(Q.real(x), Q.negative(x)) == True assert ask(Q.real(x), Q.integer(x)) == True assert ask(Q.real(x), Q.even(x)) == True assert ask(Q.real(x), Q.prime(x)) == True assert ask(Q.real(x / sqrt(2)), Q.real(x)) == True assert ask(Q.real(x / sqrt(-2)), Q.real(x)) == False I = S.ImaginaryUnit assert ask(Q.real(x + 1), Q.real(x)) == True assert ask(Q.real(x + I), Q.real(x)) == False assert ask(Q.real(x + I), Q.complex(x)) == None assert ask(Q.real(2 * x), Q.real(x)) == True assert ask(Q.real(I * x), Q.real(x)) == False assert ask(Q.real(I * x), Q.imaginary(x)) == True assert ask(Q.real(I * x), Q.complex(x)) == None assert ask(Q.real(x**2), Q.real(x)) == True assert ask(Q.real(sqrt(x)), Q.negative(x)) == False assert ask(Q.real(x**y), Q.real(x) & Q.integer(y)) == True assert ask(Q.real(x**y), Q.real(x) & Q.real(y)) == None assert ask(Q.real(x**y), Q.positive(x) & Q.real(y)) == True # trigonometric functions assert ask(Q.real(sin(x))) == None assert ask(Q.real(cos(x))) == None assert ask(Q.real(sin(x)), Q.real(x)) == True assert ask(Q.real(cos(x)), Q.real(x)) == True # exponential function assert ask(Q.real(exp(x))) == None assert ask(Q.real(exp(x)), Q.real(x)) == True assert ask(Q.real(x + exp(x)), Q.real(x)) == True # Q.complexes assert ask(Q.real(re(x))) == True assert ask(Q.real(im(x))) == True
def test_real(): x, y = symbols('x,y') assert ask(Q.real(x)) == None assert ask(Q.real(x), Q.real(x)) == True assert ask(Q.real(x), Q.nonzero(x)) == True assert ask(Q.real(x), Q.positive(x)) == True assert ask(Q.real(x), Q.negative(x)) == True assert ask(Q.real(x), Q.integer(x)) == True assert ask(Q.real(x), Q.even(x)) == True assert ask(Q.real(x), Q.prime(x)) == True assert ask(Q.real(x/sqrt(2)), Q.real(x)) == True assert ask(Q.real(x/sqrt(-2)), Q.real(x)) == False I = S.ImaginaryUnit assert ask(Q.real(x+1), Q.real(x)) == True assert ask(Q.real(x+I), Q.real(x)) == False assert ask(Q.real(x+I), Q.complex(x)) == None assert ask(Q.real(2*x), Q.real(x)) == True assert ask(Q.real(I*x), Q.real(x)) == False assert ask(Q.real(I*x), Q.imaginary(x)) == True assert ask(Q.real(I*x), Q.complex(x)) == None assert ask(Q.real(x**2), Q.real(x)) == True assert ask(Q.real(sqrt(x)), Q.negative(x)) == False assert ask(Q.real(x**y), Q.real(x) & Q.integer(y)) == True assert ask(Q.real(x**y), Q.real(x) & Q.real(y)) == None assert ask(Q.real(x**y), Q.positive(x) & Q.real(y)) == True # trigonometric functions assert ask(Q.real(sin(x))) == None assert ask(Q.real(cos(x))) == None assert ask(Q.real(sin(x)), Q.real(x)) == True assert ask(Q.real(cos(x)), Q.real(x)) == True # exponential function assert ask(Q.real(exp(x))) == None assert ask(Q.real(exp(x)), Q.real(x)) == True assert ask(Q.real(x + exp(x)), Q.real(x)) == True # Q.complexes assert ask(Q.real(re(x))) == True assert ask(Q.real(im(x))) == True
def test_type_extensibility(): """test that new types can be added to the ask system at runtime We create a custom type MyType, and override ask Q.prime=True with handler MyAskHandler for this type TODO: test incompatible resolutors """ from sympy.core import Basic class MyType(Basic): pass class MyAskHandler(AskHandler): @staticmethod def MyType(expr, assumptions): return True a = MyType() register_handler(Q.prime, MyAskHandler) assert ask(Q.prime(a)) == True
def test_integer(): x = symbols('x') assert ask(Q.integer(x)) == None assert ask(Q.integer(x), Q.integer(x)) == True assert ask(Q.integer(x), ~Q.integer(x)) == False assert ask(Q.integer(x), ~Q.real(x)) == False assert ask(Q.integer(x), ~Q.positive(x)) == None assert ask(Q.integer(x), Q.even(x) | Q.odd(x)) == True assert ask(Q.integer(2*x), Q.integer(x)) == True assert ask(Q.integer(2*x), Q.even(x)) == True assert ask(Q.integer(2*x), Q.prime(x)) == True assert ask(Q.integer(2*x), Q.rational(x)) == None assert ask(Q.integer(2*x), Q.real(x)) == None assert ask(Q.integer(sqrt(2)*x), Q.integer(x)) == False assert ask(Q.integer(x/2), Q.odd(x)) == False assert ask(Q.integer(x/2), Q.even(x)) == True assert ask(Q.integer(x/3), Q.odd(x)) == None assert ask(Q.integer(x/3), Q.even(x)) == None
def test_integer(): x = symbols('x') assert ask(Q.integer(x)) == None assert ask(Q.integer(x), Q.integer(x)) == True assert ask(Q.integer(x), ~Q.integer(x)) == False assert ask(Q.integer(x), ~Q.real(x)) == False assert ask(Q.integer(x), ~Q.positive(x)) == None assert ask(Q.integer(x), Q.even(x) | Q.odd(x)) == True assert ask(Q.integer(2 * x), Q.integer(x)) == True assert ask(Q.integer(2 * x), Q.even(x)) == True assert ask(Q.integer(2 * x), Q.prime(x)) == True assert ask(Q.integer(2 * x), Q.rational(x)) == None assert ask(Q.integer(2 * x), Q.real(x)) == None assert ask(Q.integer(sqrt(2) * x), Q.integer(x)) == False assert ask(Q.integer(x / 2), Q.odd(x)) == False assert ask(Q.integer(x / 2), Q.even(x)) == True assert ask(Q.integer(x / 3), Q.odd(x)) == None assert ask(Q.integer(x / 3), Q.even(x)) == None
def test_imaginary(): x, y, z = symbols('x,y,z') I = S.ImaginaryUnit assert ask(Q.imaginary(x)) == None assert ask(Q.imaginary(x), Q.real(x)) == False assert ask(Q.imaginary(x), Q.prime(x)) == False assert ask(Q.imaginary(x+1), Q.real(x)) == False assert ask(Q.imaginary(x+1), Q.imaginary(x)) == False assert ask(Q.imaginary(x+I), Q.real(x)) == False assert ask(Q.imaginary(x+I), Q.imaginary(x)) == True assert ask(Q.imaginary(x+y), Q.imaginary(x) & Q.imaginary(y)) == True assert ask(Q.imaginary(x+y), Q.real(x) & Q.real(y)) == False assert ask(Q.imaginary(x+y), Q.imaginary(x) & Q.real(y)) == False assert ask(Q.imaginary(x+y), Q.complex(x) & Q.real(y)) == None assert ask(Q.imaginary(I*x), Q.real(x)) == True assert ask(Q.imaginary(I*x), Q.imaginary(x)) == False assert ask(Q.imaginary(I*x), Q.complex(x)) == None assert ask(Q.imaginary(x*y), Q.imaginary(x) & Q.real(y)) == True assert ask(Q.imaginary(x+y+z), Q.real(x) & Q.real(y) & Q.real(z)) == False assert ask(Q.imaginary(x+y+z), Q.real(x) & Q.real(y) & Q.imaginary(z)) == None assert ask(Q.imaginary(x+y+z), Q.real(x) & Q.imaginary(y) & Q.imaginary(z)) == False
def test_pi(): z = S.Pi assert ask(Q.commutative(z)) == True assert ask(Q.integer(z)) == False assert ask(Q.rational(z)) == False assert ask(Q.real(z)) == True assert ask(Q.complex(z)) == True assert ask(Q.irrational(z)) == True assert ask(Q.imaginary(z)) == False assert ask(Q.positive(z)) == True assert ask(Q.negative(z)) == False assert ask(Q.even(z)) == False assert ask(Q.odd(z)) == False assert ask(Q.bounded(z)) == True assert ask(Q.infinitesimal(z)) == False assert ask(Q.prime(z)) == False assert ask(Q.composite(z)) == False z = S.Pi + 1 assert ask(Q.commutative(z)) == True assert ask(Q.integer(z)) == False assert ask(Q.rational(z)) == False assert ask(Q.real(z)) == True assert ask(Q.complex(z)) == True assert ask(Q.irrational(z)) == True assert ask(Q.imaginary(z)) == False assert ask(Q.positive(z)) == True assert ask(Q.negative(z)) == False assert ask(Q.even(z)) == False assert ask(Q.odd(z)) == False assert ask(Q.bounded(z)) == True assert ask(Q.infinitesimal(z)) == False assert ask(Q.prime(z)) == False assert ask(Q.composite(z)) == False z = 2*S.Pi assert ask(Q.commutative(z)) == True assert ask(Q.integer(z)) == False assert ask(Q.rational(z)) == False assert ask(Q.real(z)) == True assert ask(Q.complex(z)) == True assert ask(Q.irrational(z)) == True assert ask(Q.imaginary(z)) == False assert ask(Q.positive(z)) == True assert ask(Q.negative(z)) == False assert ask(Q.even(z)) == False assert ask(Q.odd(z)) == False assert ask(Q.bounded(z)) == True assert ask(Q.infinitesimal(z)) == False assert ask(Q.prime(z)) == False assert ask(Q.composite(z)) == False z = S.Pi ** 2 assert ask(Q.commutative(z)) == True assert ask(Q.integer(z)) == False assert ask(Q.rational(z)) == False assert ask(Q.real(z)) == True assert ask(Q.complex(z)) == True assert ask(Q.irrational(z)) == True assert ask(Q.imaginary(z)) == False assert ask(Q.positive(z)) == True assert ask(Q.negative(z)) == False assert ask(Q.even(z)) == False assert ask(Q.odd(z)) == False assert ask(Q.bounded(z)) == True assert ask(Q.infinitesimal(z)) == False assert ask(Q.prime(z)) == False assert ask(Q.composite(z)) == False z = (1+S.Pi) ** 2 assert ask(Q.commutative(z)) == True assert ask(Q.integer(z)) == False assert ask(Q.rational(z)) == False assert ask(Q.real(z)) == True assert ask(Q.complex(z)) == True assert ask(Q.irrational(z)) == True assert ask(Q.imaginary(z)) == False assert ask(Q.positive(z)) == True assert ask(Q.negative(z)) == False assert ask(Q.even(z)) == False assert ask(Q.odd(z)) == False assert ask(Q.bounded(z)) == True assert ask(Q.infinitesimal(z)) == False assert ask(Q.prime(z)) == False assert ask(Q.composite(z)) == False
def test_pi(): z = S.Pi assert ask(Q.commutative(z)) == True assert ask(Q.integer(z)) == False assert ask(Q.rational(z)) == False assert ask(Q.real(z)) == True assert ask(Q.complex(z)) == True assert ask(Q.irrational(z)) == True assert ask(Q.imaginary(z)) == False assert ask(Q.positive(z)) == True assert ask(Q.negative(z)) == False assert ask(Q.even(z)) == False assert ask(Q.odd(z)) == False assert ask(Q.bounded(z)) == True assert ask(Q.infinitesimal(z)) == False assert ask(Q.prime(z)) == False assert ask(Q.composite(z)) == False z = S.Pi + 1 assert ask(Q.commutative(z)) == True assert ask(Q.integer(z)) == False assert ask(Q.rational(z)) == False assert ask(Q.real(z)) == True assert ask(Q.complex(z)) == True assert ask(Q.irrational(z)) == True assert ask(Q.imaginary(z)) == False assert ask(Q.positive(z)) == True assert ask(Q.negative(z)) == False assert ask(Q.even(z)) == False assert ask(Q.odd(z)) == False assert ask(Q.bounded(z)) == True assert ask(Q.infinitesimal(z)) == False assert ask(Q.prime(z)) == False assert ask(Q.composite(z)) == False z = 2 * S.Pi assert ask(Q.commutative(z)) == True assert ask(Q.integer(z)) == False assert ask(Q.rational(z)) == False assert ask(Q.real(z)) == True assert ask(Q.complex(z)) == True assert ask(Q.irrational(z)) == True assert ask(Q.imaginary(z)) == False assert ask(Q.positive(z)) == True assert ask(Q.negative(z)) == False assert ask(Q.even(z)) == False assert ask(Q.odd(z)) == False assert ask(Q.bounded(z)) == True assert ask(Q.infinitesimal(z)) == False assert ask(Q.prime(z)) == False assert ask(Q.composite(z)) == False z = S.Pi**2 assert ask(Q.commutative(z)) == True assert ask(Q.integer(z)) == False assert ask(Q.rational(z)) == False assert ask(Q.real(z)) == True assert ask(Q.complex(z)) == True assert ask(Q.irrational(z)) == True assert ask(Q.imaginary(z)) == False assert ask(Q.positive(z)) == True assert ask(Q.negative(z)) == False assert ask(Q.even(z)) == False assert ask(Q.odd(z)) == False assert ask(Q.bounded(z)) == True assert ask(Q.infinitesimal(z)) == False assert ask(Q.prime(z)) == False assert ask(Q.composite(z)) == False z = (1 + S.Pi)**2 assert ask(Q.commutative(z)) == True assert ask(Q.integer(z)) == False assert ask(Q.rational(z)) == False assert ask(Q.real(z)) == True assert ask(Q.complex(z)) == True assert ask(Q.irrational(z)) == True assert ask(Q.imaginary(z)) == False assert ask(Q.positive(z)) == True assert ask(Q.negative(z)) == False assert ask(Q.even(z)) == False assert ask(Q.odd(z)) == False assert ask(Q.bounded(z)) == True assert ask(Q.infinitesimal(z)) == False assert ask(Q.prime(z)) == False assert ask(Q.composite(z)) == False
def get_known_facts(x=None): """ Facts between unary predicates. Parameters ========== x : Symbol, optional Placeholder symbol for unary facts. Default is ``Symbol('x')``. Returns ======= fact : Known facts in conjugated normal form. """ if x is None: x = Symbol('x') fact = And( # primitive predicates for extended real exclude each other. Exclusive(Q.negative_infinite(x), Q.negative(x), Q.zero(x), Q.positive(x), Q.positive_infinite(x)), # build complex plane Exclusive(Q.real(x), Q.imaginary(x)), Implies(Q.real(x) | Q.imaginary(x), Q.complex(x)), # other subsets of complex Exclusive(Q.transcendental(x), Q.algebraic(x)), Equivalent(Q.real(x), Q.rational(x) | Q.irrational(x)), Exclusive(Q.irrational(x), Q.rational(x)), Implies(Q.rational(x), Q.algebraic(x)), # integers Exclusive(Q.even(x), Q.odd(x)), Implies(Q.integer(x), Q.rational(x)), Implies(Q.zero(x), Q.even(x)), Exclusive(Q.composite(x), Q.prime(x)), Implies(Q.composite(x) | Q.prime(x), Q.integer(x) & Q.positive(x)), Implies(Q.even(x) & Q.positive(x) & ~Q.prime(x), Q.composite(x)), # hermitian and antihermitian Implies(Q.real(x), Q.hermitian(x)), Implies(Q.imaginary(x), Q.antihermitian(x)), Implies(Q.zero(x), Q.hermitian(x) | Q.antihermitian(x)), # define finity and infinity, and build extended real line Exclusive(Q.infinite(x), Q.finite(x)), Implies(Q.complex(x), Q.finite(x)), Implies( Q.negative_infinite(x) | Q.positive_infinite(x), Q.infinite(x)), # commutativity Implies(Q.finite(x) | Q.infinite(x), Q.commutative(x)), # matrices Implies(Q.orthogonal(x), Q.positive_definite(x)), Implies(Q.orthogonal(x), Q.unitary(x)), Implies(Q.unitary(x) & Q.real_elements(x), Q.orthogonal(x)), Implies(Q.unitary(x), Q.normal(x)), Implies(Q.unitary(x), Q.invertible(x)), Implies(Q.normal(x), Q.square(x)), Implies(Q.diagonal(x), Q.normal(x)), Implies(Q.positive_definite(x), Q.invertible(x)), Implies(Q.diagonal(x), Q.upper_triangular(x)), Implies(Q.diagonal(x), Q.lower_triangular(x)), Implies(Q.lower_triangular(x), Q.triangular(x)), Implies(Q.upper_triangular(x), Q.triangular(x)), Implies(Q.triangular(x), Q.upper_triangular(x) | Q.lower_triangular(x)), Implies(Q.upper_triangular(x) & Q.lower_triangular(x), Q.diagonal(x)), Implies(Q.diagonal(x), Q.symmetric(x)), Implies(Q.unit_triangular(x), Q.triangular(x)), Implies(Q.invertible(x), Q.fullrank(x)), Implies(Q.invertible(x), Q.square(x)), Implies(Q.symmetric(x), Q.square(x)), Implies(Q.fullrank(x) & Q.square(x), Q.invertible(x)), Equivalent(Q.invertible(x), ~Q.singular(x)), Implies(Q.integer_elements(x), Q.real_elements(x)), Implies(Q.real_elements(x), Q.complex_elements(x)), ) return fact
def test_prime(): x, y = symbols('x,y') assert ask(Q.prime(x), Q.prime(x)) == True assert ask(Q.prime(x), ~Q.prime(x)) == False assert ask(Q.prime(x), Q.integer(x)) == None assert ask(Q.prime(x), ~Q.integer(x)) == False assert ask(Q.prime(2 * x), Q.integer(x)) == False assert ask(Q.prime(x * y)) == None assert ask(Q.prime(x * y), Q.prime(x)) == None assert ask(Q.prime(x * y), Q.integer(x) & Q.integer(y)) == False assert ask(Q.prime(x**2), Q.integer(x)) == False assert ask(Q.prime(x**2), Q.prime(x)) == False assert ask(Q.prime(x**y), Q.integer(x) & Q.integer(y)) == False
def test_prime(): x, y = symbols('x,y') assert ask(Q.prime(x), Q.prime(x)) == True assert ask(Q.prime(x), ~Q.prime(x)) == False assert ask(Q.prime(x), Q.integer(x)) == None assert ask(Q.prime(x), ~Q.integer(x)) == False assert ask(Q.prime(2*x), Q.integer(x)) == False assert ask(Q.prime(x*y)) == None assert ask(Q.prime(x*y), Q.prime(x)) == None assert ask(Q.prime(x*y), Q.integer(x) & Q.integer(y)) == False assert ask(Q.prime(x**2), Q.integer(x)) == False assert ask(Q.prime(x**2), Q.prime(x)) == False assert ask(Q.prime(x**y), Q.integer(x) & Q.integer(y)) == False