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_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_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 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 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_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 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 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 _(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_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_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_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
