def refine_abs(expr, assumptions):
"""
Handler for the absolute value.
Examples
========
>>> from sympy import Symbol, Q, refine, Abs
>>> from sympy.assumptions.refine import refine_abs
>>> from sympy.abc import x
>>> refine_abs(Abs(x), Q.real(x))
>>> refine_abs(Abs(x), Q.positive(x))
x
>>> refine_abs(Abs(x), Q.negative(x))
-x
"""
from sympy.core.logic import fuzzy_not
arg = expr.args[0]
if ask(Q.real(arg), assumptions) and fuzzy_not(ask(Q.negative(arg), assumptions)):
# if it's nonnegative
return arg
if ask(Q.negative(arg), assumptions):
return -arg
def refine_atan2(expr, assumptions):
"""
Handler for the atan2 function
Examples
========
>>> from sympy import Symbol, Q, refine, atan2
>>> from sympy.assumptions.refine import refine_atan2
>>> from sympy.abc import x, y
>>> refine_atan2(atan2(y,x), Q.real(y) & Q.positive(x))
atan(y/x)
>>> refine_atan2(atan2(y,x), Q.negative(y) & Q.negative(x))
atan(y/x) - pi
>>> refine_atan2(atan2(y,x), Q.positive(y) & Q.negative(x))
atan(y/x) + pi
"""
from sympy.functions.elementary.complexes import atan
from sympy.core import S
y, x = expr.args
if ask(Q.real(y) & Q.positive(x), assumptions):
return atan(y / x)
elif ask(Q.negative(y) & Q.negative(x), assumptions):
return atan(y / x) - S.Pi
elif ask(Q.positive(y) & Q.negative(x), assumptions):
return atan(y / x) + S.Pi
else:
return expr
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 Pow(expr, assumptions):
"""
Imaginary**integer -> Imaginary if integer % 2 == 1
Imaginary**integer -> real if integer % 2 == 0
Imaginary**Imaginary -> ?
Imaginary**Real -> ?
"""
if expr.is_number:
return AskImaginaryHandler._number(expr, assumptions)
if ask(Q.imaginary(expr.base), assumptions):
if ask(Q.real(expr.exp), assumptions):
if ask(Q.odd(expr.exp), assumptions):
return True
elif ask(Q.even(expr.exp), assumptions):
return False
elif ask(Q.real(expr.base), assumptions):
if ask(Q.real(expr.exp), assumptions):
if expr.exp.is_Rational and \
ask(Q.even(expr.exp.q), assumptions):
return ask(Q.negative(expr.base),assumptions)
elif ask(Q.integer(expr.exp), assumptions):
return False
elif ask(Q.positive(expr.base), assumptions):
return False
elif ask(Q.negative(expr.base), assumptions):
return True
def test_refine():
m0 = OperationsOnlyMatrix([[Abs(x)**2, sqrt(x**2)],
[sqrt(x**2)*Abs(y)**2, sqrt(y**2)*Abs(x)**2]])
m1 = m0.refine(Q.real(x) & Q.real(y))
assert m1 == Matrix([[x**2, Abs(x)], [y**2*Abs(x), x**2*Abs(y)]])
m1 = m0.refine(Q.positive(x) & Q.positive(y))
assert m1 == Matrix([[x**2, x], [x*y**2, x**2*y]])
m1 = m0.refine(Q.negative(x) & Q.negative(y))
assert m1 == Matrix([[x**2, -x], [-x*y**2, -x**2*y]])
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_extended_real():
x = symbols('x')
assert ask(Q.extended_real(x), Q.positive(x)) == True
assert ask(Q.extended_real(-x), Q.positive(x)) == True
assert ask(Q.extended_real(-x), Q.negative(x)) == True
assert ask(Q.extended_real(x + S.Infinity), Q.real(x)) == True
def Pow(expr, assumptions):
"""
Unbounded ** NonZero -> Unbounded
Bounded ** Bounded -> Bounded
Abs()<=1 ** Positive -> Bounded
Abs()>=1 ** Negative -> Bounded
Otherwise unknown
"""
base_bounded = ask(Q.bounded(expr.base), assumptions)
exp_bounded = ask(Q.bounded(expr.exp), assumptions)
if base_bounded is None and exp_bounded is None: # Common Case
return None
if base_bounded is False and ask(Q.nonzero(expr.exp), assumptions):
return False
if base_bounded and exp_bounded:
return True
if (abs(expr.base) <= 1) is True and ask(Q.positive(expr.exp),
assumptions):
return True
if (abs(expr.base) >= 1) is True and ask(Q.negative(expr.exp),
assumptions):
return True
if (abs(expr.base) >= 1) is True and exp_bounded is False:
return False
return None
def test_composite_proposition():
from sympy.logic.boolalg import Equivalent, Implies
x = symbols('x')
assert ask(True) is True
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
def test_functions_in_assumptions():
from sympy.logic.boolalg import Equivalent, Xor
x = symbols("x")
assert ask(x, Q.negative, Q.real(x) >> Q.positive(x)) is False
assert ask(x, Q.negative, Equivalent(Q.real(x), Q.positive(x))) is False
assert ask(x, Q.negative, Xor(Q.real(x), Q.negative(x))) is False
def test_extended_real():
x = symbols('x')
assert ask(Q.extended_real(x), Q.positive(x)) == True
assert ask(Q.extended_real(-x), Q.positive(x)) == True
assert ask(Q.extended_real(-x), Q.negative(x)) == True
assert ask(Q.extended_real(x+S.Infinity), Q.real(x)) == True
def Pow(expr, assumptions):
"""
Real**Integer -> Real
Positive**Real -> Real
Real**(Integer/Even) -> Real if base is nonnegative
Real**(Integer/Odd) -> Real
Real**Imaginary -> ?
Imaginary**Real -> ?
Real**Real -> ?
"""
if expr.is_number:
return AskRealHandler._number(expr, assumptions)
if ask(Q.imaginary(expr.base), assumptions):
if ask(Q.real(expr.exp), assumptions):
if ask(Q.odd(expr.exp), assumptions):
return False
elif ask(Q.even(expr.exp), assumptions):
return True
elif ask(Q.real(expr.base), assumptions):
if ask(Q.real(expr.exp), assumptions):
if expr.exp.is_Rational and \
ask(Q.even(expr.exp.q), assumptions):
return ask(Q.positive(expr.base), assumptions)
elif ask(Q.integer(expr.exp), assumptions):
return True
elif ask(Q.positive(expr.base), assumptions):
return True
elif ask(Q.negative(expr.base), assumptions):
return False
def _(expr, assumptions):
"""
* Unbounded ** NonZero -> Unbounded
* Bounded ** Bounded -> Bounded
* Abs()<=1 ** Positive -> Bounded
* Abs()>=1 ** Negative -> Bounded
* Otherwise unknown
"""
if expr.base == E:
return ask(Q.finite(expr.exp), assumptions)
base_bounded = ask(Q.finite(expr.base), assumptions)
exp_bounded = ask(Q.finite(expr.exp), assumptions)
if base_bounded is None and exp_bounded is None: # Common Case
return None
if base_bounded is False and ask(Q.nonzero(expr.exp), assumptions):
return False
if base_bounded and exp_bounded:
return True
if (abs(expr.base) <= 1) == True and ask(Q.positive(expr.exp),
assumptions):
return True
if (abs(expr.base) >= 1) == True and ask(Q.negative(expr.exp),
assumptions):
return True
if (abs(expr.base) >= 1) == True and exp_bounded is False:
return False
return None
def _(expr, assumptions):
return ask(Q.negative_infinite(expr)
| Q.negative(expr)
| Q.zero(expr)
| Q.positive(expr)
| Q.positive_infinite(expr),
assumptions)
def MatPow(expr, assumptions):
# only for integer powers
base, exp = expr.args
int_exp = ask(Q.integer(exp), assumptions)
if int_exp and ask(~Q.negative(exp), assumptions):
return ask(Q.fullrank(base), assumptions)
return None
def test_composite_proposition():
from sympy.logic.boolalg import Equivalent, Implies
x = symbols('x')
assert ask(True) is True
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
def MatPow(expr, assumptions):
# only for integer powers
base, exp = expr.args
int_exp = ask(Q.integer(exp), assumptions)
if int_exp and ask(~Q.negative(exp), assumptions):
return ask(Q.fullrank(base), assumptions)
return None
def Pow(expr, assumptions):
"""
Imaginary**integer/odd -> Imaginary
Imaginary**integer/even -> Real if integer % 2 == 0
b**Imaginary -> !Imaginary if exponent is an integer multiple of I*pi/log(b)
Imaginary**Real -> ?
Negative**even root -> Imaginary
Negative**odd root -> Real
Negative**Real -> Imaginary
Real**Integer -> Real
Real**Positive -> Real
"""
if expr.is_number:
return AskImaginaryHandler._number(expr, assumptions)
if expr.base.func == C.exp:
if ask(Q.imaginary(expr.base.args[0]), assumptions):
if ask(Q.imaginary(expr.exp), assumptions):
return False
i = expr.base.args[0] / I / pi
if ask(Q.integer(2 * i), assumptions):
return ask(Q.imaginary(((-1)**i)**expr.exp), assumptions)
if ask(Q.imaginary(expr.base), assumptions):
if ask(Q.integer(expr.exp), assumptions):
odd = ask(Q.odd(expr.exp), assumptions)
if odd is not None:
return odd
return
if ask(Q.imaginary(expr.exp), assumptions):
imlog = ask(Q.imaginary(C.log(expr.base)), assumptions)
if imlog is not None:
return False # I**i -> real; (2*I)**i -> complex ==> not imaginary
if ask(Q.real(expr.base), assumptions):
if ask(Q.real(expr.exp), assumptions):
if ask(
Q.rational(expr.exp) & Q.even(denom(expr.exp)),
assumptions):
return ask(Q.negative(expr.base), assumptions)
elif ask(Q.integer(expr.exp), assumptions):
return False
elif ask(Q.positive(expr.base), assumptions):
return False
elif ask(Q.negative(expr.base), assumptions):
return True
def log(expr, assumptions):
r = ask(Q.real(expr.args[0]), assumptions)
if r is not True:
return r
if ask(Q.positive(expr.args[0] - 1), assumptions):
return True
if ask(Q.negative(expr.args[0] - 1), assumptions):
return False
def log(expr, assumptions):
r = ask(Q.real(expr.args[0]), assumptions)
if r is not True:
return r
if ask(Q.positive(expr.args[0] - 1), assumptions):
return True
if ask(Q.negative(expr.args[0] - 1), assumptions):
return False
def Pow(expr, assumptions):
"""
Imaginary**integer/odd -> Imaginary
Imaginary**integer/even -> Real if integer % 2 == 0
b**Imaginary -> !Imaginary if exponent is an integer multiple of I*pi/log(b)
Imaginary**Real -> ?
Negative**even root -> Imaginary
Negative**odd root -> Real
Negative**Real -> Imaginary
Real**Integer -> Real
Real**Positive -> Real
"""
if expr.is_number:
return AskImaginaryHandler._number(expr, assumptions)
if expr.base.func == C.exp:
if ask(Q.imaginary(expr.base.args[0]), assumptions):
if ask(Q.imaginary(expr.exp), assumptions):
return False
i = expr.base.args[0]/I/pi
if ask(Q.integer(2*i), assumptions):
return ask(Q.imaginary(((-1)**i)**expr.exp), assumptions)
if ask(Q.imaginary(expr.base), assumptions):
if ask(Q.integer(expr.exp), assumptions):
odd = ask(Q.odd(expr.exp), assumptions)
if odd is not None:
return odd
return
if ask(Q.imaginary(expr.exp), assumptions):
imlog = ask(Q.imaginary(C.log(expr.base)), assumptions)
if imlog is not None:
return False # I**i -> real; (2*I)**i -> complex ==> not imaginary
if ask(Q.real(expr.base), assumptions):
if ask(Q.real(expr.exp), assumptions):
if ask(Q.rational(expr.exp) & Q.even(denom(expr.exp)), assumptions):
return ask(Q.negative(expr.base), assumptions)
elif ask(Q.integer(expr.exp), assumptions):
return False
elif ask(Q.positive(expr.base), assumptions):
return False
elif ask(Q.negative(expr.base), assumptions):
return True
def refine_atan2(expr, assumptions):
"""
Handler for the atan2 function
Examples
========
>>> from sympy import Symbol, Q, refine, atan2
>>> from sympy.assumptions.refine import refine_atan2
>>> from sympy.abc import x, y
>>> refine_atan2(atan2(y,x), Q.real(y) & Q.positive(x))
atan(y/x)
>>> refine_atan2(atan2(y,x), Q.negative(y) & Q.negative(x))
atan(y/x) - pi
>>> refine_atan2(atan2(y,x), Q.positive(y) & Q.negative(x))
atan(y/x) + pi
>>> refine_atan2(atan2(y,x), Q.zero(y) & Q.negative(x))
pi
>>> refine_atan2(atan2(y,x), Q.positive(y) & Q.zero(x))
pi/2
>>> refine_atan2(atan2(y,x), Q.negative(y) & Q.zero(x))
-pi/2
>>> refine_atan2(atan2(y,x), Q.zero(y) & Q.zero(x))
nan
"""
from sympy.functions.elementary.trigonometric import atan
from sympy.core import S
y, x = expr.args
if ask(Q.real(y) & Q.positive(x), assumptions):
return atan(y / x)
elif ask(Q.negative(y) & Q.negative(x), assumptions):
return atan(y / x) - S.Pi
elif ask(Q.positive(y) & Q.negative(x), assumptions):
return atan(y / x) + S.Pi
elif ask(Q.zero(y) & Q.negative(x), assumptions):
return S.Pi
elif ask(Q.positive(y) & Q.zero(x), assumptions):
return S.Pi / 2
elif ask(Q.negative(y) & Q.zero(x), assumptions):
return -S.Pi / 2
elif ask(Q.zero(y) & Q.zero(x), assumptions):
return S.NaN
else:
return expr
def Pow(expr, assumptions):
if expr.is_number: return expr.evalf() > 0
if ask(Q.positive(expr.base), assumptions):
return True
if ask(Q.negative(expr.base), assumptions):
if ask(Q.even(expr.exp), assumptions):
return True
if ask(Q.even(expr.exp), assumptions):
return False
def _(expr, assumptions):
"""
* Real**Integer -> Real
* Positive**Real -> Real
* Real**(Integer/Even) -> Real if base is nonnegative
* Real**(Integer/Odd) -> Real
* Imaginary**(Integer/Even) -> Real
* Imaginary**(Integer/Odd) -> not Real
* Imaginary**Real -> ? since Real could be 0 (giving real)
or 1 (giving imaginary)
* b**Imaginary -> Real if log(b) is imaginary and b != 0
and exponent != integer multiple of
I*pi/log(b)
* Real**Real -> ? e.g. sqrt(-1) is imaginary and
sqrt(2) is not
"""
if expr.is_number:
return _RealPredicate_number(expr, assumptions)
if expr.base.func == exp:
if ask(Q.imaginary(expr.base.args[0]), assumptions):
if ask(Q.imaginary(expr.exp), assumptions):
return True
# If the i = (exp's arg)/(I*pi) is an integer or half-integer
# multiple of I*pi then 2*i will be an integer. In addition,
# exp(i*I*pi) = (-1)**i so the overall realness of the expr
# can be determined by replacing exp(i*I*pi) with (-1)**i.
i = expr.base.args[0] / I / pi
if ask(Q.integer(2 * i), assumptions):
return ask(Q.real(((-1)**i)**expr.exp), assumptions)
return
if ask(Q.imaginary(expr.base), assumptions):
if ask(Q.integer(expr.exp), assumptions):
odd = ask(Q.odd(expr.exp), assumptions)
if odd is not None:
return not odd
return
if ask(Q.imaginary(expr.exp), assumptions):
imlog = ask(Q.imaginary(log(expr.base)), assumptions)
if imlog is not None:
# I**i -> real, log(I) is imag;
# (2*I)**i -> complex, log(2*I) is not imag
return imlog
if ask(Q.real(expr.base), assumptions):
if ask(Q.real(expr.exp), assumptions):
if expr.exp.is_Rational and \
ask(Q.even(expr.exp.q), assumptions):
return ask(Q.positive(expr.base), assumptions)
elif ask(Q.integer(expr.exp), assumptions):
return True
elif ask(Q.positive(expr.base), assumptions):
return True
elif ask(Q.negative(expr.base), assumptions):
return False
def Pow(expr, assumptions):
if expr.is_number: return expr.evalf() > 0
if ask(Q.positive(expr.base), assumptions):
return True
if ask(Q.negative(expr.base), assumptions):
if ask(Q.even(expr.exp), assumptions):
return True
if ask(Q.even(expr.exp), assumptions):
return 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 Pow(expr, assumptions):
if expr.is_number:
return AskEvenHandler._number(expr, assumptions)
if ask(Q.integer(expr.exp), assumptions):
if ask(Q.positive(expr.exp), assumptions):
return ask(Q.even(expr.base), assumptions)
elif ask(~Q.negative(expr.exp) & Q.odd(expr.base), assumptions):
return False
elif expr.base is S.NegativeOne:
return False
def Pow(expr, assumptions):
if expr.is_number:
return AskEvenHandler._number(expr, assumptions)
if ask(Q.integer(expr.exp), assumptions):
if ask(Q.positive(expr.exp), assumptions):
return ask(Q.even(expr.base), assumptions)
elif ask(~Q.negative(expr.exp) & Q.odd(expr.base), assumptions):
return False
elif expr.base is S.NegativeOne:
return False
def Mul(expr, assumptions):
if expr.is_number:
return AskPositiveHandler._number(expr, assumptions)
result = True
for arg in expr.args:
if ask(Q.positive(arg), assumptions): continue
elif ask(Q.negative(arg), assumptions):
result = result ^ True
else: return
return result
def _(expr, assumptions):
if expr.is_number:
return _EvenPredicate_number(expr, assumptions)
if ask(Q.integer(expr.exp), assumptions):
if ask(Q.positive(expr.exp), assumptions):
return ask(Q.even(expr.base), assumptions)
elif ask(~Q.negative(expr.exp) & Q.odd(expr.base), assumptions):
return False
elif expr.base is S.NegativeOne:
return False
def _(expr, assumptions):
"""
* Imaginary**Odd -> Imaginary
* Imaginary**Even -> Real
* b**Imaginary -> !Imaginary if exponent is an integer
multiple of I*pi/log(b)
* Imaginary**Real -> ?
* Positive**Real -> Real
* Negative**Integer -> Real
* Negative**(Integer/2) -> Imaginary
* Negative**Real -> not Imaginary if exponent is not Rational
"""
if expr.is_number:
return _Imaginary_number(expr, assumptions)
if expr.base == E:
a = expr.exp / I / pi
return ask(Q.integer(2 * a) & ~Q.integer(a), assumptions)
if expr.base.func == exp or (expr.base.is_Pow and expr.base.base == E):
if ask(Q.imaginary(expr.base.exp), assumptions):
if ask(Q.imaginary(expr.exp), assumptions):
return False
i = expr.base.exp / I / pi
if ask(Q.integer(2 * i), assumptions):
return ask(Q.imaginary((S.NegativeOne**i)**expr.exp),
assumptions)
if ask(Q.imaginary(expr.base), assumptions):
if ask(Q.integer(expr.exp), assumptions):
odd = ask(Q.odd(expr.exp), assumptions)
if odd is not None:
return odd
return
if ask(Q.imaginary(expr.exp), assumptions):
imlog = ask(Q.imaginary(log(expr.base)), assumptions)
if imlog is not None:
# I**i -> real; (2*I)**i -> complex ==> not imaginary
return False
if ask(Q.real(expr.base) & Q.real(expr.exp), assumptions):
if ask(Q.positive(expr.base), assumptions):
return False
else:
rat = ask(Q.rational(expr.exp), assumptions)
if not rat:
return rat
if ask(Q.integer(expr.exp), assumptions):
return False
else:
half = ask(Q.integer(2 * expr.exp), assumptions)
if half:
return ask(Q.negative(expr.base), assumptions)
return half
def MatPow(expr, assumptions):
# only for integer powers
base, exp = expr.args
int_exp = ask(Q.integer(exp), assumptions)
if not int_exp:
return None
non_negative = ask(~Q.negative(exp), assumptions)
if (non_negative or non_negative == False
and ask(Q.invertible(base), assumptions)):
return ask(Q.complex_elements(base), assumptions)
return None
def Pow(expr, assumptions):
if expr.is_number:
return AskPositiveHandler._number(expr, assumptions)
if ask(Q.positive(expr.base), assumptions):
if ask(Q.real(expr.exp), assumptions):
return True
if ask(Q.negative(expr.base), assumptions):
if ask(Q.even(expr.exp), assumptions):
return True
if ask(Q.odd(expr.exp), assumptions):
return False
def MatPow(expr, assumptions):
# only for integer powers
base, exp = expr.args
int_exp = ask(Q.integer(exp), assumptions)
if not int_exp:
return None
non_negative = ask(~Q.negative(exp), assumptions)
if (non_negative or non_negative == False
and ask(Q.invertible(base), assumptions)):
return ask(Q.complex_elements(base), assumptions)
return None
def _(expr, assumptions):
if expr.is_number:
return _PositivePredicate_number(expr, assumptions)
if ask(Q.positive(expr.base), assumptions):
if ask(Q.real(expr.exp), assumptions):
return True
if ask(Q.negative(expr.base), assumptions):
if ask(Q.even(expr.exp), assumptions):
return True
if ask(Q.odd(expr.exp), assumptions):
return False
def Pow(expr, assumptions):
if expr.is_number:
return AskPositiveHandler._number(expr, assumptions)
if ask(Q.positive(expr.base), assumptions):
if ask(Q.real(expr.exp), assumptions):
return True
if ask(Q.negative(expr.base), assumptions):
if ask(Q.even(expr.exp), assumptions):
return True
if ask(Q.odd(expr.exp), assumptions):
return False
def Mul(expr, assumptions):
if expr.is_number:
return AskPositiveHandler._number(expr, assumptions)
result = True
for arg in expr.args:
if ask(Q.positive(arg), assumptions): continue
elif ask(Q.negative(arg), assumptions):
result = result ^ True
else:
return
return result
def _(expr, assumptions):
# only for integer powers
base, exp = expr.args
int_exp = ask(Q.integer(exp), assumptions)
if not int_exp:
return None
non_negative = ask(~Q.negative(exp), assumptions)
if (non_negative
or non_negative == False and ask(Q.invertible(base), assumptions)):
return ask(Q.diagonal(base), assumptions)
return None
def _(expr, assumptions):
if expr.is_number:
return _PositivePredicate_number(expr, assumptions)
result = True
for arg in expr.args:
if ask(Q.positive(arg), assumptions):
continue
elif ask(Q.negative(arg), assumptions):
result = result ^ True
else:
return
return result
def Pow(expr, assumptions):
"""
Real**Integer -> Real
Positive**Real -> Real
Real**(Integer/Even) -> Real if base is nonnegative
Real**(Integer/Odd) -> Real
Imaginary**(Integer/Even) -> Real
Imaginary**(Integer/Odd) -> not Real
Imaginary**Real -> ? since Real could be 0 (giving real) or 1 (giving imaginary)
b**Imaginary -> Real if log(b) is imaginary and b != 0 and exponent != integer multiple of I*pi/log(b)
Real**Real -> ? e.g. sqrt(-1) is imaginary and sqrt(2) is not
"""
if expr.is_number:
return AskRealHandler._number(expr, assumptions)
if expr.base.func == exp:
if ask(Q.imaginary(expr.base.args[0]), assumptions):
if ask(Q.imaginary(expr.exp), assumptions):
return True
# If the i = (exp's arg)/(I*pi) is an integer or half-integer
# multiple of I*pi then 2*i will be an integer. In addition,
# exp(i*I*pi) = (-1)**i so the overall realness of the expr
# can be determined by replacing exp(i*I*pi) with (-1)**i.
i = expr.base.args[0]/I/pi
if ask(Q.integer(2*i), assumptions):
return ask(Q.real(((-1)**i)**expr.exp), assumptions)
return
if ask(Q.imaginary(expr.base), assumptions):
if ask(Q.integer(expr.exp), assumptions):
odd = ask(Q.odd(expr.exp), assumptions)
if odd is not None:
return not odd
return
if ask(Q.imaginary(expr.exp), assumptions):
imlog = ask(Q.imaginary(log(expr.base)), assumptions)
if imlog is not None:
# I**i -> real, log(I) is imag;
# (2*I)**i -> complex, log(2*I) is not imag
return imlog
if ask(Q.real(expr.base), assumptions):
if ask(Q.real(expr.exp), assumptions):
if expr.exp.is_Rational and \
ask(Q.even(expr.exp.q), assumptions):
return ask(Q.positive(expr.base), assumptions)
elif ask(Q.integer(expr.exp), assumptions):
return True
elif ask(Q.positive(expr.base), assumptions):
return True
elif ask(Q.negative(expr.base), assumptions):
return False
def Mul(expr, assumptions):
if expr.is_number:
return AskNegativeHandler._number(expr, assumptions)
result = None
for arg in expr.args:
if result is None: result = False
if ask(Q.negative(arg), assumptions):
result = not result
elif ask(Q.positive(arg), assumptions):
pass
else: return
return result
def test_positive():
x, y, z, w = symbols('x,y,z,w')
assert ask(Q.positive(x), Q.positive(x)) == True
assert ask(Q.positive(x), Q.negative(x)) == False
assert ask(Q.positive(x), Q.nonzero(x)) == None
assert ask(Q.positive(-x), Q.positive(x)) == False
assert ask(Q.positive(-x), Q.negative(x)) == True
assert ask(Q.positive(x + y), Q.positive(x) & Q.positive(y)) == True
assert ask(Q.positive(x + y), Q.positive(x) & Q.negative(y)) == None
assert ask(Q.positive(2 * x), Q.positive(x)) == True
assumptions = Q.positive(x) & Q.negative(y) & Q.negative(z) & Q.positive(w)
assert ask(Q.positive(x * y * z)) == None
assert ask(Q.positive(x * y * z), assumptions) == True
assert ask(Q.positive(-x * y * z), assumptions) == False
assert ask(Q.positive(x**2), Q.positive(x)) == True
assert ask(Q.positive(x**2), Q.negative(x)) == True
#exponential
assert ask(Q.positive(exp(x)), Q.real(x)) == True
assert ask(Q.positive(x + exp(x)), Q.real(x)) == None
#absolute value
assert ask(Q.positive(Abs(x))) == None # Abs(0) = 0
assert ask(Q.positive(Abs(x)), Q.positive(x)) == True
def test_positive():
x, y, z, w = symbols('x,y,z,w')
assert ask(Q.positive(x), Q.positive(x)) == True
assert ask(Q.positive(x), Q.negative(x)) == False
assert ask(Q.positive(x), Q.nonzero(x)) == None
assert ask(Q.positive(-x), Q.positive(x)) == False
assert ask(Q.positive(-x), Q.negative(x)) == True
assert ask(Q.positive(x+y), Q.positive(x) & Q.positive(y)) == True
assert ask(Q.positive(x+y), Q.positive(x) & Q.negative(y)) == None
assert ask(Q.positive(2*x), Q.positive(x)) == True
assumptions = Q.positive(x) & Q.negative(y) & Q.negative(z) & Q.positive(w)
assert ask(Q.positive(x*y*z)) == None
assert ask(Q.positive(x*y*z), assumptions) == True
assert ask(Q.positive(-x*y*z), assumptions) == False
assert ask(Q.positive(x**2), Q.positive(x)) == True
assert ask(Q.positive(x**2), Q.negative(x)) == True
#exponential
assert ask(Q.positive(exp(x)), Q.real(x)) == True
assert ask(Q.positive(x + exp(x)), Q.real(x)) == None
#absolute value
assert ask(Q.positive(Abs(x))) == None # Abs(0) = 0
assert ask(Q.positive(Abs(x)), Q.positive(x)) == True
def Add(expr, assumptions):
"""
Positive + Positive -> Positive,
Negative + Negative -> Negative
"""
if expr.is_number:
return AskNegativeHandler._number(expr, assumptions)
for arg in expr.args:
if not ask(Q.negative(arg), assumptions):
break
else:
# if all argument's are negative
return True
def Add(expr, assumptions):
if expr.is_number:
return AskPositiveHandler._number(expr, assumptions)
nonneg = 0
for arg in expr.args:
if ask(Q.positive(arg), assumptions) is not True:
if ask(Q.negative(arg), assumptions) is False:
nonneg += 1
else:
break
else:
if nonneg < len(expr.args):
return True
def Mul(expr, assumptions):
if expr.is_number:
return AskNegativeHandler._number(expr, assumptions)
result = None
for arg in expr.args:
if result is None: result = False
if ask(Q.negative(arg), assumptions):
result = not result
elif ask(Q.positive(arg), assumptions):
pass
else:
return
return result
def Add(expr, assumptions):
"""
Positive + Positive -> Positive,
Negative + Negative -> Negative
"""
if expr.is_number:
return AskNegativeHandler._number(expr, assumptions)
for arg in expr.args:
if not ask(Q.negative(arg), assumptions):
break
else:
# if all argument's are negative
return True
def refine_sign(expr, assumptions):
"""
Handler for sign.
Examples
========
>>> from sympy.assumptions.refine import refine_sign
>>> from sympy import Symbol, Q, sign, im
>>> x = Symbol('x', real = True)
>>> expr = sign(x)
>>> refine_sign(expr, Q.positive(x) & Q.nonzero(x))
1
>>> refine_sign(expr, Q.negative(x) & Q.nonzero(x))
-1
>>> refine_sign(expr, Q.zero(x))
0
>>> y = Symbol('y', imaginary = True)
>>> expr = sign(y)
>>> refine_sign(expr, Q.positive(im(y)))
I
>>> refine_sign(expr, Q.negative(im(y)))
-I
"""
arg = expr.args[0]
if ask(Q.zero(arg), assumptions):
return S.Zero
if ask(Q.real(arg)):
if ask(Q.positive(arg), assumptions):
return S.One
if ask(Q.negative(arg), assumptions):
return S.NegativeOne
if ask(Q.imaginary(arg)):
arg_re, arg_im = arg.as_real_imag()
if ask(Q.positive(arg_im), assumptions):
return S.ImaginaryUnit
if ask(Q.negative(arg_im), assumptions):
return -S.ImaginaryUnit
return expr
def Add(expr, assumptions):
if expr.is_number:
return AskPositiveHandler._number(expr, assumptions)
nonneg = 0
for arg in expr.args:
if ask(Q.positive(arg), assumptions) is not True:
if ask(Q.negative(arg), assumptions) is False:
nonneg += 1
else:
break
else:
if nonneg < len(expr.args):
return True
def _(expr, assumptions):
if expr.is_number:
return _NegativePredicate_number(expr, assumptions)
result = None
for arg in expr.args:
if result is None:
result = False
if ask(Q.negative(arg), assumptions):
result = not result
elif ask(Q.positive(arg), assumptions):
pass
else:
return
return result
def refine_abs(expr, assumptions):
"""
Handler for the absolute value.
Examples::
>>> from sympy import Symbol, Q, refine, Abs
>>> from sympy.assumptions.refine import refine_abs
>>> from sympy.abc import x
>>> refine_abs(Abs(x), Q.real(x))
>>> refine_abs(Abs(x), Q.positive(x))
x
>>> refine_abs(Abs(x), Q.negative(x))
-x
"""
arg = expr.args[0]
if ask(Q.real(arg), assumptions) and \
fuzzy_not(ask(Q.negative(arg), assumptions)):
# if it's nonnegative
return arg
if ask(Q.negative(arg), assumptions):
return -arg
def refine_abs(expr, assumptions):
"""
Handler for the absolute value.
Examples
========
>>> from sympy import Symbol, Q, refine, Abs
>>> from sympy.assumptions.refine import refine_abs
>>> from sympy.abc import x
>>> refine_abs(Abs(x), Q.real(x))
>>> refine_abs(Abs(x), Q.positive(x))
x
>>> refine_abs(Abs(x), Q.negative(x))
-x
"""
from sympy.core.logic import fuzzy_not
from sympy import Abs
arg = expr.args[0]
if ask(Q.real(arg), assumptions) and \
fuzzy_not(ask(Q.negative(arg), assumptions)):
# if it's nonnegative
return arg
if ask(Q.negative(arg), assumptions):
return -arg
# arg is Mul
if isinstance(arg, Mul):
r = [refine(abs(a), assumptions) for a in arg.args]
non_abs = []
in_abs = []
for i in r:
if isinstance(i, Abs):
in_abs.append(i.args[0])
else:
non_abs.append(i)
return Mul(*non_abs) * Abs(Mul(*in_abs))
def Pow(expr, assumptions):
"""
Real ** Even -> NonNegative
Real ** Odd -> same_as_base
NonNegative ** Positive -> NonNegative
"""
if expr.is_number:
return AskNegativeHandler._number(expr, assumptions)
if ask(Q.real(expr.base), assumptions):
if ask(Q.positive(expr.base), assumptions):
return False
if ask(Q.even(expr.exp), assumptions):
return False
if ask(Q.odd(expr.exp), assumptions):
return ask(Q.negative(expr.base), assumptions)
def Pow(expr, assumptions):
"""
Imaginary**Odd -> Imaginary
Imaginary**Even -> Real
b**Imaginary -> !Imaginary if exponent is an integer multiple of I*pi/log(b)
Imaginary**Real -> ?
Positive**Real -> Real
Negative**Integer -> Real
Negative**(Integer/2) -> Imaginary
Negative**Real -> not Imaginary if exponent is not Rational
"""
if expr.is_number:
return AskImaginaryHandler._number(expr, assumptions)
if expr.base.func == exp:
if ask(Q.imaginary(expr.base.args[0]), assumptions):
if ask(Q.imaginary(expr.exp), assumptions):
return False
i = expr.base.args[0]/I/pi
if ask(Q.integer(2*i), assumptions):
return ask(Q.imaginary(((-1)**i)**expr.exp), assumptions)
if ask(Q.imaginary(expr.base), assumptions):
if ask(Q.integer(expr.exp), assumptions):
odd = ask(Q.odd(expr.exp), assumptions)
if odd is not None:
return odd
return
if ask(Q.imaginary(expr.exp), assumptions):
imlog = ask(Q.imaginary(log(expr.base)), assumptions)
if imlog is not None:
return False # I**i -> real; (2*I)**i -> complex ==> not imaginary
if ask(Q.real(expr.base) & Q.real(expr.exp), assumptions):
if ask(Q.positive(expr.base), assumptions):
return False
else:
rat = ask(Q.rational(expr.exp), assumptions)
if not rat:
return rat
if ask(Q.integer(expr.exp), assumptions):
return False
else:
half = ask(Q.integer(2*expr.exp), assumptions)
if half:
return ask(Q.negative(expr.base), assumptions)
return half
def test_bounded():
x, y = symbols('x,y')
assert ask(Q.bounded(x)) == False
assert ask(Q.bounded(x), Q.bounded(x)) == True
assert ask(Q.bounded(x), Q.bounded(y)) == False
assert ask(Q.bounded(x), Q.complex(x)) == False
assert ask(Q.bounded(x+1)) == False
assert ask(Q.bounded(x+1), Q.bounded(x)) == True
assert ask(Q.bounded(x+y)) == None
assert ask(Q.bounded(x+y), Q.bounded(x)) == False
assert ask(Q.bounded(x+1), Q.bounded(x) & Q.bounded(y)) == True
assert ask(Q.bounded(2*x)) == False
assert ask(Q.bounded(2*x), Q.bounded(x)) == True
assert ask(Q.bounded(x*y)) == None
assert ask(Q.bounded(x*y), Q.bounded(x)) == False
assert ask(Q.bounded(x*y), Q.bounded(x) & Q.bounded(y)) == True
assert ask(Q.bounded(x**2)) == False
assert ask(Q.bounded(2**x)) == False
assert ask(Q.bounded(2**x), Q.bounded(x)) == True
assert ask(Q.bounded(x**x)) == False
assert ask(Q.bounded(Rational(1,2) ** x)) == None
assert ask(Q.bounded(Rational(1,2) ** x), Q.positive(x)) == True
assert ask(Q.bounded(Rational(1,2) ** x), Q.negative(x)) == False
assert ask(Q.bounded(sqrt(x))) == False
# sign function
assert ask(Q.bounded(sign(x))) == True
assert ask(Q.bounded(sign(x)), ~Q.bounded(x)) == True
# exponential functions
assert ask(Q.bounded(log(x))) == False
assert ask(Q.bounded(log(x)), Q.bounded(x)) == True
assert ask(Q.bounded(exp(x))) == False
assert ask(Q.bounded(exp(x)), Q.bounded(x)) == True
assert ask(Q.bounded(exp(2))) == True
# trigonometric functions
assert ask(Q.bounded(sin(x))) == True
assert ask(Q.bounded(sin(x)), ~Q.bounded(x)) == True
assert ask(Q.bounded(cos(x))) == True
assert ask(Q.bounded(cos(x)), ~Q.bounded(x)) == True
assert ask(Q.bounded(2*sin(x))) == True
assert ask(Q.bounded(sin(x)**2)) == True
assert ask(Q.bounded(cos(x)**2)) == True
assert ask(Q.bounded(cos(x) + sin(x))) == True
def Pow(expr, assumptions):
"""
Unbounded ** Whatever -> Unbounded
Bounded ** Unbounded -> Unbounded if base > 1
Bounded ** Unbounded -> Unbounded if base < 1
"""
base_bounded = ask(Q.bounded(expr.base), assumptions)
if not base_bounded:
return False
if ask(Q.bounded(expr.exp), assumptions):# and base_bounded:
return True
if expr.base.is_number:# and base_bounded and not exp_bounded:
# We need to implement relations for this
if abs(expr.base) > 1:
return False
return ask(~Q.negative(expr.exp), assumptions)
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
