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_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):
"""
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_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 Basic(expr, assumptions):
_real = ask(Q.real(expr), assumptions)
if _real:
_rational = ask(Q.rational(expr), assumptions)
if _rational is None: return None
return not _rational
else: return _real
def test_odd():
x, y, z, t = symbols('x,y,z,t')
assert ask(Q.odd(x)) == None
assert ask(Q.odd(x), Q.odd(x)) == True
assert ask(Q.odd(x), Q.integer(x)) == None
assert ask(Q.odd(x), ~Q.integer(x)) == False
assert ask(Q.odd(x), Q.rational(x)) == None
assert ask(Q.odd(x), Q.positive(x)) == None
assert ask(Q.odd(-x), Q.odd(x)) == True
assert ask(Q.odd(2*x)) == None
assert ask(Q.odd(2*x), Q.integer(x)) == False
assert ask(Q.odd(2*x), Q.odd(x)) == False
assert ask(Q.odd(2*x), Q.irrational(x)) == False
assert ask(Q.odd(2*x), ~Q.integer(x)) == None
assert ask(Q.odd(3*x), Q.integer(x)) == None
assert ask(Q.odd(x/3), Q.odd(x)) == None
assert ask(Q.odd(x/3), Q.even(x)) == None
assert ask(Q.odd(x+1), Q.even(x)) == True
assert ask(Q.odd(x+2), Q.even(x)) == False
assert ask(Q.odd(x+2), Q.odd(x)) == True
assert ask(Q.odd(3-x), Q.odd(x)) == False
assert ask(Q.odd(3-x), Q.even(x)) == True
assert ask(Q.odd(3+x), Q.odd(x)) == False
assert ask(Q.odd(3+x), Q.even(x)) == True
assert ask(Q.odd(x+y), Q.odd(x) & Q.odd(y)) == False
assert ask(Q.odd(x+y), Q.odd(x) & Q.even(y)) == True
assert ask(Q.odd(x-y), Q.even(x) & Q.odd(y)) == True
assert ask(Q.odd(x-y), Q.odd(x) & Q.odd(y)) == False
assert ask(Q.odd(x+y+z), Q.odd(x) & Q.odd(y) & Q.even(z)) == False
assert ask(Q.odd(x+y+z+t),
Q.odd(x) & Q.odd(y) & Q.even(z) & Q.integer(t)) == None
assert ask(Q.odd(2*x + 1), Q.integer(x)) == True
assert ask(Q.odd(2*x + y), Q.integer(x) & Q.odd(y)) == True
assert ask(Q.odd(2*x + y), Q.integer(x) & Q.even(y)) == False
assert ask(Q.odd(2*x + y), Q.integer(x) & Q.integer(y)) == None
assert ask(Q.odd(x*y), Q.odd(x) & Q.even(y)) == False
assert ask(Q.odd(x*y), Q.odd(x) & Q.odd(y)) == True
assert ask(Q.odd(2*x*y), Q.rational(x) & Q.rational(x)) == None
assert ask(Q.odd(2*x*y), Q.irrational(x) & Q.irrational(x)) == None
assert ask(Q.odd(Abs(x)), Q.odd(x)) == True
def test_odd():
x, y, z, t = symbols('x,y,z,t')
assert ask(Q.odd(x)) == None
assert ask(Q.odd(x), Q.odd(x)) == True
assert ask(Q.odd(x), Q.integer(x)) == None
assert ask(Q.odd(x), ~Q.integer(x)) == False
assert ask(Q.odd(x), Q.rational(x)) == None
assert ask(Q.odd(x), Q.positive(x)) == None
assert ask(Q.odd(-x), Q.odd(x)) == True
assert ask(Q.odd(2 * x)) == None
assert ask(Q.odd(2 * x), Q.integer(x)) == False
assert ask(Q.odd(2 * x), Q.odd(x)) == False
assert ask(Q.odd(2 * x), Q.irrational(x)) == False
assert ask(Q.odd(2 * x), ~Q.integer(x)) == None
assert ask(Q.odd(3 * x), Q.integer(x)) == None
assert ask(Q.odd(x / 3), Q.odd(x)) == None
assert ask(Q.odd(x / 3), Q.even(x)) == None
assert ask(Q.odd(x + 1), Q.even(x)) == True
assert ask(Q.odd(x + 2), Q.even(x)) == False
assert ask(Q.odd(x + 2), Q.odd(x)) == True
assert ask(Q.odd(3 - x), Q.odd(x)) == False
assert ask(Q.odd(3 - x), Q.even(x)) == True
assert ask(Q.odd(3 + x), Q.odd(x)) == False
assert ask(Q.odd(3 + x), Q.even(x)) == True
assert ask(Q.odd(x + y), Q.odd(x) & Q.odd(y)) == False
assert ask(Q.odd(x + y), Q.odd(x) & Q.even(y)) == True
assert ask(Q.odd(x - y), Q.even(x) & Q.odd(y)) == True
assert ask(Q.odd(x - y), Q.odd(x) & Q.odd(y)) == False
assert ask(Q.odd(x + y + z), Q.odd(x) & Q.odd(y) & Q.even(z)) == False
assert ask(Q.odd(x + y + z + t),
Q.odd(x) & Q.odd(y) & Q.even(z) & Q.integer(t)) == None
assert ask(Q.odd(2 * x + 1), Q.integer(x)) == True
assert ask(Q.odd(2 * x + y), Q.integer(x) & Q.odd(y)) == True
assert ask(Q.odd(2 * x + y), Q.integer(x) & Q.even(y)) == False
assert ask(Q.odd(2 * x + y), Q.integer(x) & Q.integer(y)) == None
assert ask(Q.odd(x * y), Q.odd(x) & Q.even(y)) == False
assert ask(Q.odd(x * y), Q.odd(x) & Q.odd(y)) == True
assert ask(Q.odd(2 * x * y), Q.rational(x) & Q.rational(x)) == None
assert ask(Q.odd(2 * x * y), Q.irrational(x) & Q.irrational(x)) == None
assert ask(Q.odd(Abs(x)), Q.odd(x)) == True
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 _(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 test_even():
x, y, z, t = symbols('x,y,z,t')
assert ask(Q.even(x)) == None
assert ask(Q.even(x), Q.integer(x)) == None
assert ask(Q.even(x), ~Q.integer(x)) == False
assert ask(Q.even(x), Q.rational(x)) == None
assert ask(Q.even(x), Q.positive(x)) == None
assert ask(Q.even(2 * x)) == None
assert ask(Q.even(2 * x), Q.integer(x)) == True
assert ask(Q.even(2 * x), Q.even(x)) == True
assert ask(Q.even(2 * x), Q.irrational(x)) == False
assert ask(Q.even(2 * x), Q.odd(x)) == True
assert ask(Q.even(2 * x), ~Q.integer(x)) == None
assert ask(Q.even(3 * x), Q.integer(x)) == None
assert ask(Q.even(3 * x), Q.even(x)) == True
assert ask(Q.even(3 * x), Q.odd(x)) == False
assert ask(Q.even(x + 1), Q.odd(x)) == True
assert ask(Q.even(x + 1), Q.even(x)) == False
assert ask(Q.even(x + 2), Q.odd(x)) == False
assert ask(Q.even(x + 2), Q.even(x)) == True
assert ask(Q.even(7 - x), Q.odd(x)) == True
assert ask(Q.even(7 + x), Q.odd(x)) == True
assert ask(Q.even(x + y), Q.odd(x) & Q.odd(y)) == True
assert ask(Q.even(x + y), Q.odd(x) & Q.even(y)) == False
assert ask(Q.even(x + y), Q.even(x) & Q.even(y)) == True
assert ask(Q.even(2 * x + 1), Q.integer(x)) == False
assert ask(Q.even(2 * x * y), Q.rational(x) & Q.rational(x)) == None
assert ask(Q.even(2 * x * y), Q.irrational(x) & Q.irrational(x)) == None
assert ask(Q.even(x + y + z), Q.odd(x) & Q.odd(y) & Q.even(z)) == True
assert ask(Q.even(x + y + z + t),
Q.odd(x) & Q.odd(y) & Q.even(z) & Q.integer(t)) == None
assert ask(Q.even(Abs(x)), Q.even(x)) == True
assert ask(Q.even(Abs(x)), ~Q.even(x)) == None
assert ask(Q.even(re(x)), Q.even(x)) == True
assert ask(Q.even(re(x)), ~Q.even(x)) == None
assert ask(Q.even(im(x)), Q.even(x)) == True
assert ask(Q.even(im(x)), Q.real(x)) == True
def test_even():
x, y, z, t = symbols('x,y,z,t')
assert ask(Q.even(x)) == None
assert ask(Q.even(x), Q.integer(x)) == None
assert ask(Q.even(x), ~Q.integer(x)) == False
assert ask(Q.even(x), Q.rational(x)) == None
assert ask(Q.even(x), Q.positive(x)) == None
assert ask(Q.even(2*x)) == None
assert ask(Q.even(2*x), Q.integer(x)) == True
assert ask(Q.even(2*x), Q.even(x)) == True
assert ask(Q.even(2*x), Q.irrational(x)) == False
assert ask(Q.even(2*x), Q.odd(x)) == True
assert ask(Q.even(2*x), ~Q.integer(x)) == None
assert ask(Q.even(3*x), Q.integer(x)) == None
assert ask(Q.even(3*x), Q.even(x)) == True
assert ask(Q.even(3*x), Q.odd(x)) == False
assert ask(Q.even(x+1), Q.odd(x)) == True
assert ask(Q.even(x+1), Q.even(x)) == False
assert ask(Q.even(x+2), Q.odd(x)) == False
assert ask(Q.even(x+2), Q.even(x)) == True
assert ask(Q.even(7-x), Q.odd(x)) == True
assert ask(Q.even(7+x), Q.odd(x)) == True
assert ask(Q.even(x+y), Q.odd(x) & Q.odd(y)) == True
assert ask(Q.even(x+y), Q.odd(x) & Q.even(y)) == False
assert ask(Q.even(x+y), Q.even(x) & Q.even(y)) == True
assert ask(Q.even(2*x + 1), Q.integer(x)) == False
assert ask(Q.even(2*x*y), Q.rational(x) & Q.rational(x)) == None
assert ask(Q.even(2*x*y), Q.irrational(x) & Q.irrational(x)) == None
assert ask(Q.even(x+y+z), Q.odd(x) & Q.odd(y) & Q.even(z)) == True
assert ask(Q.even(x+y+z+t),
Q.odd(x) & Q.odd(y) & Q.even(z) & Q.integer(t)) == None
assert ask(Q.even(Abs(x)), Q.even(x)) == True
assert ask(Q.even(Abs(x)), ~Q.even(x)) == None
assert ask(Q.even(re(x)), Q.even(x)) == True
assert ask(Q.even(re(x)), ~Q.even(x)) == None
assert ask(Q.even(im(x)), Q.even(x)) == True
assert ask(Q.even(im(x)), Q.real(x)) == True
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 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 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 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_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_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 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 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_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_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_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_complex():
x, y = symbols('x,y')
assert ask(Q.complex(x)) == None
assert ask(Q.complex(x), Q.complex(x)) == True
assert ask(Q.complex(x), Q.complex(y)) == None
assert ask(Q.complex(x), ~Q.complex(x)) == False
assert ask(Q.complex(x), Q.real(x)) == True
assert ask(Q.complex(x), ~Q.real(x)) == None
assert ask(Q.complex(x), Q.rational(x)) == True
assert ask(Q.complex(x), Q.irrational(x)) == True
assert ask(Q.complex(x), Q.positive(x)) == True
assert ask(Q.complex(x), Q.imaginary(x)) == True
# a+b
assert ask(Q.complex(x + 1), Q.complex(x)) == True
assert ask(Q.complex(x + 1), Q.real(x)) == True
assert ask(Q.complex(x + 1), Q.rational(x)) == True
assert ask(Q.complex(x + 1), Q.irrational(x)) == True
assert ask(Q.complex(x + 1), Q.imaginary(x)) == True
assert ask(Q.complex(x + 1), Q.integer(x)) == True
assert ask(Q.complex(x + 1), Q.even(x)) == True
assert ask(Q.complex(x + 1), Q.odd(x)) == True
assert ask(Q.complex(x + y), Q.complex(x) & Q.complex(y)) == True
assert ask(Q.complex(x + y), Q.real(x) & Q.imaginary(y)) == True
# a*x +b
assert ask(Q.complex(2 * x + 1), Q.complex(x)) == True
assert ask(Q.complex(2 * x + 1), Q.real(x)) == True
assert ask(Q.complex(2 * x + 1), Q.positive(x)) == True
assert ask(Q.complex(2 * x + 1), Q.rational(x)) == True
assert ask(Q.complex(2 * x + 1), Q.irrational(x)) == True
assert ask(Q.complex(2 * x + 1), Q.imaginary(x)) == True
assert ask(Q.complex(2 * x + 1), Q.integer(x)) == True
assert ask(Q.complex(2 * x + 1), Q.even(x)) == True
assert ask(Q.complex(2 * x + 1), Q.odd(x)) == True
# x**2
assert ask(Q.complex(x**2), Q.complex(x)) == True
assert ask(Q.complex(x**2), Q.real(x)) == True
assert ask(Q.complex(x**2), Q.positive(x)) == True
assert ask(Q.complex(x**2), Q.rational(x)) == True
assert ask(Q.complex(x**2), Q.irrational(x)) == True
assert ask(Q.complex(x**2), Q.imaginary(x)) == True
assert ask(Q.complex(x**2), Q.integer(x)) == True
assert ask(Q.complex(x**2), Q.even(x)) == True
assert ask(Q.complex(x**2), Q.odd(x)) == True
# 2**x
assert ask(Q.complex(2**x), Q.complex(x)) == True
assert ask(Q.complex(2**x), Q.real(x)) == True
assert ask(Q.complex(2**x), Q.positive(x)) == True
assert ask(Q.complex(2**x), Q.rational(x)) == True
assert ask(Q.complex(2**x), Q.irrational(x)) == True
assert ask(Q.complex(2**x), Q.imaginary(x)) == True
assert ask(Q.complex(2**x), Q.integer(x)) == True
assert ask(Q.complex(2**x), Q.even(x)) == True
assert ask(Q.complex(2**x), Q.odd(x)) == True
assert ask(Q.complex(x**y), Q.complex(x) & Q.complex(y)) == True
# trigonometric expressions
assert ask(Q.complex(sin(x))) == True
assert ask(Q.complex(sin(2 * x + 1))) == True
assert ask(Q.complex(cos(x))) == True
assert ask(Q.complex(cos(2 * x + 1))) == True
# exponential
assert ask(Q.complex(exp(x))) == True
assert ask(Q.complex(exp(x))) == True
# Q.complexes
assert ask(Q.complex(Abs(x))) == True
assert ask(Q.complex(re(x))) == True
assert ask(Q.complex(im(x))) == True
def _(expr, assumptions):
x = expr.args[0]
if ask(Q.rational(x), assumptions):
return ask(~Q.nonzero(x - 1), assumptions)
def _(expr, assumptions):
x = expr.args[0]
if ask(Q.rational(x), assumptions):
return False
def _(expr, assumptions):
x = expr.exp
if ask(Q.rational(x), assumptions):
return ask(~Q.nonzero(x), assumptions)
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 log(expr, assumptions):
x = expr.args[0]
if ask(Q.rational(x), assumptions):
return ask(~Q.nonzero(x - 1), assumptions)
def cot(expr, assumptions):
x = expr.args[0]
if ask(Q.rational(x), assumptions):
return 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_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_rational():
x, y = symbols('x,y')
assert ask(Q.rational(x), Q.integer(x)) == True
assert ask(Q.rational(x), Q.irrational(x)) == False
assert ask(Q.rational(x), Q.real(x)) == None
assert ask(Q.rational(x), Q.positive(x)) == None
assert ask(Q.rational(x), Q.negative(x)) == None
assert ask(Q.rational(x), Q.nonzero(x)) == None
assert ask(Q.rational(2 * x), Q.rational(x)) == True
assert ask(Q.rational(2 * x), Q.integer(x)) == True
assert ask(Q.rational(2 * x), Q.even(x)) == True
assert ask(Q.rational(2 * x), Q.odd(x)) == True
assert ask(Q.rational(2 * x), Q.irrational(x)) == False
assert ask(Q.rational(x / 2), Q.rational(x)) == True
assert ask(Q.rational(x / 2), Q.integer(x)) == True
assert ask(Q.rational(x / 2), Q.even(x)) == True
assert ask(Q.rational(x / 2), Q.odd(x)) == True
assert ask(Q.rational(x / 2), Q.irrational(x)) == False
assert ask(Q.rational(1 / x), Q.rational(x)) == True
assert ask(Q.rational(1 / x), Q.integer(x)) == True
assert ask(Q.rational(1 / x), Q.even(x)) == True
assert ask(Q.rational(1 / x), Q.odd(x)) == True
assert ask(Q.rational(1 / x), Q.irrational(x)) == False
assert ask(Q.rational(2 / x), Q.rational(x)) == True
assert ask(Q.rational(2 / x), Q.integer(x)) == True
assert ask(Q.rational(2 / x), Q.even(x)) == True
assert ask(Q.rational(2 / x), Q.odd(x)) == True
assert ask(Q.rational(2 / x), Q.irrational(x)) == False
# with multiple symbols
assert ask(Q.rational(x * y), Q.irrational(x) & Q.irrational(y)) == None
assert ask(Q.rational(y / x), Q.rational(x) & Q.rational(y)) == True
assert ask(Q.rational(y / x), Q.integer(x) & Q.rational(y)) == True
assert ask(Q.rational(y / x), Q.even(x) & Q.rational(y)) == True
assert ask(Q.rational(y / x), Q.odd(x) & Q.rational(y)) == True
assert ask(Q.rational(y / x), Q.irrational(x) & Q.rational(y)) == False
def test_rational():
x, y = symbols('x,y')
assert ask(Q.rational(x), Q.integer(x)) == True
assert ask(Q.rational(x), Q.irrational(x)) == False
assert ask(Q.rational(x), Q.real(x)) == None
assert ask(Q.rational(x), Q.positive(x)) == None
assert ask(Q.rational(x), Q.negative(x)) == None
assert ask(Q.rational(x), Q.nonzero(x)) == None
assert ask(Q.rational(2*x), Q.rational(x)) == True
assert ask(Q.rational(2*x), Q.integer(x)) == True
assert ask(Q.rational(2*x), Q.even(x)) == True
assert ask(Q.rational(2*x), Q.odd(x)) == True
assert ask(Q.rational(2*x), Q.irrational(x)) == False
assert ask(Q.rational(x/2), Q.rational(x)) == True
assert ask(Q.rational(x/2), Q.integer(x)) == True
assert ask(Q.rational(x/2), Q.even(x)) == True
assert ask(Q.rational(x/2), Q.odd(x)) == True
assert ask(Q.rational(x/2), Q.irrational(x)) == False
assert ask(Q.rational(1/x), Q.rational(x)) == True
assert ask(Q.rational(1/x), Q.integer(x)) == True
assert ask(Q.rational(1/x), Q.even(x)) == True
assert ask(Q.rational(1/x), Q.odd(x)) == True
assert ask(Q.rational(1/x), Q.irrational(x)) == False
assert ask(Q.rational(2/x), Q.rational(x)) == True
assert ask(Q.rational(2/x), Q.integer(x)) == True
assert ask(Q.rational(2/x), Q.even(x)) == True
assert ask(Q.rational(2/x), Q.odd(x)) == True
assert ask(Q.rational(2/x), Q.irrational(x)) == False
# with multiple symbols
assert ask(Q.rational(x*y), Q.irrational(x) & Q.irrational(y)) == None
assert ask(Q.rational(y/x), Q.rational(x) & Q.rational(y)) == True
assert ask(Q.rational(y/x), Q.integer(x) & Q.rational(y)) == True
assert ask(Q.rational(y/x), Q.even(x) & Q.rational(y)) == True
assert ask(Q.rational(y/x), Q.odd(x) & Q.rational(y)) == True
assert ask(Q.rational(y/x), Q.irrational(x) & Q.rational(y)) == False
def test_complex():
x, y = symbols('x,y')
assert ask(Q.complex(x)) == None
assert ask(Q.complex(x), Q.complex(x)) == True
assert ask(Q.complex(x), Q.complex(y)) == None
assert ask(Q.complex(x), ~Q.complex(x)) == False
assert ask(Q.complex(x), Q.real(x)) == True
assert ask(Q.complex(x), ~Q.real(x)) == None
assert ask(Q.complex(x), Q.rational(x)) == True
assert ask(Q.complex(x), Q.irrational(x)) == True
assert ask(Q.complex(x), Q.positive(x)) == True
assert ask(Q.complex(x), Q.imaginary(x)) == True
# a+b
assert ask(Q.complex(x+1), Q.complex(x)) == True
assert ask(Q.complex(x+1), Q.real(x)) == True
assert ask(Q.complex(x+1), Q.rational(x)) == True
assert ask(Q.complex(x+1), Q.irrational(x)) == True
assert ask(Q.complex(x+1), Q.imaginary(x)) == True
assert ask(Q.complex(x+1), Q.integer(x)) == True
assert ask(Q.complex(x+1), Q.even(x)) == True
assert ask(Q.complex(x+1), Q.odd(x)) == True
assert ask(Q.complex(x+y), Q.complex(x) & Q.complex(y)) == True
assert ask(Q.complex(x+y), Q.real(x) & Q.imaginary(y)) == True
# a*x +b
assert ask(Q.complex(2*x+1), Q.complex(x)) == True
assert ask(Q.complex(2*x+1), Q.real(x)) == True
assert ask(Q.complex(2*x+1), Q.positive(x)) == True
assert ask(Q.complex(2*x+1), Q.rational(x)) == True
assert ask(Q.complex(2*x+1), Q.irrational(x)) == True
assert ask(Q.complex(2*x+1), Q.imaginary(x)) == True
assert ask(Q.complex(2*x+1), Q.integer(x)) == True
assert ask(Q.complex(2*x+1), Q.even(x)) == True
assert ask(Q.complex(2*x+1), Q.odd(x)) == True
# x**2
assert ask(Q.complex(x**2), Q.complex(x)) == True
assert ask(Q.complex(x**2), Q.real(x)) == True
assert ask(Q.complex(x**2), Q.positive(x)) == True
assert ask(Q.complex(x**2), Q.rational(x)) == True
assert ask(Q.complex(x**2), Q.irrational(x)) == True
assert ask(Q.complex(x**2), Q.imaginary(x)) == True
assert ask(Q.complex(x**2), Q.integer(x)) == True
assert ask(Q.complex(x**2), Q.even(x)) == True
assert ask(Q.complex(x**2), Q.odd(x)) == True
# 2**x
assert ask(Q.complex(2**x), Q.complex(x)) == True
assert ask(Q.complex(2**x), Q.real(x)) == True
assert ask(Q.complex(2**x), Q.positive(x)) == True
assert ask(Q.complex(2**x), Q.rational(x)) == True
assert ask(Q.complex(2**x), Q.irrational(x)) == True
assert ask(Q.complex(2**x), Q.imaginary(x)) == True
assert ask(Q.complex(2**x), Q.integer(x)) == True
assert ask(Q.complex(2**x), Q.even(x)) == True
assert ask(Q.complex(2**x), Q.odd(x)) == True
assert ask(Q.complex(x**y), Q.complex(x) & Q.complex(y)) == True
# trigonometric expressions
assert ask(Q.complex(sin(x))) == True
assert ask(Q.complex(sin(2*x + 1))) == True
assert ask(Q.complex(cos(x))) == True
assert ask(Q.complex(cos(2*x+1))) == True
# exponential
assert ask(Q.complex(exp(x))) == True
assert ask(Q.complex(exp(x))) == True
# Q.complexes
assert ask(Q.complex(Abs(x))) == True
assert ask(Q.complex(re(x))) == True
assert ask(Q.complex(im(x))) == True
