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_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_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