def test_pow():
x, y, z = symbols('x,y,z')
assert refine((-1)**x, Q.even(x)) == 1
assert refine((-1)**x, Q.odd(x)) == -1
assert refine((-2)**x, Q.even(x)) == 2**x
# nested powers
assert refine(sqrt(x**2)) != Abs(x)
assert refine(sqrt(x**2), Q.complex(x)) != Abs(x)
assert refine(sqrt(x**2), Q.real(x)) == Abs(x)
assert refine(sqrt(x**2), Q.positive(x)) == x
assert refine((x**3)**(S(1)/3)) != x
assert refine((x**3)**(S(1)/3), Q.real(x)) != x
assert refine((x**3)**(S(1)/3), Q.positive(x)) == x
assert refine(sqrt(1/x), Q.real(x)) != 1/sqrt(x)
assert refine(sqrt(1/x), Q.positive(x)) == 1/sqrt(x)
# powers of (-1)
assert refine((-1)**(x+y), Q.even(x)) == (-1)**y
assert refine((-1)**(x+y+z), Q.odd(x) & Q.odd(z))==(-1)**y
assert refine((-1)**(x+y+1), Q.odd(x))==(-1)**y
assert refine((-1)**(x+y+2), Q.odd(x))==(-1)**(y+1)
assert refine((-1)**(x+3)) == (-1)**(x+1)
def test_pow1():
assert refine((-1)**x, Q.even(x)) == 1
assert refine((-1)**x, Q.odd(x)) == -1
assert refine((-2)**x, Q.even(x)) == 2**x
# nested powers
assert refine(sqrt(x**2)) != Abs(x)
assert refine(sqrt(x**2), Q.complex(x)) != Abs(x)
assert refine(sqrt(x**2), Q.real(x)) == Abs(x)
assert refine(sqrt(x**2), Q.positive(x)) == x
assert refine((x**3)**Rational(1, 3)) != x
assert refine((x**3)**Rational(1, 3), Q.real(x)) != x
assert refine((x**3)**Rational(1, 3), Q.positive(x)) == x
assert refine(sqrt(1/x), Q.real(x)) != 1/sqrt(x)
assert refine(sqrt(1/x), Q.positive(x)) == 1/sqrt(x)
# powers of (-1)
assert refine((-1)**(x + y), Q.even(x)) == (-1)**y
assert refine((-1)**(x + y + z), Q.odd(x) & Q.odd(z)) == (-1)**y
assert refine((-1)**(x + y + 1), Q.odd(x)) == (-1)**y
assert refine((-1)**(x + y + 2), Q.odd(x)) == (-1)**(y + 1)
assert refine((-1)**(x + 3)) == (-1)**(x + 1)
# continuation
assert refine((-1)**((-1)**x/2 - S.Half), Q.integer(x)) == (-1)**x
assert refine((-1)**((-1)**x/2 + S.Half), Q.integer(x)) == (-1)**(x + 1)
assert refine((-1)**((-1)**x/2 + 5*S.Half), Q.integer(x)) == (-1)**(x + 1)
def test_pow():
x, y, z = symbols('x,y,z')
assert refine((-1)**x, Q.even(x)) == 1
assert refine((-1)**x, Q.odd(x)) == -1
assert refine((-2)**x, Q.even(x)) == 2**x
# nested powers
assert refine(sqrt(x**2)) != Abs(x)
assert refine(sqrt(x**2), Q.complex(x)) != Abs(x)
assert refine(sqrt(x**2), Q.real(x)) == Abs(x)
assert refine(sqrt(x**2), Q.positive(x)) == x
assert refine((x**3)**(S(1) / 3)) != x
assert refine((x**3)**(S(1) / 3), Q.real(x)) != x
assert refine((x**3)**(S(1) / 3), Q.positive(x)) == x
assert refine(sqrt(1 / x), Q.real(x)) != 1 / sqrt(x)
assert refine(sqrt(1 / x), Q.positive(x)) == 1 / sqrt(x)
# powers of (-1)
assert refine((-1)**(x + y), Q.even(x)) == (-1)**y
assert refine((-1)**(x + y + z), Q.odd(x) & Q.odd(z)) == (-1)**y
assert refine((-1)**(x + y + 1), Q.odd(x)) == (-1)**y
assert refine((-1)**(x + y + 2), Q.odd(x)) == (-1)**(y + 1)
assert refine((-1)**(x + 3)) == (-1)**(x + 1)
def test_pow():
assert refine((-1) ** x, Q.even(x)) == 1
assert refine((-1) ** x, Q.odd(x)) == -1
assert refine((-2) ** x, Q.even(x)) == 2 ** x
# nested powers
assert refine(sqrt(x ** 2)) != Abs(x)
assert refine(sqrt(x ** 2), Q.complex(x)) != Abs(x)
assert refine(sqrt(x ** 2), Q.real(x)) == Abs(x)
assert refine(sqrt(x ** 2), Q.positive(x)) == x
assert refine((x ** 3) ** (S(1) / 3)) != x
assert refine((x ** 3) ** (S(1) / 3), Q.real(x)) != x
assert refine((x ** 3) ** (S(1) / 3), Q.positive(x)) == x
assert refine(sqrt(1 / x), Q.real(x)) != 1 / sqrt(x)
assert refine(sqrt(1 / x), Q.positive(x)) == 1 / sqrt(x)
# powers of (-1)
assert refine((-1) ** (x + y), Q.even(x)) == (-1) ** y
assert refine((-1) ** (x + y + z), Q.odd(x) & Q.odd(z)) == (-1) ** y
assert refine((-1) ** (x + y + 1), Q.odd(x)) == (-1) ** y
assert refine((-1) ** (x + y + 2), Q.odd(x)) == (-1) ** (y + 1)
assert refine((-1) ** (x + 3)) == (-1) ** (x + 1)
assert refine((-1) ** ((-1) ** x / 2 - S.Half), Q.integer(x)) == (-1) ** x
assert refine((-1) ** ((-1) ** x / 2 + S.Half), Q.integer(x)) == (-1) ** (x + 1)
assert refine((-1) ** ((-1) ** x / 2 + 5 * S.Half), Q.integer(x)) == (-1) ** (x + 1)
assert refine((-1) ** ((-1) ** x / 2 - 7 * S.Half), Q.integer(x)) == (-1) ** (x + 1)
assert refine((-1) ** ((-1) ** x / 2 - 9 * S.Half), Q.integer(x)) == (-1) ** x
# powers of Abs
assert refine(Abs(x) ** 2, Q.real(x)) == x ** 2
assert refine(Abs(x) ** 3, Q.real(x)) == Abs(x) ** 3
assert refine(Abs(x) ** 2) == Abs(x) ** 2
def test_pow2():
# powers of (-1)
assert refine((-1)**(x + y), Q.even(x)) == (-1)**y
assert refine((-1)**(x + y + z), Q.odd(x) & Q.odd(z)) == (-1)**y
assert refine((-1)**(x + y + 1), Q.odd(x)) == (-1)**y
assert refine((-1)**(x + y + 2), Q.odd(x)) == (-1)**(y + 1)
assert refine((-1)**(x + 3)) == (-1)**(x + 1)
def test_pow2():
# powers of (-1)
assert refine((-1)**(x + y), Q.even(x)) == (-1)**y
assert refine((-1)**(x + y + z), Q.odd(x) & Q.odd(z)) == (-1)**y
assert refine((-1)**(x + y + 1), Q.odd(x)) == (-1)**y
assert refine((-1)**(x + y + 2), Q.odd(x)) == (-1)**(y + 1)
assert refine((-1)**(x + 3)) == (-1)**(x + 1)
def test_pow1():
assert refine((-1)**x, Q.even(x)) == 1
assert refine((-1)**x, Q.odd(x)) == -1
assert refine((-2)**x, Q.even(x)) == 2**x
# nested powers
assert refine(sqrt(x**2)) != Abs(x)
assert refine(sqrt(x**2), Q.complex(x)) != Abs(x)
assert refine(sqrt(x**2), Q.real(x)) == Abs(x)
assert refine(sqrt(x**2), Q.positive(x)) == x
assert refine((x**3)**(S(1)/3)) != x
assert refine((x**3)**(S(1)/3), Q.real(x)) != x
assert refine((x**3)**(S(1)/3), Q.positive(x)) == x
assert refine(sqrt(1/x), Q.real(x)) != 1/sqrt(x)
assert refine(sqrt(1/x), Q.positive(x)) == 1/sqrt(x)
def test_get_relevant_clsfacts():
exprs = {Abs(x*y)}
exprs, facts = get_relevant_clsfacts(exprs)
assert exprs == {x*y}
assert facts.clauses == \
{frozenset({Literal(Q.odd(Abs(x*y)), False), Literal(Q.odd(x*y), True)}),
frozenset({Literal(Q.zero(Abs(x*y)), False), Literal(Q.zero(x*y), True)}),
frozenset({Literal(Q.even(Abs(x*y)), False), Literal(Q.even(x*y), True)}),
frozenset({Literal(Q.zero(Abs(x*y)), True), Literal(Q.zero(x*y), False)}),
frozenset({Literal(Q.even(Abs(x*y)), False),
Literal(Q.odd(Abs(x*y)), False),
Literal(Q.odd(x*y), True)}),
frozenset({Literal(Q.even(Abs(x*y)), False),
Literal(Q.even(x*y), True),
Literal(Q.odd(Abs(x*y)), False)}),
frozenset({Literal(Q.positive(Abs(x*y)), False),
Literal(Q.zero(Abs(x*y)), False)})}
def test_pow1():
assert refine((-1)**x, Q.even(x)) == 1
assert refine((-1)**x, Q.odd(x)) == -1
assert refine((-2)**x, Q.even(x)) == 2**x
# nested powers
assert refine(sqrt(x**2)) != Abs(x)
assert refine(sqrt(x**2), Q.complex(x)) != Abs(x)
assert refine(sqrt(x**2), Q.real(x)) == Abs(x)
assert refine(sqrt(x**2), Q.positive(x)) == x
assert refine((x**3)**(S(1)/3)) != x
assert refine((x**3)**(S(1)/3), Q.real(x)) != x
assert refine((x**3)**(S(1)/3), Q.positive(x)) == x
assert refine(sqrt(1/x), Q.real(x)) != 1/sqrt(x)
assert refine(sqrt(1/x), Q.positive(x)) == 1/sqrt(x)
def test_pow():
x = Symbol('x', even=True)
assert refine((-1)**x) == 1
x = Symbol('x', odd=True)
assert refine((-1)**x) == -1
x = Symbol('x', even=True)
assert refine((-2)**x) == 2**x
# nested powers
x = Symbol('x')
assert refine(sqrt(x**2)) != Abs(x)
x = Symbol('x', complex=True)
assert refine(sqrt(x**2)) != Abs(x)
x = Symbol('x', real=True)
assert refine(sqrt(x**2)) == Abs(x)
p = Symbol('p', positive=True)
assert refine(sqrt(p**2)) == p
x = Symbol('x')
assert refine((x**3)**(S(1) / 3)) != x
x = Symbol('x', real=True)
assert refine((x**3)**(S(1) / 3)) != x
x = Symbol('x', positive=True)
assert refine((x**3)**(S(1) / 3)) == x
x = Symbol('x', real=True)
assert refine(sqrt(1 / x)) != 1 / sqrt(x)
x = Symbol('x', positive=True)
assert refine(sqrt(1 / x)) == 1 / sqrt(x)
# powers of (-1)
x = Symbol('x', even=True)
assert refine((-1)**(x + y), Q.even(x)) == (-1)**y
x = Symbol('x', odd=True)
z = Symbol('z', odd=True)
assert refine((-1)**(x + y + z), Q.odd(x) & Q.odd(z)) == (-1)**y
assert refine((-1)**(x + y + 1), Q.odd(x)) == (-1)**y
assert refine((-1)**(x + y + 2), Q.odd(x)) == (-1)**(y + 1)
x = Symbol('x')
assert refine((-1)**(x + 3)) == (-1)**(x + 1)
x = Symbol('x', integer=True)
assert refine((-1)**((-1)**x / 2 - S.Half), Q.integer(x)) == (-1)**x
assert refine((-1)**((-1)**x / 2 + S.Half), Q.integer(x)) == (-1)**(x + 1)
assert refine((-1)**((-1)**x / 2 + 5 * S.Half),
Q.integer(x)) == (-1)**(x + 1)
assert refine((-1)**((-1)**x / 2 - 7 * S.Half),
Q.integer(x)) == (-1)**(x + 1)
assert refine((-1)**((-1)**x / 2 - 9 * S.Half), Q.integer(x)) == (-1)**x
# powers of Abs
x = Symbol('x', real=True)
assert refine(Abs(x)**2, Q.real(x)) == x**2
assert refine(Abs(x)**3, Q.real(x)) == Abs(x)**3
x = Symbol('x')
assert refine(Abs(x)**2) == Abs(x)**2
def test_pow():
x = Symbol('x', even=True)
assert refine((-1)**x) == 1
x = Symbol('x', odd=True)
assert refine((-1)**x) == -1
x = Symbol('x', even=True)
assert refine((-2)**x) == 2**x
# nested powers
x = Symbol('x')
assert refine(sqrt(x**2)) != Abs(x)
x = Symbol('x', complex=True)
assert refine(sqrt(x**2)) != Abs(x)
x = Symbol('x', real=True)
assert refine(sqrt(x**2)) == Abs(x)
p = Symbol('p', positive=True)
assert refine(sqrt(p**2)) == p
x = Symbol('x')
assert refine((x**3)**(S(1)/3)) != x
x = Symbol('x', real=True)
assert refine((x**3)**(S(1)/3)) != x
x = Symbol('x', positive=True)
assert refine((x**3)**(S(1)/3)) == x
x = Symbol('x', real=True)
assert refine(sqrt(1/x)) != 1/sqrt(x)
x = Symbol('x', positive=True)
assert refine(sqrt(1/x)) == 1/sqrt(x)
# powers of (-1)
x = Symbol('x', even=True)
assert refine((-1)**(x + y), Q.even(x)) == (-1)**y
x = Symbol('x', odd=True)
z = Symbol('z', odd=True)
assert refine((-1)**(x + y + z), Q.odd(x) & Q.odd(z)) == (-1)**y
assert refine((-1)**(x + y + 1), Q.odd(x)) == (-1)**y
assert refine((-1)**(x + y + 2), Q.odd(x)) == (-1)**(y + 1)
x = Symbol('x')
assert refine((-1)**(x + 3)) == (-1)**(x + 1)
x = Symbol('x', integer=True)
assert refine((-1)**((-1)**x/2 - S.Half), Q.integer(x)) == (-1)**x
assert refine((-1)**((-1)**x/2 + S.Half), Q.integer(x)) == (-1)**(x + 1)
assert refine((-1)**((-1)**x/2 + 5*S.Half), Q.integer(x)) == (-1)**(x + 1)
assert refine((-1)**((-1)**x/2 - 7*S.Half), Q.integer(x)) == (-1)**(x + 1)
assert refine((-1)**((-1)**x/2 - 9*S.Half), Q.integer(x)) == (-1)**x
# powers of Abs
x = Symbol('x', real=True)
assert refine(Abs(x)**2, Q.real(x)) == x**2
assert refine(Abs(x)**3, Q.real(x)) == Abs(x)**3
x = Symbol('x')
assert refine(Abs(x)**2) == Abs(x)**2
def test_pow():
assert refine((-1)**x, Q.even(x)) == 1
assert refine((-1)**x, Q.odd(x)) == -1
assert refine((-2)**x, Q.even(x)) == 2**x
# nested powers
assert refine(sqrt(x**2)) != Abs(x)
assert refine(sqrt(x**2), Q.complex(x)) != Abs(x)
assert refine(sqrt(x**2), Q.real(x)) == Abs(x)
assert refine(sqrt(x**2), Q.positive(x)) == x
assert refine((x**3)**(S(1) / 3)) != x
assert refine((x**3)**(S(1) / 3), Q.real(x)) != x
assert refine((x**3)**(S(1) / 3), Q.positive(x)) == x
assert refine(sqrt(1 / x), Q.real(x)) != 1 / sqrt(x)
assert refine(sqrt(1 / x), Q.positive(x)) == 1 / sqrt(x)
# powers of (-1)
assert refine((-1)**(x + y), Q.even(x)) == (-1)**y
assert refine((-1)**(x + y + z), Q.odd(x) & Q.odd(z)) == (-1)**y
assert refine((-1)**(x + y + 1), Q.odd(x)) == (-1)**y
assert refine((-1)**(x + y + 2), Q.odd(x)) == (-1)**(y + 1)
assert refine((-1)**(x + 3)) == (-1)**(x + 1)
assert refine((-1)**((-1)**x / 2 - S.Half), Q.integer(x)) == (-1)**x
assert refine((-1)**((-1)**x / 2 + S.Half), Q.integer(x)) == (-1)**(x + 1)
assert refine((-1)**((-1)**x / 2 + 5 * S.Half),
Q.integer(x)) == (-1)**(x + 1)
assert refine((-1)**((-1)**x / 2 - 7 * S.Half),
Q.integer(x)) == (-1)**(x + 1)
assert refine((-1)**((-1)**x / 2 - 9 * S.Half), Q.integer(x)) == (-1)**x
# powers of Abs
assert refine(Abs(x)**2, Q.real(x)) == x**2
assert refine(Abs(x)**3, Q.real(x)) == Abs(x)**3
assert refine(Abs(x)**2) == Abs(x)**2
def test_pow2():
# powers of (-1)
assert refine((-1)**(x + y), Q.even(x)) == (-1)**y
assert refine((-1)**(x + y + z), Q.odd(x) & Q.odd(z)) == (-1)**y
assert refine((-1)**(x + y + 1), Q.odd(x)) == (-1)**y
assert refine((-1)**(x + y + 2), Q.odd(x)) == (-1)**(y + 1)
assert refine((-1)**(x + 3)) == (-1)**(x + 1)
assert refine((-1)**((-1)**x/2 - S.Half), Q.integer(x)) == (-1)**x
assert refine((-1)**((-1)**x/2 + S.Half), Q.integer(x)) == (-1)**(x + 1)
assert refine((-1)**((-1)**x/2 + 5*S.Half), Q.integer(x)) == (-1)**(x + 1)
assert refine((-1)**((-1)**x/2 - 7*S.Half), Q.integer(x)) == (-1)**(x + 1)
assert refine((-1)**((-1)**x/2 - 9*S.Half), Q.integer(x)) == (-1)**x
# powers of Abs
assert refine(Abs(x)**2, Q.real(x)) == x**2
assert refine(Abs(x)**3, Q.real(x)) == Abs(x)**3
assert refine(Abs(x)**2) == Abs(x)**2
def test_odd():
assert satask(Q.odd(2)) is False
assert satask(Q.odd(3)) is True
assert satask(Q.odd(x*y), Q.even(x) & Q.odd(y)) is False
assert satask(Q.odd(x*y), Q.even(x) & Q.integer(y)) is False
assert satask(Q.odd(x*y), Q.even(x) & Q.even(y)) is False
assert satask(Q.odd(x*y), Q.odd(x) & Q.odd(y)) is True
assert satask(Q.odd(x*y), Q.even(x)) is None
assert satask(Q.odd(x*y), Q.odd(x) & Q.integer(y)) is None
assert satask(Q.odd(x*y), Q.odd(x) & Q.odd(y)) is True
assert satask(Q.odd(abs(x)), Q.even(x)) is False
assert satask(Q.odd(abs(x)), Q.odd(x)) is True
assert satask(Q.odd(x), Q.odd(abs(x))) is None # x could be complex
def test_odd():
assert satask(Q.odd(2)) is False
assert satask(Q.odd(3)) is True
assert satask(Q.odd(x * y), Q.even(x) & Q.odd(y)) is False
assert satask(Q.odd(x * y), Q.even(x) & Q.integer(y)) is False
assert satask(Q.odd(x * y), Q.even(x) & Q.even(y)) is False
assert satask(Q.odd(x * y), Q.odd(x) & Q.odd(y)) is True
assert satask(Q.odd(x * y), Q.even(x)) is None
assert satask(Q.odd(x * y), Q.odd(x) & Q.integer(y)) is None
assert satask(Q.odd(x * y), Q.odd(x) & Q.odd(y)) is True
assert satask(Q.odd(abs(x)), Q.even(x)) is False
assert satask(Q.odd(abs(x)), Q.odd(x)) is True
assert satask(Q.odd(x), Q.odd(abs(x))) is None # x could be complex
def test_AppliedPredicate():
assert sstr(Q.even(x)) == 'Q.even(x)'
from sympy import ask, Q
print("ask(Q.even(2)):", ask(Q.even(2)))
print("ask(Q.odd(2)):", ask(Q.odd(2)))
print("ask(Q.even(3)):", ask(Q.even(3)))
print("ask(Q.odd(3)):", ask(Q.odd(3)))
print("ask(Q.even(4)):", ask(Q.even(4)))
print("ask(Q.odd(4)):", ask(Q.odd(4)))
print("-----------------------")
print("ask(Q.prime(2)):", ask(Q.prime(2)))
print("ask(Q.prime(3)):", ask(Q.prime(3)))
print("ask(Q.prime(4)):", ask(Q.prime(4)))
"""
ask(Q.even(2)): True
ask(Q.odd(2)): False
ask(Q.even(3)): False
ask(Q.odd(3)): True
ask(Q.even(4)): True
ask(Q.odd(4)): False
-----------------------
ask(Q.prime(2)): True
ask(Q.prime(3)): True
ask(Q.prime(4)): False
"""
def test_AppliedPredicate():
sT(Q.even(Symbol('z')), "AppliedPredicate(Q.even, Symbol('z'))")
