def test_abs():
assert satask(Q.nonnegative(abs(x))) is True
assert satask(Q.positive(abs(x)), ~Q.zero(x)) is True
assert satask(Q.zero(x), ~Q.zero(abs(x))) is False
assert satask(Q.zero(x), Q.zero(abs(x))) is True
assert satask(Q.nonzero(x), ~Q.zero(abs(x))) is None # x could be complex
assert satask(Q.zero(abs(x)), Q.zero(x)) is True
def _(expr):
arg = expr.args[0]
return [Q.nonnegative(expr),
Equivalent(~Q.zero(arg), ~Q.zero(expr)),
Q.even(arg) >> Q.even(expr),
Q.odd(arg) >> Q.odd(expr),
Q.integer(arg) >> Q.integer(expr),
]
def _(expr):
base, exp = expr.base, expr.exp
return [
(Q.real(base) & Q.even(exp) & Q.nonnegative(exp)) >> Q.nonnegative(expr),
(Q.nonnegative(base) & Q.odd(exp) & Q.nonnegative(exp)) >> Q.nonnegative(expr),
(Q.nonpositive(base) & Q.odd(exp) & Q.nonnegative(exp)) >> Q.nonpositive(expr),
Equivalent(Q.zero(expr), Q.zero(base) & Q.positive(exp))
]
def test_old_assump():
assert satask(Q.positive(1)) is True
assert satask(Q.positive(-1)) is False
assert satask(Q.positive(0)) is False
assert satask(Q.positive(I)) is False
assert satask(Q.positive(pi)) is True
assert satask(Q.negative(1)) is False
assert satask(Q.negative(-1)) is True
assert satask(Q.negative(0)) is False
assert satask(Q.negative(I)) is False
assert satask(Q.negative(pi)) is False
assert satask(Q.zero(1)) is False
assert satask(Q.zero(-1)) is False
assert satask(Q.zero(0)) is True
assert satask(Q.zero(I)) is False
assert satask(Q.zero(pi)) is False
assert satask(Q.nonzero(1)) is True
assert satask(Q.nonzero(-1)) is True
assert satask(Q.nonzero(0)) is False
assert satask(Q.nonzero(I)) is False
assert satask(Q.nonzero(pi)) is True
assert satask(Q.nonpositive(1)) is False
assert satask(Q.nonpositive(-1)) is True
assert satask(Q.nonpositive(0)) is True
assert satask(Q.nonpositive(I)) is False
assert satask(Q.nonpositive(pi)) is False
assert satask(Q.nonnegative(1)) is True
assert satask(Q.nonnegative(-1)) is False
assert satask(Q.nonnegative(0)) is True
assert satask(Q.nonnegative(I)) is False
assert satask(Q.nonnegative(pi)) is True
def test_is_ge_le():
# test assumptions
assert is_ge(x, S(0), Q.nonnegative(x)) is True
assert is_ge(x, S(0), Q.negative(x)) is False
# test registration
class PowTest(Expr):
def __new__(cls, base, exp):
return Basic.__new__(cls, _sympify(base), _sympify(exp))
@dispatch(PowTest, PowTest)
def _eval_is_ge(lhs, rhs):
if type(lhs) == PowTest and type(rhs) == PowTest:
return fuzzy_and([
is_ge(lhs.args[0], rhs.args[0]),
is_ge(lhs.args[1], rhs.args[1])
])
assert is_ge(PowTest(3, 9), PowTest(3, 2))
assert is_gt(PowTest(3, 9), PowTest(3, 2))
assert is_le(PowTest(3, 2), PowTest(3, 9))
assert is_lt(PowTest(3, 2), PowTest(3, 9))
def register_fact(klass, fact, registry=fact_registry):
registry[klass] |= {fact}
for klass, fact in [
(Mul, Equivalent(Q.zero, AnyArgs(Q.zero))),
(MatMul, Implies(AllArgs(Q.square), Equivalent(Q.invertible, AllArgs(Q.invertible)))),
(Add, Implies(AllArgs(Q.positive), Q.positive)),
(Add, Implies(AllArgs(Q.negative), Q.negative)),
(Mul, Implies(AllArgs(Q.positive), Q.positive)),
(Mul, Implies(AllArgs(Q.commutative), Q.commutative)),
(Mul, Implies(AllArgs(Q.real), Q.commutative)),
(Pow, CustomLambda(lambda power: Implies(Q.real(power.base) &
Q.even(power.exp) & Q.nonnegative(power.exp), Q.nonnegative(power)))),
(Pow, CustomLambda(lambda power: Implies(Q.nonnegative(power.base) & Q.odd(power.exp) & Q.nonnegative(power.exp), Q.nonnegative(power)))),
(Pow, CustomLambda(lambda power: Implies(Q.nonpositive(power.base) & Q.odd(power.exp) & Q.nonnegative(power.exp), Q.nonpositive(power)))),
# This one can still be made easier to read. I think we need basic pattern
# matching, so that we can just write Equivalent(Q.zero(x**y), Q.zero(x) & Q.positive(y))
(Pow, CustomLambda(lambda power: Equivalent(Q.zero(power), Q.zero(power.base) & Q.positive(power.exp)))),
(Integer, CheckIsPrime(Q.prime)),
# Implicitly assumes Mul has more than one arg
# Would be AllArgs(Q.prime | Q.composite) except 1 is composite
(Mul, Implies(AllArgs(Q.prime), ~Q.prime)),
# More advanced prime assumptions will require inequalities, as 1 provides
# a corner case.
(Mul, Implies(AllArgs(Q.imaginary | Q.real), Implies(ExactlyOneArg(Q.imaginary), Q.imaginary))),
(Mul, Implies(AllArgs(Q.real), Q.real)),
(Add, Implies(AllArgs(Q.real), Q.real)),
registry[klass] |= {fact}
for klass, fact in [
(Mul, Equivalent(Q.zero, AnyArgs(Q.zero))),
(MatMul,
Implies(AllArgs(Q.square), Equivalent(Q.invertible,
AllArgs(Q.invertible)))),
(Add, Implies(AllArgs(Q.positive), Q.positive)),
(Add, Implies(AllArgs(Q.negative), Q.negative)),
(Mul, Implies(AllArgs(Q.positive), Q.positive)),
(Mul, Implies(AllArgs(Q.commutative), Q.commutative)),
(Mul, Implies(AllArgs(Q.real), Q.commutative)),
(Pow,
CustomLambda(lambda power: Implies(
Q.real(power.base) & Q.even(power.exp) & Q.nonnegative(power.exp),
Q.nonnegative(power)))),
(Pow,
CustomLambda(lambda power: Implies(
Q.nonnegative(power.base) & Q.odd(power.exp) & Q.nonnegative(
power.exp), Q.nonnegative(power)))),
(Pow,
CustomLambda(lambda power: Implies(
Q.nonpositive(power.base) & Q.odd(power.exp) & Q.nonnegative(
power.exp), Q.nonpositive(power)))),
# This one can still be made easier to read. I think we need basic pattern
# matching, so that we can just write Equivalent(Q.zero(x**y), Q.zero(x) & Q.positive(y))
(Pow,
CustomLambda(
lambda power: Equivalent(Q.zero(power),
def test_pow_pos_neg():
assert satask(Q.nonnegative(x**2), Q.positive(x)) is True
assert satask(Q.nonpositive(x**2), Q.positive(x)) is False
assert satask(Q.positive(x**2), Q.positive(x)) is True
assert satask(Q.negative(x**2), Q.positive(x)) is False
assert satask(Q.real(x**2), Q.positive(x)) is True
assert satask(Q.nonnegative(x**2), Q.negative(x)) is True
assert satask(Q.nonpositive(x**2), Q.negative(x)) is False
assert satask(Q.positive(x**2), Q.negative(x)) is True
assert satask(Q.negative(x**2), Q.negative(x)) is False
assert satask(Q.real(x**2), Q.negative(x)) is True
assert satask(Q.nonnegative(x**2), Q.nonnegative(x)) is True
assert satask(Q.nonpositive(x**2), Q.nonnegative(x)) is None
assert satask(Q.positive(x**2), Q.nonnegative(x)) is None
assert satask(Q.negative(x**2), Q.nonnegative(x)) is False
assert satask(Q.real(x**2), Q.nonnegative(x)) is True
assert satask(Q.nonnegative(x**2), Q.nonpositive(x)) is True
assert satask(Q.nonpositive(x**2), Q.nonpositive(x)) is None
assert satask(Q.positive(x**2), Q.nonpositive(x)) is None
assert satask(Q.negative(x**2), Q.nonpositive(x)) is False
assert satask(Q.real(x**2), Q.nonpositive(x)) is True
assert satask(Q.nonnegative(x**3), Q.positive(x)) is True
assert satask(Q.nonpositive(x**3), Q.positive(x)) is False
assert satask(Q.positive(x**3), Q.positive(x)) is True
assert satask(Q.negative(x**3), Q.positive(x)) is False
assert satask(Q.real(x**3), Q.positive(x)) is True
assert satask(Q.nonnegative(x**3), Q.negative(x)) is False
assert satask(Q.nonpositive(x**3), Q.negative(x)) is True
assert satask(Q.positive(x**3), Q.negative(x)) is False
assert satask(Q.negative(x**3), Q.negative(x)) is True
assert satask(Q.real(x**3), Q.negative(x)) is True
assert satask(Q.nonnegative(x**3), Q.nonnegative(x)) is True
assert satask(Q.nonpositive(x**3), Q.nonnegative(x)) is None
assert satask(Q.positive(x**3), Q.nonnegative(x)) is None
assert satask(Q.negative(x**3), Q.nonnegative(x)) is False
assert satask(Q.real(x**3), Q.nonnegative(x)) is True
assert satask(Q.nonnegative(x**3), Q.nonpositive(x)) is None
assert satask(Q.nonpositive(x**3), Q.nonpositive(x)) is True
assert satask(Q.positive(x**3), Q.nonpositive(x)) is False
assert satask(Q.negative(x**3), Q.nonpositive(x)) is None
assert satask(Q.real(x**3), Q.nonpositive(x)) is True
# If x is zero, x**negative is not real.
assert satask(Q.nonnegative(x**-2), Q.nonpositive(x)) is None
assert satask(Q.nonpositive(x**-2), Q.nonpositive(x)) is None
assert satask(Q.positive(x**-2), Q.nonpositive(x)) is None
assert satask(Q.negative(x**-2), Q.nonpositive(x)) is None
assert satask(Q.real(x**-2), Q.nonpositive(x)) is None
