def test_is_infinite():
x = Symbol('x', infinite=True)
y = Symbol('y', infinite=False)
z = Symbol('z')
assert is_infinite(x)
assert not is_infinite(y)
assert is_infinite(z) is None
assert is_infinite(z, Q.infinite(z))
def test_is_eq():
# test assumptions
assert is_eq(x, y, Q.infinite(x) & Q.finite(y)) is False
assert is_eq(
x, y,
Q.infinite(x) & Q.infinite(y) & Q.extended_real(x)
& ~Q.extended_real(y)) is False
assert is_eq(
x, y,
Q.infinite(x) & Q.infinite(y) & Q.extended_positive(x)
& Q.extended_negative(y)) is False
assert is_eq(x + I, y + I, Q.infinite(x) & Q.finite(y)) is False
assert is_eq(1 + x * I, 1 + y * I, Q.infinite(x) & Q.finite(y)) is False
assert is_eq(x, S(0), assumptions=Q.zero(x))
assert is_eq(x, S(0), assumptions=~Q.zero(x)) is False
assert is_eq(x, S(0), assumptions=Q.nonzero(x)) is False
assert is_neq(x, S(0), assumptions=Q.zero(x)) is False
assert is_neq(x, S(0), assumptions=~Q.zero(x))
assert is_neq(x, S(0), assumptions=Q.nonzero(x))
# test registration
class PowTest(Expr):
def __new__(cls, base, exp):
return Basic.__new__(cls, _sympify(base), _sympify(exp))
@dispatch(PowTest, PowTest)
def _eval_is_eq(lhs, rhs):
if type(lhs) == PowTest and type(rhs) == PowTest:
return fuzzy_and([
is_eq(lhs.args[0], rhs.args[0]),
is_eq(lhs.args[1], rhs.args[1])
])
assert is_eq(PowTest(3, 4), PowTest(3, 4))
assert is_eq(PowTest(3, 4), _sympify(4)) is None
assert is_neq(PowTest(3, 4), PowTest(3, 7))
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