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_refine():
# relational
assert not refine(x < 0, ~Q.is_true(x < 0))
assert refine(x < 0, Q.is_true(x < 0))
assert refine(x < 0, Q.is_true(0 > x)) == True
assert refine(x < 0, Q.is_true(y < 0)) == (x < 0)
assert not refine(x <= 0, ~Q.is_true(x <= 0))
assert refine(x <= 0, Q.is_true(x <= 0))
assert refine(x <= 0, Q.is_true(0 >= x)) == True
assert refine(x <= 0, Q.is_true(y <= 0)) == (x <= 0)
assert not refine(x > 0, ~Q.is_true(x > 0))
assert refine(x > 0, Q.is_true(x > 0))
assert refine(x > 0, Q.is_true(0 < x)) == True
assert refine(x > 0, Q.is_true(y > 0)) == (x > 0)
assert not refine(x >= 0, ~Q.is_true(x >= 0))
assert refine(x >= 0, Q.is_true(x >= 0))
assert refine(x >= 0, Q.is_true(0 <= x)) == True
assert refine(x >= 0, Q.is_true(y >= 0)) == (x >= 0)
assert not refine(Eq(x, 0), ~Q.is_true(Eq(x, 0)))
assert refine(Eq(x, 0), Q.is_true(Eq(x, 0)))
assert refine(Eq(x, 0), Q.is_true(Eq(0, x))) == True
assert refine(Eq(x, 0), Q.is_true(Eq(y, 0))) == Eq(x, 0)
assert not refine(Ne(x, 0), ~Q.is_true(Ne(x, 0)))
assert refine(Ne(x, 0), Q.is_true(Ne(0, x))) == True
assert refine(Ne(x, 0), Q.is_true(Ne(x, 0)))
assert refine(Ne(x, 0), Q.is_true(Ne(y, 0))) == (Ne(x, 0))
# boolean functions
assert refine(And(x > 0, y > 0), Q.is_true(x > 0)) == (y > 0)
assert refine(And(x > 0, y > 0), Q.is_true(x > 0) & Q.is_true(y > 0)) == True
# predicates
assert refine(Q.positive(x), Q.positive(x)) == True
assert refine(Q.positive(x), Q.negative(x)) == False
assert refine(Q.positive(x), Q.real(x)) == Q.positive(x)
def test_Abs():
assert refine(Abs(x), Q.positive(x)) == x
assert refine(1 + Abs(x), Q.positive(x)) == 1 + x
assert refine(Abs(x), Q.negative(x)) == -x
assert refine(1 + Abs(x), Q.negative(x)) == 1 - x
assert refine(Abs(x**2)) != x**2
assert refine(Abs(x**2), Q.real(x)) == x**2
def test_refine_issue_12724():
expr1 = refine(Abs(x * y), Q.positive(x))
expr2 = refine(Abs(x * y * z), Q.positive(x))
assert expr1 == x * Abs(y)
assert expr2 == x * Abs(y * z)
y1 = Symbol('y1', real = True)
expr3 = refine(Abs(x * y1**2 * z), Q.positive(x))
assert expr3 == x * y1**2 * Abs(z)
def _(expr):
return [allarg(x, Q.positive(x), expr) >> Q.positive(expr),
allarg(x, Q.negative(x), expr) >> Q.negative(expr),
allarg(x, Q.real(x), expr) >> Q.real(expr),
allarg(x, Q.rational(x), expr) >> Q.rational(expr),
allarg(x, Q.integer(x), expr) >> Q.integer(expr),
exactlyonearg(x, ~Q.integer(x), expr) >> ~Q.integer(expr),
]
def _(expr):
return [Equivalent(Q.zero(expr), anyarg(x, Q.zero(x), expr)),
allarg(x, Q.positive(x), expr) >> Q.positive(expr),
allarg(x, Q.real(x), expr) >> Q.real(expr),
allarg(x, Q.rational(x), expr) >> Q.rational(expr),
allarg(x, Q.integer(x), expr) >> Q.integer(expr),
exactlyonearg(x, ~Q.rational(x), expr) >> ~Q.integer(expr),
allarg(x, Q.commutative(x), expr) >> Q.commutative(expr),
]
def test_atan2():
assert refine(atan2(y, x), Q.real(y) & Q.positive(x)) == atan(y/x)
assert refine(atan2(y, x), Q.negative(y) & Q.positive(x)) == atan(y/x)
assert refine(atan2(y, x), Q.negative(y) & Q.negative(x)) == atan(y/x) - pi
assert refine(atan2(y, x), Q.positive(y) & Q.negative(x)) == atan(y/x) + pi
assert refine(atan2(y, x), Q.zero(y) & Q.negative(x)) == pi
assert refine(atan2(y, x), Q.positive(y) & Q.zero(x)) == pi/2
assert refine(atan2(y, x), Q.negative(y) & Q.zero(x)) == -pi/2
assert refine(atan2(y, x), Q.zero(y) & Q.zero(x)) is nan
def test_exactlyonearg():
assert exactlyonearg(x, Q.zero(x), x*y) == \
Or(Q.zero(x) & ~Q.zero(y), Q.zero(y) & ~Q.zero(x))
assert exactlyonearg(x, Q.zero(x), x*y*z) == \
Or(Q.zero(x) & ~Q.zero(y) & ~Q.zero(z), Q.zero(y)
& ~Q.zero(x) & ~Q.zero(z), Q.zero(z) & ~Q.zero(x) & ~Q.zero(y))
assert exactlyonearg(x, Q.positive(x) | Q.negative(x), x*y) == \
Or((Q.positive(x) | Q.negative(x)) &
~(Q.positive(y) | Q.negative(y)), (Q.positive(y) | Q.negative(y)) &
~(Q.positive(x) | Q.negative(x)))
def test_pos_neg():
assert satask(~Q.positive(x), Q.negative(x)) is True
assert satask(~Q.negative(x), Q.positive(x)) is True
assert satask(Q.positive(x + y), Q.positive(x) & Q.positive(y)) is True
assert satask(Q.negative(x + y), Q.negative(x) & Q.negative(y)) is True
assert satask(Q.positive(x + y), Q.negative(x) & Q.negative(y)) is False
assert satask(Q.negative(x + y), Q.positive(x) & Q.positive(y)) is False
def test_sign():
x = Symbol('x', real = True)
assert refine(sign(x), Q.positive(x)) == 1
assert refine(sign(x), Q.negative(x)) == -1
assert refine(sign(x), Q.zero(x)) == 0
assert refine(sign(x), True) == sign(x)
assert refine(sign(Abs(x)), Q.nonzero(x)) == 1
x = Symbol('x', imaginary=True)
assert refine(sign(x), Q.positive(im(x))) == S.ImaginaryUnit
assert refine(sign(x), Q.negative(im(x))) == -S.ImaginaryUnit
assert refine(sign(x), True) == sign(x)
x = Symbol('x', complex=True)
assert refine(sign(x), Q.zero(x)) == 0
def test_zero_pow():
assert satask(Q.zero(x**y), Q.zero(x) & Q.positive(y)) is True
assert satask(Q.zero(x**y), Q.nonzero(x) & Q.zero(y)) is False
assert satask(Q.zero(x), Q.zero(x**y)) is True
assert satask(Q.zero(x**y), Q.zero(x)) is None
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):
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_prime_composite():
assert satask(Q.prime(x), Q.composite(x)) is False
assert satask(Q.composite(x), Q.prime(x)) is False
assert satask(Q.composite(x), ~Q.prime(x)) is None
assert satask(Q.prime(x), ~Q.composite(x)) is None
# since 1 is neither prime nor composite the following should hold
assert satask(Q.prime(x), Q.integer(x) & Q.positive(x) & ~Q.composite(x)) is None
assert satask(Q.prime(2)) is True
assert satask(Q.prime(4)) is False
assert satask(Q.prime(1)) is False
assert satask(Q.composite(1)) is False
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_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_call():
x, y = symbols('x y')
# See the long history of this in issues 5026 and 5105.
raises(TypeError, lambda: sin(x)({x: 1, sin(x): 2}))
raises(TypeError, lambda: sin(x)(1))
# No effect as there are no callables
assert sin(x).rcall(1) == sin(x)
assert (1 + sin(x)).rcall(1) == 1 + sin(x)
# Effect in the pressence of callables
l = Lambda(x, 2 * x)
assert (l + x).rcall(y) == 2 * y + x
assert (x**l).rcall(2) == x**4
# TODO UndefinedFunction does not subclass Expr
#f = Function('f')
#assert (2*f)(x) == 2*f(x)
assert (Q.real & Q.positive).rcall(x) == Q.real(x) & Q.positive(x)
def test_positive_definite():
assert ask(Q.positive_definite(X), Q.positive_definite(X))
assert ask(Q.positive_definite(X.T), Q.positive_definite(X)) is True
assert ask(Q.positive_definite(X.I), Q.positive_definite(X)) is True
assert ask(Q.positive_definite(Y)) is False
assert ask(Q.positive_definite(X)) is None
assert ask(Q.positive_definite(X**3), Q.positive_definite(X))
assert ask(Q.positive_definite(X * Z * X),
Q.positive_definite(X) & Q.positive_definite(Z)) is True
assert ask(Q.positive_definite(X), Q.orthogonal(X))
assert ask(Q.positive_definite(Y.T * X * Y),
Q.positive_definite(X) & Q.fullrank(Y)) is True
assert not ask(Q.positive_definite(Y.T * X * Y), Q.positive_definite(X))
assert ask(Q.positive_definite(Identity(3))) is True
assert ask(Q.positive_definite(ZeroMatrix(3, 3))) is False
assert ask(Q.positive_definite(OneMatrix(1, 1))) is True
assert ask(Q.positive_definite(OneMatrix(3, 3))) is False
assert ask(Q.positive_definite(X + Z),
Q.positive_definite(X) & Q.positive_definite(Z)) is True
assert not ask(Q.positive_definite(-X), Q.positive_definite(X))
assert ask(Q.positive(X[1, 1]), Q.positive_definite(X))
def test_satask():
# No relevant facts
assert satask(Q.real(x), Q.real(x)) is True
assert satask(Q.real(x), ~Q.real(x)) is False
assert satask(Q.real(x)) is None
assert satask(Q.real(x), Q.positive(x)) is True
assert satask(Q.positive(x), Q.real(x)) is None
assert satask(Q.real(x), ~Q.positive(x)) is None
assert satask(Q.positive(x), ~Q.real(x)) is False
raises(ValueError, lambda: satask(Q.real(x), Q.real(x) & ~Q.real(x)))
with assuming(Q.positive(x)):
assert satask(Q.real(x)) is True
assert satask(~Q.positive(x)) is False
raises(ValueError, lambda: satask(Q.real(x), ~Q.positive(x)))
assert satask(Q.zero(x), Q.nonzero(x)) is False
assert satask(Q.positive(x), Q.zero(x)) is False
assert satask(Q.real(x), Q.zero(x)) is True
assert satask(Q.zero(x), Q.zero(x*y)) is None
assert satask(Q.zero(x*y), Q.zero(x))
def _contains(self, other):
from sympy.assumptions.ask import ask, Q
if ask(Q.positive(other)) and ask(Q.integer(other)):
return True
return False
def _contains(self, other):
if ask(Q.positive(other)) and ask(Q.integer(other)):
return True
return False
def test_arg():
x = Symbol('x', complex = True)
assert refine(arg(x), Q.positive(x)) == 0
assert refine(arg(x), Q.negative(x)) == pi
def test_allargs():
assert allargs(x, Q.zero(x), x * y) == And(Q.zero(x), Q.zero(y))
assert allargs(x,
Q.positive(x) | Q.negative(x), x * y) == And(
Q.positive(x) | Q.negative(x),
Q.positive(y) | Q.negative(y))
def test_anyarg():
assert anyarg(x, Q.zero(x), x * y) == Or(Q.zero(x), Q.zero(y))
assert anyarg(x, Q.positive(x) & Q.negative(x), x*y) == \
Or(Q.positive(x) & Q.negative(x), Q.positive(y) & Q.negative(y))
def _contains(self, other):
from sympy.assumptions.ask import ask, Q
if ask(Q.positive(other)) and ask(Q.integer(other)):
return True
return False
def test_composite_predicates():
pred = Q.integer | ~Q.positive
assert type(pred(x)) is Or
assert pred(x) == Q.integer(x) | ~Q.positive(x)
(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)),
# General Case: Odd number of imaginary args implies mul is imaginary(To be implemented)
(Mul, Implies(AllArgs(Q.real), Implies(ExactlyOneArg(Q.irrational),
Q.irrational))),
(Add, Implies(AllArgs(Q.real), Implies(ExactlyOneArg(Q.irrational),
Q.irrational))),
(Mul, Implies(AllArgs(Q.rational), Q.rational)),
def test_pretty():
assert pretty(Q.positive(x)) == "Q.positive(x)"
assert pretty(set([Q.positive, Q.integer])) == "set([Q.integer, Q.positive])"
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)),
# General Case: Odd number of imaginary args implies mul is imaginary(To be implemented)
(Mul,
Implies(AllArgs(Q.real), Implies(ExactlyOneArg(Q.irrational),
def test_equal():
"""Test for equality"""
x = symbols('x')
assert Q.positive(x) == Q.positive(x)
assert Q.positive(x) != ~Q.positive(x)
assert ~Q.positive(x) == ~Q.positive(x)
def test_pretty():
x = symbols('x')
assert pretty(Q.positive(x)) == "Q.positive(x)"
def register_fact(klass, fact, registry=fact_registry):
registry[klass] |= set([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)),
# 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)),
#General Case: Odd number of imaginary args implies mul is imaginary(To be implemented)
(Mul, Implies(AllArgs(Q.real), Implies(ExactlyOneArg(Q.irrational),
Q.irrational))),
(Add, Implies(AllArgs(Q.real), Implies(ExactlyOneArg(Q.irrational),
Q.irrational))),
(Mul, Implies(AllArgs(Q.rational), Q.rational)),
def test_equal():
"""Test for equality"""
assert Q.positive(x) == Q.positive(x)
assert Q.positive(x) != ~Q.positive(x)
assert ~Q.positive(x) == ~Q.positive(x)
def test_equal():
"""Test for equality"""
x = symbols('x')
assert Q.positive(x) == Q.positive(x)
assert Q.positive(x) != ~Q.positive(x)
assert ~Q.positive(x) == ~Q.positive(x)
def test_pretty():
assert pretty(Q.positive(x)) == "Q.positive(x)"
assert pretty(set([Q.positive,
Q.integer])) == "set([Q.integer, Q.positive])"
def test_equal():
"""Test for equality"""
assert Q.positive(x) == Q.positive(x)
assert Q.positive(x) != ~Q.positive(x)
assert ~Q.positive(x) == ~Q.positive(x)
def test_pretty():
assert pretty(Q.positive(x)) == "Q.positive(x)"
def test_composite_predicates():
pred = Q.integer | ~Q.positive
assert type(pred(x)) is Or
assert pred(x) == Q.integer(x) | ~Q.positive(x)
def test_pretty():
assert pretty(Q.positive(x)) == "Q.positive(x)"