def test_fuzzy_or():
assert fuzzy_or([T, T]) == T
assert fuzzy_or([T, F]) == T
assert fuzzy_or([T, U]) == T
assert fuzzy_or([F, F]) == F
assert fuzzy_or([F, U]) == U
assert fuzzy_or([U, U]) == U
assert [fuzzy_or([w]) for w in [U, T, F]] == [U, T, F]
assert fuzzy_or([T, F, U]) == T
assert fuzzy_or([]) == F
raises(TypeError, lambda: fuzzy_or())
def is_monotonic(f, interval=S.Reals, symbol=None):
"""
Returns if a function is monotonic or not, in the given
``Interval``.
Examples
========
>>> from sympy import is_monotonic
>>> from sympy.abc import x, y
>>> from sympy import S, Interval, oo
>>> is_monotonic(1/(x**2 - 3*x), Interval.open(1.5, 3))
True
>>> is_monotonic(1/(x**2 - 3*x), Interval.Lopen(3, oo))
True
>>> is_monotonic(x**3 - 3*x**2 + 4*x, S.Reals)
True
>>> is_monotonic(-x**2, S.Reals)
False
>>> is_monotonic(x**2 + y + 1, Interval(1, 2), x)
True
"""
from sympy.core.logic import fuzzy_or
f = sympify(f)
free_sym = f.free_symbols
if symbol is None and len(free_sym) > 1:
raise NotImplementedError('is_monotonic has not yet been '
'for all multivariate expressions')
inc = is_increasing(f, interval, symbol)
dec = is_decreasing(f, interval, symbol)
return fuzzy_or([inc, dec])
def is_monotonic(f, interval=S.Reals):
"""
Returns if a function is monotonic or not, in the given
``Interval``.
Examples
========
>>> from sympy import is_monotonic
>>> from sympy.abc import x
>>> from sympy import S, Interval, oo
>>> is_monotonic(1/(x**2 - 3*x), Interval.open(1.5, 3))
True
>>> is_monotonic(1/(x**2 - 3*x), Interval.Lopen(3, oo))
True
>>> is_monotonic(x**3 - 3*x**2 + 4*x, S.Reals)
True
>>> is_monotonic(-x**2, S.Reals)
False
"""
from sympy.core.logic import fuzzy_or
if len(f.free_symbols) > 1:
raise NotImplementedError('is_monotonic has not yet been '
'implemented for multivariate expressions')
inc = is_increasing(f, interval)
dec = is_decreasing(f, interval)
return fuzzy_or([inc, dec])
def Basic(expr, assumptions):
if expr.is_number:
# if there are no symbols just evalf
i = expr.evalf(2)
def nonz(i):
if i._prec != 1:
return i != 0
return fuzzy_or(nonz(i) for i in i.as_real_imag())
def Basic(expr, assumptions):
if ask(Q.real(expr)) is False:
return False
if expr.is_number:
# if there are no symbols just evalf
i = expr.evalf(2)
def nonz(i):
if i._prec != 1:
return i != 0
return fuzzy_or(nonz(i) for i in i.as_real_imag())
def _eval_is_positive(self):
arg = self.args[0]
if arg.is_real:
return True
re, im = arg.as_real_imag()
im_mod = im % (2*pi)
if im_mod == 0:
return True
if re == 0:
if im_mod < pi/2 or im_mod > 3*pi/2:
return True
elif im_mod >= pi/2 or im_mod <= 3*pi/2:
return False
return fuzzy_or([fuzzy_and([fuzzy_bool(Eq(re, 0)),
fuzzy_or([fuzzy_bool(im_mod < pi/2),
fuzzy_bool(im_mod > 3*pi/2)])]),
fuzzy_bool(Eq(im_mod, 0))])
def _contains(self, other):
from sympy.functions import arg, Abs
from sympy.core.containers import Tuple
other = sympify(other)
isTuple = isinstance(other, Tuple)
if isTuple and len(other) != 2:
raise ValueError("expecting Tuple of length 2")
# If the other is not an Expression, and neither a Tuple
if not isinstance(other, Expr) and not isinstance(other, Tuple):
return S.false
# self in rectangular form
if not self.polar:
re, im = other if isTuple else other.as_real_imag()
return fuzzy_or(
fuzzy_and(
[pset.args[0]._contains(re), pset.args[1]._contains(im)])
for pset in self.psets)
# self in polar form
elif self.polar:
if other.is_zero:
# ignore undefined complex argument
return fuzzy_or(pset.args[0]._contains(S.Zero)
for pset in self.psets)
if isTuple:
r, theta = other
else:
r, theta = Abs(other), arg(other)
if theta.is_real and theta.is_number:
# angles in psets are normalized to [0, 2pi)
theta %= 2 * S.Pi
return fuzzy_or(
fuzzy_and([
pset.args[0]._contains(r), pset.args[1]._contains(
theta)
]) for pset in self.psets)
def is_subset_sets(a_interval, b_u): # noqa:F811
if all(isinstance(s, (Interval, FiniteSet)) for s in b_u.args):
intervals = [s for s in b_u.args if isinstance(s, Interval)]
if all(fuzzy_bool(a_interval.start < s.start) for s in intervals):
return False
if all(fuzzy_bool(a_interval.end > s.end) for s in intervals):
return False
if a_interval.measure.is_nonzero:
no_overlap = lambda s1, s2: fuzzy_or([
fuzzy_bool(s1.end <= s2.start),
fuzzy_bool(s1.start >= s2.end),
])
if all(no_overlap(s, a_interval) for s in intervals):
return False
def _eval_is_nonnegative(self):
z = self.args[0]
x, y = z.as_real_imag()
ymod = y % (2 * pi)
yzero = ymod.is_zero
# shortcut if ymod is zero
if yzero:
return True
xzero = x.is_zero
# shortcut x is not zero
if xzero is False:
return yzero
return fuzzy_or([
# Case 1:
yzero,
# Case 2:
fuzzy_and([xzero,
fuzzy_or([ymod <= pi / 2, ymod >= 3 * pi / 2])])
])
def arc_coplanar(self, other):
"""
Returns True if the transformation arcs represented by the input quaternions happen in the same plane.
Explanation
===========
Two quaternions are said to be coplanar (in this arc sense) when their axes are parallel.
The plane of a quaternion is the one normal to its axis.
Parameters
==========
other : a Quaternion
Returns
=======
True : if the planes of the two quaternions are the same, apart from its orientation/sign.
False : if the planes of the two quaternions are not the same, apart from its orientation/sign.
None : if plane of either of the quaternion is unknown.
Examples
========
>>> from sympy.algebras.quaternion import Quaternion
>>> q1 = Quaternion(1, 4, 4, 4)
>>> q2 = Quaternion(3, 8, 8, 8)
>>> Quaternion.arc_coplanar(q1, q2)
True
>>> q1 = Quaternion(2, 8, 13, 12)
>>> Quaternion.arc_coplanar(q1, q2)
False
See Also
========
vector_coplanar
is_pure
"""
if (self.is_zero_quaternion()) or (other.is_zero_quaternion()):
raise ValueError('Neither of the given quaternions can be 0')
return fuzzy_or([(self.axis() - other.axis()).is_zero_quaternion(),
(self.axis() + other.axis()).is_zero_quaternion()])
def __new__(cls, flambda, *sets):
if not isinstance(flambda, Lambda):
raise ValueError('first argument must be a Lambda')
if flambda is S.IdentityFunction:
if len(sets) != 1:
raise ValueError('identify function requires a single set')
return sets[0]
if not set(flambda.variables) & flambda.expr.free_symbols:
emptyprod = fuzzy_or(s.is_empty for s in sets)
if emptyprod == True:
return S.EmptySet
elif emptyprod == False:
return FiniteSet(flambda.expr)
return Basic.__new__(cls, flambda, *sets)
def _old_assump_replacer(obj):
# Things to be careful of:
# - real means real or infinite in the old assumptions.
# - nonzero does not imply real in the old assumptions.
# - finite means finite and not zero in the old assumptions.
if not isinstance(obj, AppliedPredicate):
return obj
e = obj.args[0]
ret = None
if obj.func == Q.positive:
ret = fuzzy_and([e.is_finite, e.is_positive])
if obj.func == Q.zero:
ret = e.is_zero
if obj.func == Q.negative:
ret = fuzzy_and([e.is_finite, e.is_negative])
if obj.func == Q.nonpositive:
ret = fuzzy_and([e.is_finite, e.is_nonpositive])
if obj.func == Q.nonzero:
ret = fuzzy_and([e.is_nonzero, e.is_finite])
if obj.func == Q.nonnegative:
ret = fuzzy_and([fuzzy_or([e.is_zero, e.is_finite]),
e.is_nonnegative])
if obj.func == Q.rational:
ret = e.is_rational
if obj.func == Q.irrational:
ret = e.is_irrational
if obj.func == Q.even:
ret = e.is_even
if obj.func == Q.odd:
ret = e.is_odd
if obj.func == Q.integer:
ret = e.is_integer
if obj.func == Q.imaginary:
ret = e.is_imaginary
if obj.func == Q.commutative:
ret = e.is_commutative
if ret is None:
return obj
return ret
def _old_assump_replacer(obj):
# Things to be careful of:
# - real means real or infinite in the old assumptions.
# - nonzero does not imply real in the old assumptions.
# - finite means finite and not zero in the old assumptions.
if not isinstance(obj, AppliedPredicate):
return obj
e = obj.args[0]
ret = None
if obj.func == Q.positive:
ret = fuzzy_and([e.is_finite, e.is_positive])
if obj.func == Q.zero:
ret = e.is_zero
if obj.func == Q.negative:
ret = fuzzy_and([e.is_finite, e.is_negative])
if obj.func == Q.nonpositive:
ret = fuzzy_and([e.is_finite, e.is_nonpositive])
if obj.func == Q.nonzero:
ret = fuzzy_and([e.is_nonzero, e.is_finite])
if obj.func == Q.nonnegative:
ret = fuzzy_and([fuzzy_or([e.is_zero, e.is_finite]),
e.is_nonnegative])
if obj.func == Q.rational:
ret = e.is_rational
if obj.func == Q.irrational:
ret = e.is_irrational
if obj.func == Q.even:
ret = e.is_even
if obj.func == Q.odd:
ret = e.is_odd
if obj.func == Q.integer:
ret = e.is_integer
if obj.func == Q.imaginary:
ret = e.is_imaginary
if obj.func == Q.commutative:
ret = e.is_commutative
if ret is None:
return obj
return ret
def __new__(cls, flambda, *sets):
if not isinstance(flambda, Lambda):
raise ValueError('First argument must be a Lambda')
sets = [_sympify(s) for s in sets]
if flambda is S.IdentityFunction:
if len(sets) != 1:
raise ValueError('Identity function requires a single set')
return sets[0]
if not all(isinstance(s, Set) for s in sets):
raise TypeError("Set arguments to ImageSet should of type Set")
sets = [s.flatten() if s.is_ProductSet else s for s in sets]
if not set(flambda.variables) & flambda.expr.free_symbols:
emptyprod = fuzzy_or(s.is_empty for s in sets)
if emptyprod == True:
return S.EmptySet
elif emptyprod == False:
return FiniteSet(flambda.expr)
return Basic.__new__(cls, flambda, *sets)
def _eval_is_complex(self):
def complex_extended_negative(arg):
yield arg.is_complex
yield arg.is_extended_negative
return fuzzy_or(complex_extended_negative(self.args[0]))
def _eval_is_extended_positive(self):
return fuzzy_or(a.is_extended_positive for a in self.args)
def _eval_is_zero(self):
# is_imaginary implies nonzero
return fuzzy_or([self.args[0].is_imaginary, self.args[0].is_zero])
def _eval_is_zero(self):
return fuzzy_or([self.args[0].is_zero, self.args[0].is_integer])
def _eval_is_zero(self):
# is_imaginary implies nonzero
return fuzzy_or([self.args[0].is_imaginary, self.args[0].is_zero])
def Mul(expr, assumptions):
# TODO: This should be deducible from the nonzero handler
return fuzzy_or(ask(Q.zero(arg), assumptions) for arg in expr.args)
def _eval_is_negative(self):
return fuzzy_or(a.is_negative for a in self.args)
def Mul(expr, assumptions):
# TODO: This should be deducible from the nonzero handler
return fuzzy_or(ask(Q.zero(arg), assumptions) for arg in expr.args)
def __getitem__(self, i):
ooslice = "cannot slice from the end with an infinite value"
zerostep = "slice step cannot be zero"
infinite = "slicing not possible on range with infinite start"
# if we had to take every other element in the following
# oo, ..., 6, 4, 2, 0
# we might get oo, ..., 4, 0 or oo, ..., 6, 2
ambiguous = "cannot unambiguously re-stride from the end " + \
"with an infinite value"
if isinstance(i, slice):
if self.size.is_finite: # validates, too
if self.start == self.stop:
return Range(0)
start, stop, step = i.indices(self.size)
n = ceiling((stop - start) / step)
if n <= 0:
return Range(0)
canonical_stop = start + n * step
end = canonical_stop - step
ss = step * self.step
return Range(self[start], self[end] + ss, ss)
else: # infinite Range
start = i.start
stop = i.stop
if i.step == 0:
raise ValueError(zerostep)
step = i.step or 1
ss = step * self.step
#---------------------
# handle infinite Range
# i.e. Range(-oo, oo) or Range(oo, -oo, -1)
# --------------------
if self.start.is_infinite and self.stop.is_infinite:
raise ValueError(infinite)
#---------------------
# handle infinite on right
# e.g. Range(0, oo) or Range(0, -oo, -1)
# --------------------
if self.stop.is_infinite:
# start and stop are not interdependent --
# they only depend on step --so we use the
# equivalent reversed values
return self.reversed[
stop if stop is None else -stop +
1:start if start is None else -start:step].reversed
#---------------------
# handle infinite on the left
# e.g. Range(oo, 0, -1) or Range(-oo, 0)
# --------------------
# consider combinations of
# start/stop {== None, < 0, == 0, > 0} and
# step {< 0, > 0}
if start is None:
if stop is None:
if step < 0:
return Range(self[-1], self.start, ss)
elif step > 1:
raise ValueError(ambiguous)
else: # == 1
return self
elif stop < 0:
if step < 0:
return Range(self[-1], self[stop], ss)
else: # > 0
return Range(self.start, self[stop], ss)
elif stop == 0:
if step > 0:
return Range(0)
else: # < 0
raise ValueError(ooslice)
elif stop == 1:
if step > 0:
raise ValueError(ooslice) # infinite singleton
else: # < 0
raise ValueError(ooslice)
else: # > 1
raise ValueError(ooslice)
elif start < 0:
if stop is None:
if step < 0:
return Range(self[start], self.start, ss)
else: # > 0
return Range(self[start], self.stop, ss)
elif stop < 0:
return Range(self[start], self[stop], ss)
elif stop == 0:
if step < 0:
raise ValueError(ooslice)
else: # > 0
return Range(0)
elif stop > 0:
raise ValueError(ooslice)
elif start == 0:
if stop is None:
if step < 0:
raise ValueError(ooslice) # infinite singleton
elif step > 1:
raise ValueError(ambiguous)
else: # == 1
return self
elif stop < 0:
if step > 1:
raise ValueError(ambiguous)
elif step == 1:
return Range(self.start, self[stop], ss)
else: # < 0
return Range(0)
else: # >= 0
raise ValueError(ooslice)
elif start > 0:
raise ValueError(ooslice)
else:
if self.start == self.stop:
raise IndexError('Range index out of range')
if not (all(i.is_integer or i.is_infinite for i in self.args) and (
(self.stop - self.start) / self.step).is_extended_positive):
raise ValueError('Invalid method for symbolic Range')
if i == 0:
if self.start.is_infinite:
raise ValueError(ooslice)
return self.start
if i == -1:
if self.stop.is_infinite:
raise ValueError(ooslice)
return self.stop - self.step
n = self.size # must be known for any other index
rv = (self.stop if i < 0 else self.start) + i * self.step
if rv.is_infinite:
raise ValueError(ooslice)
val = (rv - self.start) / self.step
rel = fuzzy_or([
val.is_infinite,
fuzzy_and([val.is_nonnegative, (n - val).is_nonnegative])
])
if rel:
return rv
if rel is None:
raise ValueError('Invalid method for symbolic Range')
raise IndexError("Range index out of range")
def _eval_is_random(self):
return fuzzy_or(arg.is_random for arg in self.args)
def _eval_is_negative(self):
return fuzzy_or(a.is_negative for a in self.args)
