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ai-content-maker/.venv/Lib/site-packages/sympy/core/relational.py

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2024-05-03 04:18:51 +03:00
from __future__ import annotations
from .basic import Atom, Basic
from .sorting import ordered
from .evalf import EvalfMixin
from .function import AppliedUndef
from .singleton import S
from .sympify import _sympify, SympifyError
from .parameters import global_parameters
from .logic import fuzzy_bool, fuzzy_xor, fuzzy_and, fuzzy_not
from sympy.logic.boolalg import Boolean, BooleanAtom
from sympy.utilities.iterables import sift
from sympy.utilities.misc import filldedent
__all__ = (
'Rel', 'Eq', 'Ne', 'Lt', 'Le', 'Gt', 'Ge',
'Relational', 'Equality', 'Unequality', 'StrictLessThan', 'LessThan',
'StrictGreaterThan', 'GreaterThan',
)
from .expr import Expr
from sympy.multipledispatch import dispatch
from .containers import Tuple
from .symbol import Symbol
def _nontrivBool(side):
return isinstance(side, Boolean) and \
not isinstance(side, Atom)
# Note, see issue 4986. Ideally, we wouldn't want to subclass both Boolean
# and Expr.
# from .. import Expr
def _canonical(cond):
# return a condition in which all relationals are canonical
reps = {r: r.canonical for r in cond.atoms(Relational)}
return cond.xreplace(reps)
# XXX: AttributeError was being caught here but it wasn't triggered by any of
# the tests so I've removed it...
def _canonical_coeff(rel):
# return -2*x + 1 < 0 as x > 1/2
# XXX make this part of Relational.canonical?
rel = rel.canonical
if not rel.is_Relational or rel.rhs.is_Boolean:
return rel # Eq(x, True)
b, l = rel.lhs.as_coeff_Add(rational=True)
m, lhs = l.as_coeff_Mul(rational=True)
rhs = (rel.rhs - b)/m
if m < 0:
return rel.reversed.func(lhs, rhs)
return rel.func(lhs, rhs)
class Relational(Boolean, EvalfMixin):
"""Base class for all relation types.
Explanation
===========
Subclasses of Relational should generally be instantiated directly, but
Relational can be instantiated with a valid ``rop`` value to dispatch to
the appropriate subclass.
Parameters
==========
rop : str or None
Indicates what subclass to instantiate. Valid values can be found
in the keys of Relational.ValidRelationOperator.
Examples
========
>>> from sympy import Rel
>>> from sympy.abc import x, y
>>> Rel(y, x + x**2, '==')
Eq(y, x**2 + x)
A relation's type can be defined upon creation using ``rop``.
The relation type of an existing expression can be obtained
using its ``rel_op`` property.
Here is a table of all the relation types, along with their
``rop`` and ``rel_op`` values:
+---------------------+----------------------------+------------+
|Relation |``rop`` |``rel_op`` |
+=====================+============================+============+
|``Equality`` |``==`` or ``eq`` or ``None``|``==`` |
+---------------------+----------------------------+------------+
|``Unequality`` |``!=`` or ``ne`` |``!=`` |
+---------------------+----------------------------+------------+
|``GreaterThan`` |``>=`` or ``ge`` |``>=`` |
+---------------------+----------------------------+------------+
|``LessThan`` |``<=`` or ``le`` |``<=`` |
+---------------------+----------------------------+------------+
|``StrictGreaterThan``|``>`` or ``gt`` |``>`` |
+---------------------+----------------------------+------------+
|``StrictLessThan`` |``<`` or ``lt`` |``<`` |
+---------------------+----------------------------+------------+
For example, setting ``rop`` to ``==`` produces an
``Equality`` relation, ``Eq()``.
So does setting ``rop`` to ``eq``, or leaving ``rop`` unspecified.
That is, the first three ``Rel()`` below all produce the same result.
Using a ``rop`` from a different row in the table produces a
different relation type.
For example, the fourth ``Rel()`` below using ``lt`` for ``rop``
produces a ``StrictLessThan`` inequality:
>>> from sympy import Rel
>>> from sympy.abc import x, y
>>> Rel(y, x + x**2, '==')
Eq(y, x**2 + x)
>>> Rel(y, x + x**2, 'eq')
Eq(y, x**2 + x)
>>> Rel(y, x + x**2)
Eq(y, x**2 + x)
>>> Rel(y, x + x**2, 'lt')
y < x**2 + x
To obtain the relation type of an existing expression,
get its ``rel_op`` property.
For example, ``rel_op`` is ``==`` for the ``Equality`` relation above,
and ``<`` for the strict less than inequality above:
>>> from sympy import Rel
>>> from sympy.abc import x, y
>>> my_equality = Rel(y, x + x**2, '==')
>>> my_equality.rel_op
'=='
>>> my_inequality = Rel(y, x + x**2, 'lt')
>>> my_inequality.rel_op
'<'
"""
__slots__ = ()
ValidRelationOperator: dict[str | None, type[Relational]] = {}
is_Relational = True
# ValidRelationOperator - Defined below, because the necessary classes
# have not yet been defined
def __new__(cls, lhs, rhs, rop=None, **assumptions):
# If called by a subclass, do nothing special and pass on to Basic.
if cls is not Relational:
return Basic.__new__(cls, lhs, rhs, **assumptions)
# XXX: Why do this? There should be a separate function to make a
# particular subclass of Relational from a string.
#
# If called directly with an operator, look up the subclass
# corresponding to that operator and delegate to it
cls = cls.ValidRelationOperator.get(rop, None)
if cls is None:
raise ValueError("Invalid relational operator symbol: %r" % rop)
if not issubclass(cls, (Eq, Ne)):
# validate that Booleans are not being used in a relational
# other than Eq/Ne;
# Note: Symbol is a subclass of Boolean but is considered
# acceptable here.
if any(map(_nontrivBool, (lhs, rhs))):
raise TypeError(filldedent('''
A Boolean argument can only be used in
Eq and Ne; all other relationals expect
real expressions.
'''))
return cls(lhs, rhs, **assumptions)
@property
def lhs(self):
"""The left-hand side of the relation."""
return self._args[0]
@property
def rhs(self):
"""The right-hand side of the relation."""
return self._args[1]
@property
def reversed(self):
"""Return the relationship with sides reversed.
Examples
========
>>> from sympy import Eq
>>> from sympy.abc import x
>>> Eq(x, 1)
Eq(x, 1)
>>> _.reversed
Eq(1, x)
>>> x < 1
x < 1
>>> _.reversed
1 > x
"""
ops = {Eq: Eq, Gt: Lt, Ge: Le, Lt: Gt, Le: Ge, Ne: Ne}
a, b = self.args
return Relational.__new__(ops.get(self.func, self.func), b, a)
@property
def reversedsign(self):
"""Return the relationship with signs reversed.
Examples
========
>>> from sympy import Eq
>>> from sympy.abc import x
>>> Eq(x, 1)
Eq(x, 1)
>>> _.reversedsign
Eq(-x, -1)
>>> x < 1
x < 1
>>> _.reversedsign
-x > -1
"""
a, b = self.args
if not (isinstance(a, BooleanAtom) or isinstance(b, BooleanAtom)):
ops = {Eq: Eq, Gt: Lt, Ge: Le, Lt: Gt, Le: Ge, Ne: Ne}
return Relational.__new__(ops.get(self.func, self.func), -a, -b)
else:
return self
@property
def negated(self):
"""Return the negated relationship.
Examples
========
>>> from sympy import Eq
>>> from sympy.abc import x
>>> Eq(x, 1)
Eq(x, 1)
>>> _.negated
Ne(x, 1)
>>> x < 1
x < 1
>>> _.negated
x >= 1
Notes
=====
This works more or less identical to ``~``/``Not``. The difference is
that ``negated`` returns the relationship even if ``evaluate=False``.
Hence, this is useful in code when checking for e.g. negated relations
to existing ones as it will not be affected by the `evaluate` flag.
"""
ops = {Eq: Ne, Ge: Lt, Gt: Le, Le: Gt, Lt: Ge, Ne: Eq}
# If there ever will be new Relational subclasses, the following line
# will work until it is properly sorted out
# return ops.get(self.func, lambda a, b, evaluate=False: ~(self.func(a,
# b, evaluate=evaluate)))(*self.args, evaluate=False)
return Relational.__new__(ops.get(self.func), *self.args)
@property
def weak(self):
"""return the non-strict version of the inequality or self
EXAMPLES
========
>>> from sympy.abc import x
>>> (x < 1).weak
x <= 1
>>> _.weak
x <= 1
"""
return self
@property
def strict(self):
"""return the strict version of the inequality or self
EXAMPLES
========
>>> from sympy.abc import x
>>> (x <= 1).strict
x < 1
>>> _.strict
x < 1
"""
return self
def _eval_evalf(self, prec):
return self.func(*[s._evalf(prec) for s in self.args])
@property
def canonical(self):
"""Return a canonical form of the relational by putting a
number on the rhs, canonically removing a sign or else
ordering the args canonically. No other simplification is
attempted.
Examples
========
>>> from sympy.abc import x, y
>>> x < 2
x < 2
>>> _.reversed.canonical
x < 2
>>> (-y < x).canonical
x > -y
>>> (-y > x).canonical
x < -y
>>> (-y < -x).canonical
x < y
The canonicalization is recursively applied:
>>> from sympy import Eq
>>> Eq(x < y, y > x).canonical
True
"""
args = tuple([i.canonical if isinstance(i, Relational) else i for i in self.args])
if args != self.args:
r = self.func(*args)
if not isinstance(r, Relational):
return r
else:
r = self
if r.rhs.is_number:
if r.rhs.is_Number and r.lhs.is_Number and r.lhs > r.rhs:
r = r.reversed
elif r.lhs.is_number:
r = r.reversed
elif tuple(ordered(args)) != args:
r = r.reversed
LHS_CEMS = getattr(r.lhs, 'could_extract_minus_sign', None)
RHS_CEMS = getattr(r.rhs, 'could_extract_minus_sign', None)
if isinstance(r.lhs, BooleanAtom) or isinstance(r.rhs, BooleanAtom):
return r
# Check if first value has negative sign
if LHS_CEMS and LHS_CEMS():
return r.reversedsign
elif not r.rhs.is_number and RHS_CEMS and RHS_CEMS():
# Right hand side has a minus, but not lhs.
# How does the expression with reversed signs behave?
# This is so that expressions of the type
# Eq(x, -y) and Eq(-x, y)
# have the same canonical representation
expr1, _ = ordered([r.lhs, -r.rhs])
if expr1 != r.lhs:
return r.reversed.reversedsign
return r
def equals(self, other, failing_expression=False):
"""Return True if the sides of the relationship are mathematically
identical and the type of relationship is the same.
If failing_expression is True, return the expression whose truth value
was unknown."""
if isinstance(other, Relational):
if other in (self, self.reversed):
return True
a, b = self, other
if a.func in (Eq, Ne) or b.func in (Eq, Ne):
if a.func != b.func:
return False
left, right = [i.equals(j,
failing_expression=failing_expression)
for i, j in zip(a.args, b.args)]
if left is True:
return right
if right is True:
return left
lr, rl = [i.equals(j, failing_expression=failing_expression)
for i, j in zip(a.args, b.reversed.args)]
if lr is True:
return rl
if rl is True:
return lr
e = (left, right, lr, rl)
if all(i is False for i in e):
return False
for i in e:
if i not in (True, False):
return i
else:
if b.func != a.func:
b = b.reversed
if a.func != b.func:
return False
left = a.lhs.equals(b.lhs,
failing_expression=failing_expression)
if left is False:
return False
right = a.rhs.equals(b.rhs,
failing_expression=failing_expression)
if right is False:
return False
if left is True:
return right
return left
def _eval_simplify(self, **kwargs):
from .add import Add
from .expr import Expr
r = self
r = r.func(*[i.simplify(**kwargs) for i in r.args])
if r.is_Relational:
if not isinstance(r.lhs, Expr) or not isinstance(r.rhs, Expr):
return r
dif = r.lhs - r.rhs
# replace dif with a valid Number that will
# allow a definitive comparison with 0
v = None
if dif.is_comparable:
v = dif.n(2)
elif dif.equals(0): # XXX this is expensive
v = S.Zero
if v is not None:
r = r.func._eval_relation(v, S.Zero)
r = r.canonical
# If there is only one symbol in the expression,
# try to write it on a simplified form
free = list(filter(lambda x: x.is_real is not False, r.free_symbols))
if len(free) == 1:
try:
from sympy.solvers.solveset import linear_coeffs
x = free.pop()
dif = r.lhs - r.rhs
m, b = linear_coeffs(dif, x)
if m.is_zero is False:
if m.is_negative:
# Dividing with a negative number, so change order of arguments
# canonical will put the symbol back on the lhs later
r = r.func(-b / m, x)
else:
r = r.func(x, -b / m)
else:
r = r.func(b, S.Zero)
except ValueError:
# maybe not a linear function, try polynomial
from sympy.polys.polyerrors import PolynomialError
from sympy.polys.polytools import gcd, Poly, poly
try:
p = poly(dif, x)
c = p.all_coeffs()
constant = c[-1]
c[-1] = 0
scale = gcd(c)
c = [ctmp / scale for ctmp in c]
r = r.func(Poly.from_list(c, x).as_expr(), -constant / scale)
except PolynomialError:
pass
elif len(free) >= 2:
try:
from sympy.solvers.solveset import linear_coeffs
from sympy.polys.polytools import gcd
free = list(ordered(free))
dif = r.lhs - r.rhs
m = linear_coeffs(dif, *free)
constant = m[-1]
del m[-1]
scale = gcd(m)
m = [mtmp / scale for mtmp in m]
nzm = list(filter(lambda f: f[0] != 0, list(zip(m, free))))
if scale.is_zero is False:
if constant != 0:
# lhs: expression, rhs: constant
newexpr = Add(*[i * j for i, j in nzm])
r = r.func(newexpr, -constant / scale)
else:
# keep first term on lhs
lhsterm = nzm[0][0] * nzm[0][1]
del nzm[0]
newexpr = Add(*[i * j for i, j in nzm])
r = r.func(lhsterm, -newexpr)
else:
r = r.func(constant, S.Zero)
except ValueError:
pass
# Did we get a simplified result?
r = r.canonical
measure = kwargs['measure']
if measure(r) < kwargs['ratio'] * measure(self):
return r
else:
return self
def _eval_trigsimp(self, **opts):
from sympy.simplify.trigsimp import trigsimp
return self.func(trigsimp(self.lhs, **opts), trigsimp(self.rhs, **opts))
def expand(self, **kwargs):
args = (arg.expand(**kwargs) for arg in self.args)
return self.func(*args)
def __bool__(self):
raise TypeError("cannot determine truth value of Relational")
def _eval_as_set(self):
# self is univariate and periodicity(self, x) in (0, None)
from sympy.solvers.inequalities import solve_univariate_inequality
from sympy.sets.conditionset import ConditionSet
syms = self.free_symbols
assert len(syms) == 1
x = syms.pop()
try:
xset = solve_univariate_inequality(self, x, relational=False)
except NotImplementedError:
# solve_univariate_inequality raises NotImplementedError for
# unsolvable equations/inequalities.
xset = ConditionSet(x, self, S.Reals)
return xset
@property
def binary_symbols(self):
# override where necessary
return set()
Rel = Relational
class Equality(Relational):
"""
An equal relation between two objects.
Explanation
===========
Represents that two objects are equal. If they can be easily shown
to be definitively equal (or unequal), this will reduce to True (or
False). Otherwise, the relation is maintained as an unevaluated
Equality object. Use the ``simplify`` function on this object for
more nontrivial evaluation of the equality relation.
As usual, the keyword argument ``evaluate=False`` can be used to
prevent any evaluation.
Examples
========
>>> from sympy import Eq, simplify, exp, cos
>>> from sympy.abc import x, y
>>> Eq(y, x + x**2)
Eq(y, x**2 + x)
>>> Eq(2, 5)
False
>>> Eq(2, 5, evaluate=False)
Eq(2, 5)
>>> _.doit()
False
>>> Eq(exp(x), exp(x).rewrite(cos))
Eq(exp(x), sinh(x) + cosh(x))
>>> simplify(_)
True
See Also
========
sympy.logic.boolalg.Equivalent : for representing equality between two
boolean expressions
Notes
=====
Python treats 1 and True (and 0 and False) as being equal; SymPy
does not. And integer will always compare as unequal to a Boolean:
>>> Eq(True, 1), True == 1
(False, True)
This class is not the same as the == operator. The == operator tests
for exact structural equality between two expressions; this class
compares expressions mathematically.
If either object defines an ``_eval_Eq`` method, it can be used in place of
the default algorithm. If ``lhs._eval_Eq(rhs)`` or ``rhs._eval_Eq(lhs)``
returns anything other than None, that return value will be substituted for
the Equality. If None is returned by ``_eval_Eq``, an Equality object will
be created as usual.
Since this object is already an expression, it does not respond to
the method ``as_expr`` if one tries to create `x - y` from ``Eq(x, y)``.
This can be done with the ``rewrite(Add)`` method.
.. deprecated:: 1.5
``Eq(expr)`` with a single argument is a shorthand for ``Eq(expr, 0)``,
but this behavior is deprecated and will be removed in a future version
of SymPy.
"""
rel_op = '=='
__slots__ = ()
is_Equality = True
def __new__(cls, lhs, rhs, **options):
evaluate = options.pop('evaluate', global_parameters.evaluate)
lhs = _sympify(lhs)
rhs = _sympify(rhs)
if evaluate:
val = is_eq(lhs, rhs)
if val is None:
return cls(lhs, rhs, evaluate=False)
else:
return _sympify(val)
return Relational.__new__(cls, lhs, rhs)
@classmethod
def _eval_relation(cls, lhs, rhs):
return _sympify(lhs == rhs)
def _eval_rewrite_as_Add(self, *args, **kwargs):
"""
return Eq(L, R) as L - R. To control the evaluation of
the result set pass `evaluate=True` to give L - R;
if `evaluate=None` then terms in L and R will not cancel
but they will be listed in canonical order; otherwise
non-canonical args will be returned. If one side is 0, the
non-zero side will be returned.
Examples
========
>>> from sympy import Eq, Add
>>> from sympy.abc import b, x
>>> eq = Eq(x + b, x - b)
>>> eq.rewrite(Add)
2*b
>>> eq.rewrite(Add, evaluate=None).args
(b, b, x, -x)
>>> eq.rewrite(Add, evaluate=False).args
(b, x, b, -x)
"""
from .add import _unevaluated_Add, Add
L, R = args
if L == 0:
return R
if R == 0:
return L
evaluate = kwargs.get('evaluate', True)
if evaluate:
# allow cancellation of args
return L - R
args = Add.make_args(L) + Add.make_args(-R)
if evaluate is None:
# no cancellation, but canonical
return _unevaluated_Add(*args)
# no cancellation, not canonical
return Add._from_args(args)
@property
def binary_symbols(self):
if S.true in self.args or S.false in self.args:
if self.lhs.is_Symbol:
return {self.lhs}
elif self.rhs.is_Symbol:
return {self.rhs}
return set()
def _eval_simplify(self, **kwargs):
# standard simplify
e = super()._eval_simplify(**kwargs)
if not isinstance(e, Equality):
return e
from .expr import Expr
if not isinstance(e.lhs, Expr) or not isinstance(e.rhs, Expr):
return e
free = self.free_symbols
if len(free) == 1:
try:
from .add import Add
from sympy.solvers.solveset import linear_coeffs
x = free.pop()
m, b = linear_coeffs(
e.rewrite(Add, evaluate=False), x)
if m.is_zero is False:
enew = e.func(x, -b / m)
else:
enew = e.func(m * x, -b)
measure = kwargs['measure']
if measure(enew) <= kwargs['ratio'] * measure(e):
e = enew
except ValueError:
pass
return e.canonical
def integrate(self, *args, **kwargs):
"""See the integrate function in sympy.integrals"""
from sympy.integrals.integrals import integrate
return integrate(self, *args, **kwargs)
def as_poly(self, *gens, **kwargs):
'''Returns lhs-rhs as a Poly
Examples
========
>>> from sympy import Eq
>>> from sympy.abc import x
>>> Eq(x**2, 1).as_poly(x)
Poly(x**2 - 1, x, domain='ZZ')
'''
return (self.lhs - self.rhs).as_poly(*gens, **kwargs)
Eq = Equality
class Unequality(Relational):
"""An unequal relation between two objects.
Explanation
===========
Represents that two objects are not equal. If they can be shown to be
definitively equal, this will reduce to False; if definitively unequal,
this will reduce to True. Otherwise, the relation is maintained as an
Unequality object.
Examples
========
>>> from sympy import Ne
>>> from sympy.abc import x, y
>>> Ne(y, x+x**2)
Ne(y, x**2 + x)
See Also
========
Equality
Notes
=====
This class is not the same as the != operator. The != operator tests
for exact structural equality between two expressions; this class
compares expressions mathematically.
This class is effectively the inverse of Equality. As such, it uses the
same algorithms, including any available `_eval_Eq` methods.
"""
rel_op = '!='
__slots__ = ()
def __new__(cls, lhs, rhs, **options):
lhs = _sympify(lhs)
rhs = _sympify(rhs)
evaluate = options.pop('evaluate', global_parameters.evaluate)
if evaluate:
val = is_neq(lhs, rhs)
if val is None:
return cls(lhs, rhs, evaluate=False)
else:
return _sympify(val)
return Relational.__new__(cls, lhs, rhs, **options)
@classmethod
def _eval_relation(cls, lhs, rhs):
return _sympify(lhs != rhs)
@property
def binary_symbols(self):
if S.true in self.args or S.false in self.args:
if self.lhs.is_Symbol:
return {self.lhs}
elif self.rhs.is_Symbol:
return {self.rhs}
return set()
def _eval_simplify(self, **kwargs):
# simplify as an equality
eq = Equality(*self.args)._eval_simplify(**kwargs)
if isinstance(eq, Equality):
# send back Ne with the new args
return self.func(*eq.args)
return eq.negated # result of Ne is the negated Eq
Ne = Unequality
class _Inequality(Relational):
"""Internal base class for all *Than types.
Each subclass must implement _eval_relation to provide the method for
comparing two real numbers.
"""
__slots__ = ()
def __new__(cls, lhs, rhs, **options):
try:
lhs = _sympify(lhs)
rhs = _sympify(rhs)
except SympifyError:
return NotImplemented
evaluate = options.pop('evaluate', global_parameters.evaluate)
if evaluate:
for me in (lhs, rhs):
if me.is_extended_real is False:
raise TypeError("Invalid comparison of non-real %s" % me)
if me is S.NaN:
raise TypeError("Invalid NaN comparison")
# First we invoke the appropriate inequality method of `lhs`
# (e.g., `lhs.__lt__`). That method will try to reduce to
# boolean or raise an exception. It may keep calling
# superclasses until it reaches `Expr` (e.g., `Expr.__lt__`).
# In some cases, `Expr` will just invoke us again (if neither it
# nor a subclass was able to reduce to boolean or raise an
# exception). In that case, it must call us with
# `evaluate=False` to prevent infinite recursion.
return cls._eval_relation(lhs, rhs, **options)
# make a "non-evaluated" Expr for the inequality
return Relational.__new__(cls, lhs, rhs, **options)
@classmethod
def _eval_relation(cls, lhs, rhs, **options):
val = cls._eval_fuzzy_relation(lhs, rhs)
if val is None:
return cls(lhs, rhs, evaluate=False)
else:
return _sympify(val)
class _Greater(_Inequality):
"""Not intended for general use
_Greater is only used so that GreaterThan and StrictGreaterThan may
subclass it for the .gts and .lts properties.
"""
__slots__ = ()
@property
def gts(self):
return self._args[0]
@property
def lts(self):
return self._args[1]
class _Less(_Inequality):
"""Not intended for general use.
_Less is only used so that LessThan and StrictLessThan may subclass it for
the .gts and .lts properties.
"""
__slots__ = ()
@property
def gts(self):
return self._args[1]
@property
def lts(self):
return self._args[0]
class GreaterThan(_Greater):
r"""Class representations of inequalities.
Explanation
===========
The ``*Than`` classes represent inequal relationships, where the left-hand
side is generally bigger or smaller than the right-hand side. For example,
the GreaterThan class represents an inequal relationship where the
left-hand side is at least as big as the right side, if not bigger. In
mathematical notation:
lhs $\ge$ rhs
In total, there are four ``*Than`` classes, to represent the four
inequalities:
+-----------------+--------+
|Class Name | Symbol |
+=================+========+
|GreaterThan | ``>=`` |
+-----------------+--------+
|LessThan | ``<=`` |
+-----------------+--------+
|StrictGreaterThan| ``>`` |
+-----------------+--------+
|StrictLessThan | ``<`` |
+-----------------+--------+
All classes take two arguments, lhs and rhs.
+----------------------------+-----------------+
|Signature Example | Math Equivalent |
+============================+=================+
|GreaterThan(lhs, rhs) | lhs $\ge$ rhs |
+----------------------------+-----------------+
|LessThan(lhs, rhs) | lhs $\le$ rhs |
+----------------------------+-----------------+
|StrictGreaterThan(lhs, rhs) | lhs $>$ rhs |
+----------------------------+-----------------+
|StrictLessThan(lhs, rhs) | lhs $<$ rhs |
+----------------------------+-----------------+
In addition to the normal .lhs and .rhs of Relations, ``*Than`` inequality
objects also have the .lts and .gts properties, which represent the "less
than side" and "greater than side" of the operator. Use of .lts and .gts
in an algorithm rather than .lhs and .rhs as an assumption of inequality
direction will make more explicit the intent of a certain section of code,
and will make it similarly more robust to client code changes:
>>> from sympy import GreaterThan, StrictGreaterThan
>>> from sympy import LessThan, StrictLessThan
>>> from sympy import And, Ge, Gt, Le, Lt, Rel, S
>>> from sympy.abc import x, y, z
>>> from sympy.core.relational import Relational
>>> e = GreaterThan(x, 1)
>>> e
x >= 1
>>> '%s >= %s is the same as %s <= %s' % (e.gts, e.lts, e.lts, e.gts)
'x >= 1 is the same as 1 <= x'
Examples
========
One generally does not instantiate these classes directly, but uses various
convenience methods:
>>> for f in [Ge, Gt, Le, Lt]: # convenience wrappers
... print(f(x, 2))
x >= 2
x > 2
x <= 2
x < 2
Another option is to use the Python inequality operators (``>=``, ``>``,
``<=``, ``<``) directly. Their main advantage over the ``Ge``, ``Gt``,
``Le``, and ``Lt`` counterparts, is that one can write a more
"mathematical looking" statement rather than littering the math with
oddball function calls. However there are certain (minor) caveats of
which to be aware (search for 'gotcha', below).
>>> x >= 2
x >= 2
>>> _ == Ge(x, 2)
True
However, it is also perfectly valid to instantiate a ``*Than`` class less
succinctly and less conveniently:
>>> Rel(x, 1, ">")
x > 1
>>> Relational(x, 1, ">")
x > 1
>>> StrictGreaterThan(x, 1)
x > 1
>>> GreaterThan(x, 1)
x >= 1
>>> LessThan(x, 1)
x <= 1
>>> StrictLessThan(x, 1)
x < 1
Notes
=====
There are a couple of "gotchas" to be aware of when using Python's
operators.
The first is that what your write is not always what you get:
>>> 1 < x
x > 1
Due to the order that Python parses a statement, it may
not immediately find two objects comparable. When ``1 < x``
is evaluated, Python recognizes that the number 1 is a native
number and that x is *not*. Because a native Python number does
not know how to compare itself with a SymPy object
Python will try the reflective operation, ``x > 1`` and that is the
form that gets evaluated, hence returned.
If the order of the statement is important (for visual output to
the console, perhaps), one can work around this annoyance in a
couple ways:
(1) "sympify" the literal before comparison
>>> S(1) < x
1 < x
(2) use one of the wrappers or less succinct methods described
above
>>> Lt(1, x)
1 < x
>>> Relational(1, x, "<")
1 < x
The second gotcha involves writing equality tests between relationals
when one or both sides of the test involve a literal relational:
>>> e = x < 1; e
x < 1
>>> e == e # neither side is a literal
True
>>> e == x < 1 # expecting True, too
False
>>> e != x < 1 # expecting False
x < 1
>>> x < 1 != x < 1 # expecting False or the same thing as before
Traceback (most recent call last):
...
TypeError: cannot determine truth value of Relational
The solution for this case is to wrap literal relationals in
parentheses:
>>> e == (x < 1)
True
>>> e != (x < 1)
False
>>> (x < 1) != (x < 1)
False
The third gotcha involves chained inequalities not involving
``==`` or ``!=``. Occasionally, one may be tempted to write:
>>> e = x < y < z
Traceback (most recent call last):
...
TypeError: symbolic boolean expression has no truth value.
Due to an implementation detail or decision of Python [1]_,
there is no way for SymPy to create a chained inequality with
that syntax so one must use And:
>>> e = And(x < y, y < z)
>>> type( e )
And
>>> e
(x < y) & (y < z)
Although this can also be done with the '&' operator, it cannot
be done with the 'and' operarator:
>>> (x < y) & (y < z)
(x < y) & (y < z)
>>> (x < y) and (y < z)
Traceback (most recent call last):
...
TypeError: cannot determine truth value of Relational
.. [1] This implementation detail is that Python provides no reliable
method to determine that a chained inequality is being built.
Chained comparison operators are evaluated pairwise, using "and"
logic (see
https://docs.python.org/3/reference/expressions.html#not-in). This
is done in an efficient way, so that each object being compared
is only evaluated once and the comparison can short-circuit. For
example, ``1 > 2 > 3`` is evaluated by Python as ``(1 > 2) and (2
> 3)``. The ``and`` operator coerces each side into a bool,
returning the object itself when it short-circuits. The bool of
the --Than operators will raise TypeError on purpose, because
SymPy cannot determine the mathematical ordering of symbolic
expressions. Thus, if we were to compute ``x > y > z``, with
``x``, ``y``, and ``z`` being Symbols, Python converts the
statement (roughly) into these steps:
(1) x > y > z
(2) (x > y) and (y > z)
(3) (GreaterThanObject) and (y > z)
(4) (GreaterThanObject.__bool__()) and (y > z)
(5) TypeError
Because of the ``and`` added at step 2, the statement gets turned into a
weak ternary statement, and the first object's ``__bool__`` method will
raise TypeError. Thus, creating a chained inequality is not possible.
In Python, there is no way to override the ``and`` operator, or to
control how it short circuits, so it is impossible to make something
like ``x > y > z`` work. There was a PEP to change this,
:pep:`335`, but it was officially closed in March, 2012.
"""
__slots__ = ()
rel_op = '>='
@classmethod
def _eval_fuzzy_relation(cls, lhs, rhs):
return is_ge(lhs, rhs)
@property
def strict(self):
return Gt(*self.args)
Ge = GreaterThan
class LessThan(_Less):
__doc__ = GreaterThan.__doc__
__slots__ = ()
rel_op = '<='
@classmethod
def _eval_fuzzy_relation(cls, lhs, rhs):
return is_le(lhs, rhs)
@property
def strict(self):
return Lt(*self.args)
Le = LessThan
class StrictGreaterThan(_Greater):
__doc__ = GreaterThan.__doc__
__slots__ = ()
rel_op = '>'
@classmethod
def _eval_fuzzy_relation(cls, lhs, rhs):
return is_gt(lhs, rhs)
@property
def weak(self):
return Ge(*self.args)
Gt = StrictGreaterThan
class StrictLessThan(_Less):
__doc__ = GreaterThan.__doc__
__slots__ = ()
rel_op = '<'
@classmethod
def _eval_fuzzy_relation(cls, lhs, rhs):
return is_lt(lhs, rhs)
@property
def weak(self):
return Le(*self.args)
Lt = StrictLessThan
# A class-specific (not object-specific) data item used for a minor speedup.
# It is defined here, rather than directly in the class, because the classes
# that it references have not been defined until now (e.g. StrictLessThan).
Relational.ValidRelationOperator = {
None: Equality,
'==': Equality,
'eq': Equality,
'!=': Unequality,
'<>': Unequality,
'ne': Unequality,
'>=': GreaterThan,
'ge': GreaterThan,
'<=': LessThan,
'le': LessThan,
'>': StrictGreaterThan,
'gt': StrictGreaterThan,
'<': StrictLessThan,
'lt': StrictLessThan,
}
def _n2(a, b):
"""Return (a - b).evalf(2) if a and b are comparable, else None.
This should only be used when a and b are already sympified.
"""
# /!\ it is very important (see issue 8245) not to
# use a re-evaluated number in the calculation of dif
if a.is_comparable and b.is_comparable:
dif = (a - b).evalf(2)
if dif.is_comparable:
return dif
@dispatch(Expr, Expr)
def _eval_is_ge(lhs, rhs):
return None
@dispatch(Basic, Basic)
def _eval_is_eq(lhs, rhs):
return None
@dispatch(Tuple, Expr) # type: ignore
def _eval_is_eq(lhs, rhs): # noqa:F811
return False
@dispatch(Tuple, AppliedUndef) # type: ignore
def _eval_is_eq(lhs, rhs): # noqa:F811
return None
@dispatch(Tuple, Symbol) # type: ignore
def _eval_is_eq(lhs, rhs): # noqa:F811
return None
@dispatch(Tuple, Tuple) # type: ignore
def _eval_is_eq(lhs, rhs): # noqa:F811
if len(lhs) != len(rhs):
return False
return fuzzy_and(fuzzy_bool(is_eq(s, o)) for s, o in zip(lhs, rhs))
def is_lt(lhs, rhs, assumptions=None):
"""Fuzzy bool for lhs is strictly less than rhs.
See the docstring for :func:`~.is_ge` for more.
"""
return fuzzy_not(is_ge(lhs, rhs, assumptions))
def is_gt(lhs, rhs, assumptions=None):
"""Fuzzy bool for lhs is strictly greater than rhs.
See the docstring for :func:`~.is_ge` for more.
"""
return fuzzy_not(is_le(lhs, rhs, assumptions))
def is_le(lhs, rhs, assumptions=None):
"""Fuzzy bool for lhs is less than or equal to rhs.
See the docstring for :func:`~.is_ge` for more.
"""
return is_ge(rhs, lhs, assumptions)
def is_ge(lhs, rhs, assumptions=None):
"""
Fuzzy bool for *lhs* is greater than or equal to *rhs*.
Parameters
==========
lhs : Expr
The left-hand side of the expression, must be sympified,
and an instance of expression. Throws an exception if
lhs is not an instance of expression.
rhs : Expr
The right-hand side of the expression, must be sympified
and an instance of expression. Throws an exception if
lhs is not an instance of expression.
assumptions: Boolean, optional
Assumptions taken to evaluate the inequality.
Returns
=======
``True`` if *lhs* is greater than or equal to *rhs*, ``False`` if *lhs*
is less than *rhs*, and ``None`` if the comparison between *lhs* and
*rhs* is indeterminate.
Explanation
===========
This function is intended to give a relatively fast determination and
deliberately does not attempt slow calculations that might help in
obtaining a determination of True or False in more difficult cases.
The four comparison functions ``is_le``, ``is_lt``, ``is_ge``, and ``is_gt`` are
each implemented in terms of ``is_ge`` in the following way:
is_ge(x, y) := is_ge(x, y)
is_le(x, y) := is_ge(y, x)
is_lt(x, y) := fuzzy_not(is_ge(x, y))
is_gt(x, y) := fuzzy_not(is_ge(y, x))
Therefore, supporting new type with this function will ensure behavior for
other three functions as well.
To maintain these equivalences in fuzzy logic it is important that in cases where
either x or y is non-real all comparisons will give None.
Examples
========
>>> from sympy import S, Q
>>> from sympy.core.relational import is_ge, is_le, is_gt, is_lt
>>> from sympy.abc import x
>>> is_ge(S(2), S(0))
True
>>> is_ge(S(0), S(2))
False
>>> is_le(S(0), S(2))
True
>>> is_gt(S(0), S(2))
False
>>> is_lt(S(2), S(0))
False
Assumptions can be passed to evaluate the quality which is otherwise
indeterminate.
>>> print(is_ge(x, S(0)))
None
>>> is_ge(x, S(0), assumptions=Q.positive(x))
True
New types can be supported by dispatching to ``_eval_is_ge``.
>>> from sympy import Expr, sympify
>>> from sympy.multipledispatch import dispatch
>>> class MyExpr(Expr):
... def __new__(cls, arg):
... return super().__new__(cls, sympify(arg))
... @property
... def value(self):
... return self.args[0]
>>> @dispatch(MyExpr, MyExpr)
... def _eval_is_ge(a, b):
... return is_ge(a.value, b.value)
>>> a = MyExpr(1)
>>> b = MyExpr(2)
>>> is_ge(b, a)
True
>>> is_le(a, b)
True
"""
from sympy.assumptions.wrapper import AssumptionsWrapper, is_extended_nonnegative
if not (isinstance(lhs, Expr) and isinstance(rhs, Expr)):
raise TypeError("Can only compare inequalities with Expr")
retval = _eval_is_ge(lhs, rhs)
if retval is not None:
return retval
else:
n2 = _n2(lhs, rhs)
if n2 is not None:
# use float comparison for infinity.
# otherwise get stuck in infinite recursion
if n2 in (S.Infinity, S.NegativeInfinity):
n2 = float(n2)
return n2 >= 0
_lhs = AssumptionsWrapper(lhs, assumptions)
_rhs = AssumptionsWrapper(rhs, assumptions)
if _lhs.is_extended_real and _rhs.is_extended_real:
if (_lhs.is_infinite and _lhs.is_extended_positive) or (_rhs.is_infinite and _rhs.is_extended_negative):
return True
diff = lhs - rhs
if diff is not S.NaN:
rv = is_extended_nonnegative(diff, assumptions)
if rv is not None:
return rv
def is_neq(lhs, rhs, assumptions=None):
"""Fuzzy bool for lhs does not equal rhs.
See the docstring for :func:`~.is_eq` for more.
"""
return fuzzy_not(is_eq(lhs, rhs, assumptions))
def is_eq(lhs, rhs, assumptions=None):
"""
Fuzzy bool representing mathematical equality between *lhs* and *rhs*.
Parameters
==========
lhs : Expr
The left-hand side of the expression, must be sympified.
rhs : Expr
The right-hand side of the expression, must be sympified.
assumptions: Boolean, optional
Assumptions taken to evaluate the equality.
Returns
=======
``True`` if *lhs* is equal to *rhs*, ``False`` is *lhs* is not equal to *rhs*,
and ``None`` if the comparison between *lhs* and *rhs* is indeterminate.
Explanation
===========
This function is intended to give a relatively fast determination and
deliberately does not attempt slow calculations that might help in
obtaining a determination of True or False in more difficult cases.
:func:`~.is_neq` calls this function to return its value, so supporting
new type with this function will ensure correct behavior for ``is_neq``
as well.
Examples
========
>>> from sympy import Q, S
>>> from sympy.core.relational import is_eq, is_neq
>>> from sympy.abc import x
>>> is_eq(S(0), S(0))
True
>>> is_neq(S(0), S(0))
False
>>> is_eq(S(0), S(2))
False
>>> is_neq(S(0), S(2))
True
Assumptions can be passed to evaluate the equality which is otherwise
indeterminate.
>>> print(is_eq(x, S(0)))
None
>>> is_eq(x, S(0), assumptions=Q.zero(x))
True
New types can be supported by dispatching to ``_eval_is_eq``.
>>> from sympy import Basic, sympify
>>> from sympy.multipledispatch import dispatch
>>> class MyBasic(Basic):
... def __new__(cls, arg):
... return Basic.__new__(cls, sympify(arg))
... @property
... def value(self):
... return self.args[0]
...
>>> @dispatch(MyBasic, MyBasic)
... def _eval_is_eq(a, b):
... return is_eq(a.value, b.value)
...
>>> a = MyBasic(1)
>>> b = MyBasic(1)
>>> is_eq(a, b)
True
>>> is_neq(a, b)
False
"""
# here, _eval_Eq is only called for backwards compatibility
# new code should use is_eq with multiple dispatch as
# outlined in the docstring
for side1, side2 in (lhs, rhs), (rhs, lhs):
eval_func = getattr(side1, '_eval_Eq', None)
if eval_func is not None:
retval = eval_func(side2)
if retval is not None:
return retval
retval = _eval_is_eq(lhs, rhs)
if retval is not None:
return retval
if dispatch(type(lhs), type(rhs)) != dispatch(type(rhs), type(lhs)):
retval = _eval_is_eq(rhs, lhs)
if retval is not None:
return retval
# retval is still None, so go through the equality logic
# If expressions have the same structure, they must be equal.
if lhs == rhs:
return True # e.g. True == True
elif all(isinstance(i, BooleanAtom) for i in (rhs, lhs)):
return False # True != False
elif not (lhs.is_Symbol or rhs.is_Symbol) and (
isinstance(lhs, Boolean) !=
isinstance(rhs, Boolean)):
return False # only Booleans can equal Booleans
from sympy.assumptions.wrapper import (AssumptionsWrapper,
is_infinite, is_extended_real)
from .add import Add
_lhs = AssumptionsWrapper(lhs, assumptions)
_rhs = AssumptionsWrapper(rhs, assumptions)
if _lhs.is_infinite or _rhs.is_infinite:
if fuzzy_xor([_lhs.is_infinite, _rhs.is_infinite]):
return False
if fuzzy_xor([_lhs.is_extended_real, _rhs.is_extended_real]):
return False
if fuzzy_and([_lhs.is_extended_real, _rhs.is_extended_real]):
return fuzzy_xor([_lhs.is_extended_positive, fuzzy_not(_rhs.is_extended_positive)])
# Try to split real/imaginary parts and equate them
I = S.ImaginaryUnit
def split_real_imag(expr):
real_imag = lambda t: (
'real' if is_extended_real(t, assumptions) else
'imag' if is_extended_real(I*t, assumptions) else None)
return sift(Add.make_args(expr), real_imag)
lhs_ri = split_real_imag(lhs)
if not lhs_ri[None]:
rhs_ri = split_real_imag(rhs)
if not rhs_ri[None]:
eq_real = is_eq(Add(*lhs_ri['real']), Add(*rhs_ri['real']), assumptions)
eq_imag = is_eq(I * Add(*lhs_ri['imag']), I * Add(*rhs_ri['imag']), assumptions)
return fuzzy_and(map(fuzzy_bool, [eq_real, eq_imag]))
from sympy.functions.elementary.complexes import arg
# Compare e.g. zoo with 1+I*oo by comparing args
arglhs = arg(lhs)
argrhs = arg(rhs)
# Guard against Eq(nan, nan) -> False
if not (arglhs == S.NaN and argrhs == S.NaN):
return fuzzy_bool(is_eq(arglhs, argrhs, assumptions))
if all(isinstance(i, Expr) for i in (lhs, rhs)):
# see if the difference evaluates
dif = lhs - rhs
_dif = AssumptionsWrapper(dif, assumptions)
z = _dif.is_zero
if z is not None:
if z is False and _dif.is_commutative: # issue 10728
return False
if z:
return True
n2 = _n2(lhs, rhs)
if n2 is not None:
return _sympify(n2 == 0)
# see if the ratio evaluates
n, d = dif.as_numer_denom()
rv = None
_n = AssumptionsWrapper(n, assumptions)
_d = AssumptionsWrapper(d, assumptions)
if _n.is_zero:
rv = _d.is_nonzero
elif _n.is_finite:
if _d.is_infinite:
rv = True
elif _n.is_zero is False:
rv = _d.is_infinite
if rv is None:
# if the condition that makes the denominator
# infinite does not make the original expression
# True then False can be returned
from sympy.simplify.simplify import clear_coefficients
l, r = clear_coefficients(d, S.Infinity)
args = [_.subs(l, r) for _ in (lhs, rhs)]
if args != [lhs, rhs]:
rv = fuzzy_bool(is_eq(*args, assumptions))
if rv is True:
rv = None
elif any(is_infinite(a, assumptions) for a in Add.make_args(n)):
# (inf or nan)/x != 0
rv = False
if rv is not None:
return rv