# Source code for sympy.core.relational

```
from __future__ import print_function, division
from .basic import S
from .compatibility import ordered
from .expr import Expr
from .evalf import EvalfMixin
from .function import _coeff_isneg
from .sympify import _sympify
from .evaluate import global_evaluate
from sympy.logic.boolalg import Boolean, BooleanAtom
__all__ = (
'Rel', 'Eq', 'Ne', 'Lt', 'Le', 'Gt', 'Ge',
'Relational', 'Equality', 'Unequality', 'StrictLessThan', 'LessThan',
'StrictGreaterThan', 'GreaterThan',
)
# Note, see issue 4986. Ideally, we wouldn't want to subclass both Boolean
# and Expr.
def _canonical(cond):
# return a condition in which all relationals are canonical
try:
reps = dict([(r, r.canonical)
for r in cond.atoms(Relational)])
return cond.xreplace(reps)
except AttributeError:
return cond
class Relational(Boolean, Expr, EvalfMixin):
"""Base class for all relation types.
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.ValidRelationalOperator.
Examples
========
>>> from sympy import Rel
>>> from sympy.abc import x, y
>>> Rel(y, x+x**2, '==')
Eq(y, x**2 + x)
"""
__slots__ = []
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 Expr.
if cls is not Relational:
return Expr.__new__(cls, lhs, rhs, **assumptions)
# If called directly with an operator, look up the subclass
# corresponding to that operator and delegate to it
try:
cls = cls.ValidRelationOperator[rop]
return cls(lhs, rhs, **assumptions)
except KeyError:
raise ValueError("Invalid relational operator symbol: %r" % rop)
@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 (and sign) 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 = {Gt: Lt, Ge: Le, Lt: Gt, Le: Ge}
a, b = self.args
return ops.get(self.func, self.func)(b, a, evaluate=False)
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 else ordering the args. 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
"""
args = self.args
r = self
if r.rhs.is_Number:
if 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
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 self == other or self.reversed == other:
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
l, r = [i.equals(j, failing_expression=failing_expression)
for i, j in zip(a.args, b.args)]
if l is True:
return r
if r is True:
return l
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 = (l, r, 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
l = a.lhs.equals(b.lhs, failing_expression=failing_expression)
if l is False:
return False
r = a.rhs.equals(b.rhs, failing_expression=failing_expression)
if r is False:
return False
if l is True:
return r
return l
def _eval_simplify(self, ratio, measure):
r = self
r = r.func(*[i.simplify(ratio=ratio, measure=measure)
for i in r.args])
if r.is_Relational:
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 measure(r) < ratio*measure(self):
return r
else:
return self
def __nonzero__(self):
raise TypeError("cannot determine truth value of Relational")
__bool__ = __nonzero__
def as_set(self):
"""
Rewrites univariate inequality in terms of real sets
Examples
========
>>> from sympy import Symbol, Eq
>>> x = Symbol('x', real=True)
>>> (x > 0).as_set()
Interval.open(0, oo)
>>> Eq(x, 0).as_set()
{0}
"""
from sympy.solvers.inequalities import solve_univariate_inequality
syms = self.free_symbols
if len(syms) == 1:
sym = syms.pop()
else:
raise NotImplementedError("Sorry, Relational.as_set procedure"
" is not yet implemented for"
" multivariate expressions")
return solve_univariate_inequality(self, sym, relational=False)
Rel = Relational
[docs]class Equality(Relational):
"""An equal relation between two objects.
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
=====
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.
"""
rel_op = '=='
__slots__ = []
is_Equality = True
def __new__(cls, lhs, rhs=0, **options):
from sympy.core.add import Add
from sympy.core.logic import fuzzy_bool
from sympy.core.expr import _n2
from sympy.simplify.simplify import clear_coefficients
lhs = _sympify(lhs)
rhs = _sympify(rhs)
evaluate = options.pop('evaluate', global_evaluate[0])
if evaluate:
# If one expression has an _eval_Eq, return its results.
if hasattr(lhs, '_eval_Eq'):
r = lhs._eval_Eq(rhs)
if r is not None:
return r
if hasattr(rhs, '_eval_Eq'):
r = rhs._eval_Eq(lhs)
if r is not None:
return r
# If expressions have the same structure, they must be equal.
if lhs == rhs:
return S.true
elif all(isinstance(i, BooleanAtom) for i in (rhs, lhs)):
return S.false
# check finiteness
fin = L, R = [i.is_finite for i in (lhs, rhs)]
if None not in fin:
if L != R:
return S.false
if L is False:
if lhs == -rhs: # Eq(oo, -oo)
return S.false
return S.true
elif None in fin and False in fin:
return Relational.__new__(cls, lhs, rhs, **options)
if all(isinstance(i, Expr) for i in (lhs, rhs)):
# see if the difference evaluates
dif = lhs - rhs
z = dif.is_zero
if z is not None:
if z is False and dif.is_commutative: # issue 10728
return S.false
if z:
return S.true
# evaluate numerically if possible
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
if n.is_zero:
rv = d.is_nonzero
elif n.is_finite:
if d.is_infinite:
rv = S.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
l, r = clear_coefficients(d, S.Infinity)
args = [_.subs(l, r) for _ in (lhs, rhs)]
if args != [lhs, rhs]:
rv = fuzzy_bool(Eq(*args))
if rv is True:
rv = None
elif any(a.is_infinite for a in Add.make_args(n)): # (inf or nan)/x != 0
rv = S.false
if rv is not None:
return _sympify(rv)
return Relational.__new__(cls, lhs, rhs, **options)
@classmethod
def _eval_relation(cls, lhs, rhs):
return _sympify(lhs == rhs)
Eq = Equality
[docs]class Unequality(Relational):
"""An unequal relation between two objects.
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_evaluate[0])
if evaluate:
is_equal = Equality(lhs, rhs)
if isinstance(is_equal, BooleanAtom):
return ~is_equal
return Relational.__new__(cls, lhs, rhs, **options)
@classmethod
def _eval_relation(cls, lhs, rhs):
return _sympify(lhs != rhs)
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):
lhs = _sympify(lhs)
rhs = _sympify(rhs)
evaluate = options.pop('evaluate', global_evaluate[0])
if evaluate:
# 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.
r = cls._eval_relation(lhs, rhs)
if r is not None:
return r
# Note: not sure r could be None, perhaps we never take this
# path? In principle, could use this to shortcut out if a
# class realizes the inequality cannot be evaluated further.
# make a "non-evaluated" Expr for the inequality
return Relational.__new__(cls, lhs, rhs, **options)
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]
[docs]class GreaterThan(_Greater):
"""Class representations of inequalities.
Extended Summary
================
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 >= 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 >= rhs |
+----------------------------+-----------------+
|LessThan(lhs, rhs) | lhs <= 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:
>>> e1 = Ge( x, 2 ) # Ge is a convenience wrapper
>>> print(e1)
x >= 2
>>> rels = Ge( x, 2 ), Gt( x, 2 ), Le( x, 2 ), Lt( x, 2 )
>>> print('%s\\n%s\\n%s\\n%s' % rels)
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).
>>> e2 = x >= 2
>>> print(e2)
x >= 2
>>> print("e1: %s, e2: %s" % (e1, e2))
e1: x >= 2, e2: x >= 2
>>> e1 == e2
True
However, it is also perfectly valid to instantiate a ``*Than`` class less
succinctly and less conveniently:
>>> rels = Rel(x, 1, '>='), Relational(x, 1, '>='), GreaterThan(x, 1)
>>> print('%s\\n%s\\n%s' % rels)
x >= 1
x >= 1
x >= 1
>>> rels = Rel(x, 1, '>'), Relational(x, 1, '>'), StrictGreaterThan(x, 1)
>>> print('%s\\n%s\\n%s' % rels)
x > 1
x > 1
x > 1
>>> rels = Rel(x, 1, '<='), Relational(x, 1, '<='), LessThan(x, 1)
>>> print("%s\\n%s\\n%s" % rels)
x <= 1
x <= 1
x <= 1
>>> rels = Rel(x, 1, '<'), Relational(x, 1, '<'), StrictLessThan(x, 1)
>>> print('%s\\n%s\\n%s' % rels)
x < 1
x < 1
x < 1
Notes
=====
There are a couple of "gotchas" when using Python's operators.
The first enters the mix when comparing against a literal number as the lhs
argument. Due to the order that Python decides to parse a statement, it may
not immediately find two objects comparable. For example, to evaluate the
statement (1 < x), Python will first recognize the number 1 as a native
number, and then that x is *not* a native number. At this point, because a
native Python number does not know how to compare itself with a SymPy object
Python will try the reflective operation, (x > 1). Unfortunately, there is
no way available to SymPy to recognize this has happened, so the statement
(1 < x) will turn silently into (x > 1).
>>> e1 = x > 1
>>> e2 = x >= 1
>>> e3 = x < 1
>>> e4 = x <= 1
>>> e5 = 1 > x
>>> e6 = 1 >= x
>>> e7 = 1 < x
>>> e8 = 1 <= x
>>> print("%s %s\\n"*4 % (e1, e2, e3, e4, e5, e6, e7, e8))
x > 1 x >= 1
x < 1 x <= 1
x < 1 x <= 1
x > 1 x >= 1
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, (2) use one of the wrappers, or (3)
use the less succinct methods described above:
>>> e1 = S(1) > x
>>> e2 = S(1) >= x
>>> e3 = S(1) < x
>>> e4 = S(1) <= x
>>> e5 = Gt(1, x)
>>> e6 = Ge(1, x)
>>> e7 = Lt(1, x)
>>> e8 = Le(1, x)
>>> print("%s %s\\n"*4 % (e1, e2, e3, e4, e5, e6, e7, e8))
1 > x 1 >= x
1 < x 1 <= x
1 > x 1 >= x
1 < x 1 <= x
The other gotcha is with chained inequalities. Occasionally, one may be
tempted to write statements like:
>>> 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 reliably create that as a chained inequality. To create a
chained inequality, the only method currently available is to make use of
And:
>>> e = And(x < y, y < z)
>>> type( e )
And
>>> e
(x < y) & (y < z)
Note that this is different than chaining an equality directly via use of
parenthesis (this is currently an open bug in SymPy [2]_):
>>> e = (x < y) < z
>>> type( e )
<class 'sympy.core.relational.StrictLessThan'>
>>> e
(x < y) < z
Any code that explicitly relies on this latter functionality will not be
robust as this behaviour is completely wrong and will be corrected at some
point. For the time being (circa Jan 2012), use And to create chained
inequalities.
.. [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
http://docs.python.org/2/reference/expressions.html#notin). 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.__nonzero__()) 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 __nonzero__ 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.
.. [2] For more information, see these two bug reports:
"Separate boolean and symbolic relationals"
`Issue 4986 <https://github.com/sympy/sympy/issues/4986>`_
"It right 0 < x < 1 ?"
`Issue 6059 <https://github.com/sympy/sympy/issues/6059>`_
"""
__slots__ = ()
rel_op = '>='
@classmethod
def _eval_relation(cls, lhs, rhs):
# We don't use the op symbol here: workaround issue #7951
return _sympify(lhs.__ge__(rhs))
Ge = GreaterThan
[docs]class LessThan(_Less):
__doc__ = GreaterThan.__doc__
__slots__ = ()
rel_op = '<='
@classmethod
def _eval_relation(cls, lhs, rhs):
# We don't use the op symbol here: workaround issue #7951
return _sympify(lhs.__le__(rhs))
Le = LessThan
[docs]class StrictGreaterThan(_Greater):
__doc__ = GreaterThan.__doc__
__slots__ = ()
rel_op = '>'
@classmethod
def _eval_relation(cls, lhs, rhs):
# We don't use the op symbol here: workaround issue #7951
return _sympify(lhs.__gt__(rhs))
Gt = StrictGreaterThan
[docs]class StrictLessThan(_Less):
__doc__ = GreaterThan.__doc__
__slots__ = ()
rel_op = '<'
@classmethod
def _eval_relation(cls, lhs, rhs):
# We don't use the op symbol here: workaround issue #7951
return _sympify(lhs.__lt__(rhs))
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,
}
```