```
from __future__ import print_function, division
from .basic import S
from .expr import Expr
from .evalf import EvalfMixin
from .symbol import Symbol
from .sympify import _sympify
from sympy.logic.boolalg import Boolean
__all__ = (
'Rel', 'Eq', 'Ne', 'Lt', 'Le', 'Gt', 'Ge',
'Relational', 'Equality', 'Unequality', 'StrictLessThan', 'LessThan',
'StrictGreaterThan', 'GreaterThan',
)
# Note, see issue 1887. Ideally, we wouldn't want to subclass both Boolean
# and Expr.
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, '==')
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]
@classmethod
def _eval_sides(cls, lhs, rhs):
"""Takes the difference between lhs and rhs, simplifies and evaluates.
If the difference can be simplified to a single real number, it will
be evaluated with ``cls._eval_relation``. If the difference does not
simplify or cannot be calculated, None will be returned.
"""
if isinstance(lhs, Expr) and isinstance(rhs, Expr):
diff = lhs - rhs
if not diff.has(Symbol):
know = diff.equals(0)
if know == True:
diff = S.Zero
elif know == False:
diff = diff.evalf()
if diff.is_Number and diff.is_real:
return cls._eval_relation(diff, S.Zero)
def _eval_evalf(self, prec):
return self.func(*[s._evalf(prec) for s in self.args])
def doit(self, **hints):
lhs = self.lhs
rhs = self.rhs
if hints.get('deep', True):
lhs = lhs.doit(**hints)
rhs = rhs.doit(**hints)
return self._eval_relation_doit(lhs, rhs)
@classmethod
def _eval_relation_doit(cls, lhs, rhs):
return cls(lhs, rhs)
def _eval_simplify(self, ratio, measure):
return self.__class__(self.lhs.simplify(ratio=ratio),
self.rhs.simplify(ratio=ratio))
def __nonzero__(self):
raise TypeError("symbolic boolean expression has no truth value.")
__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()
(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 shown to be
definitively equal, this will reduce to True; if definitively unequal,
this will reduce to False. Otherwise, the relation is maintained as an
Equality object.
Examples
========
>>> from sympy import Eq
>>> from sympy.abc import x, y
>>> Eq(y, x+x**2)
y == x**2 + x
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.
"""
rel_op = '=='
__slots__ = []
is_Equality = True
def __new__(cls, lhs, rhs=0, **assumptions):
lhs = _sympify(lhs)
rhs = _sympify(rhs)
# If expressions have the same structure, they must be equal.
if lhs == rhs:
return S.true
# If one side is real and the other complex, they must be unequal.
elif (lhs.is_real != rhs.is_real and
None not in (lhs.is_real, rhs.is_real)):
return S.false
# Otherwise, see if the difference can be evaluated.
r = cls._eval_sides(lhs, rhs)
if r is not None:
return r
# If not, pass arguments to Relational.
return Relational.__new__(cls, lhs, rhs, **assumptions)
@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)
y != x**2 + x
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.
"""
rel_op = '!='
__slots__ = []
def __new__(cls, lhs, rhs, **assumptions):
lhs = _sympify(lhs)
rhs = _sympify(rhs)
is_equal = Equality(lhs, rhs)
if is_equal == True or is_equal == False:
return ~is_equal
return Relational.__new__(cls, lhs, rhs, **assumptions)
@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, **assumptions):
lhs = _sympify(lhs)
rhs = _sympify(rhs)
# Try to evaluate the difference between sides.
r = cls._eval_sides(lhs, rhs)
if r is not None:
return r
# If that fails, pass arguments to Relational.
return Relational.__new__(cls, lhs, rhs, **assumptions)
@classmethod
def _eval_relation_doit(cls, lhs, rhs):
return cls._eval_relation(lhs, rhs)
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
And(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/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 1887 <http://code.google.com/p/sympy/issues/detail?id=1887>`_
"It right 0 < x < 1 ?"
`Issue 2960 <http://code.google.com/p/sympy/issues/detail?id=2960>`_
"""
rel_op = '>='
__slots__ = ()
@classmethod
def _eval_relation(cls, lhs, rhs):
return _sympify(lhs >= rhs)
Ge = GreaterThan
[docs]class LessThan(_Less):
__doc__ = GreaterThan.__doc__
__slots__ = ()
rel_op = '<='
@classmethod
def _eval_relation(cls, lhs, rhs):
return _sympify(lhs <= rhs)
Le = LessThan
[docs]class StrictGreaterThan(_Greater):
__doc__ = GreaterThan.__doc__
__slots__ = ()
rel_op = '>'
@classmethod
def _eval_relation(cls, lhs, rhs):
return _sympify(lhs > rhs)
Gt = StrictGreaterThan
[docs]class StrictLessThan(_Less):
__doc__ = GreaterThan.__doc__
__slots__ = ()
rel_op = '<'
@classmethod
def _eval_relation(cls, lhs, rhs):
return _sympify(lhs < 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,
}
```