```
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',
)
[docs]def Rel(a, b, op):
"""
A handy wrapper around the Relational class.
Rel(a,b, op)
Examples
========
>>> from sympy import Rel
>>> from sympy.abc import x, y
>>> Rel(y, x+x**2, '==')
y == x**2 + x
"""
return Relational(a, b, op)
[docs]def Eq(a, b=0):
"""
A handy wrapper around the Relational class.
Eq(a,b)
Examples
========
>>> from sympy import Eq
>>> from sympy.abc import x, y
>>> Eq(y, x+x**2)
y == x**2 + x
"""
return Relational(a, b, '==')
[docs]def Ne(a, b):
"""
A handy wrapper around the Relational class.
Ne(a,b)
Examples
========
>>> from sympy import Ne
>>> from sympy.abc import x, y
>>> Ne(y, x+x**2)
y != x**2 + x
"""
return Relational(a, b, '!=')
[docs]def Lt(a, b):
"""
A handy wrapper around the Relational class.
Lt(a,b)
Examples
========
>>> from sympy import Lt
>>> from sympy.abc import x, y
>>> Lt(y, x+x**2)
y < x**2 + x
"""
return Relational(a, b, '<')
[docs]def Le(a, b):
"""
A handy wrapper around the Relational class.
Le(a,b)
Examples
========
>>> from sympy import Le
>>> from sympy.abc import x, y
>>> Le(y, x+x**2)
y <= x**2 + x
"""
return Relational(a, b, '<=')
[docs]def Gt(a, b):
"""
A handy wrapper around the Relational class.
Gt(a,b)
Examples
========
>>> from sympy import Gt
>>> from sympy.abc import x, y
>>> Gt(y, x + x**2)
y > x**2 + x
"""
return Relational(a, b, '>')
[docs]def Ge(a, b):
"""
A handy wrapper around the Relational class.
Ge(a,b)
Examples
========
>>> from sympy import Ge
>>> from sympy.abc import x, y
>>> Ge(y, x + x**2)
y >= x**2 + x
"""
return Relational(a, b, '>=')
# Note, see issue 1887. Ideally, we wouldn't want to subclass both Boolean
# and Expr.
class Relational(Boolean, Expr, EvalfMixin):
__slots__ = []
is_Relational = True
# ValidRelationOperator - Defined below, because the necessary classes
# have not yet been defined
def __new__(cls, lhs, rhs, rop=None, **assumptions):
lhs = _sympify(lhs)
rhs = _sympify(rhs)
if cls is not Relational:
rop_cls = cls
else:
try:
rop_cls = Relational.ValidRelationOperator[ rop ]
except KeyError:
msg = "Invalid relational operator symbol: '%r'"
raise ValueError(msg % repr(rop))
diff = S.NaN
if isinstance(lhs, Expr) and isinstance(rhs, Expr):
diff = lhs - rhs
if not (diff is S.NaN or diff.has(Symbol)):
know = diff.equals(0, failing_expression=True)
if know is True: # exclude failing expression case
diff = S.Zero
elif know is False:
diff = diff.n()
if rop_cls is Equality:
if (lhs == rhs) is True or (diff == S.Zero) is True:
return True
elif diff is S.NaN:
pass
elif diff.is_Number or diff.is_Float:
return False
elif lhs.is_real is not rhs.is_real and \
lhs.is_real is not None and \
rhs.is_real is not None:
return False
elif rop_cls is Unequality:
if (lhs == rhs) is True or (diff == S.Zero) is True:
return False
elif diff is S.NaN:
pass
elif diff.is_Number or diff.is_Float:
return True
elif lhs.is_real is not rhs.is_real and \
lhs.is_real is not None and \
rhs.is_real is not None:
return True
elif diff.is_Number and diff.is_real:
return rop_cls._eval_relation(diff, S.Zero)
obj = Expr.__new__(rop_cls, lhs, rhs, **assumptions)
return obj
@property
def lhs(self):
return self._args[0]
@property
def rhs(self):
return self._args[1]
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._eval_relation(lhs, rhs)
def _eval_simplify(self, ratio, measure):
return self.__class__(self.lhs.simplify(ratio=ratio),
self.rhs.simplify(ratio=ratio))
[docs]class Equality(Relational):
rel_op = '=='
__slots__ = []
is_Equality = True
@classmethod
def _eval_relation(cls, lhs, rhs):
return lhs == rhs
@classmethod
def _eval_relation_doit(cls, lhs, rhs):
return Eq(lhs, rhs)
def __nonzero__(self):
return self.lhs.compare(self.rhs) == 0
[docs]class Unequality(Relational):
rel_op = '!='
__slots__ = []
@classmethod
def _eval_relation(cls, lhs, rhs):
return lhs != rhs
@classmethod
def _eval_relation_doit(cls, lhs, rhs):
return Ne(lhs, rhs)
def __nonzero__(self):
return self.lhs.compare(self.rhs) != 0
class _Greater(Relational):
"""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(Relational):
"""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 # silent error! Where did ``x`` go?
>>> e #doctest: +SKIP
y < z
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. Currently, the bool of the --Than
operators will give True or False arbitrarily. 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) (True) and (y > z)
(6) (y > z)
(7) LessThanObject
Because of the "and" added at step 2, the statement gets turned into a
weak ternary statement. If the first object evalutes __nonzero__ as
True, then the second object, (y > z) is returned. If the first object
evaluates __nonzero__ as False (step 5), then (x > y) is returned.
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 is an open PEP to change this,
:pep:`335`, but until that is implemented, this cannot be made to work.
.. [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 lhs >= rhs
def __nonzero__(self):
return self.lhs.compare( self.rhs ) >= 0
[docs]class LessThan(_Less):
__doc__ = GreaterThan.__doc__
__slots__ = ()
rel_op = '<='
@classmethod
def _eval_relation(cls, lhs, rhs):
return lhs <= rhs
def __nonzero__(self):
return self.lhs.compare( self.rhs ) <= 0
[docs]class StrictGreaterThan(_Greater):
__doc__ = GreaterThan.__doc__
__slots__ = ()
rel_op = '>'
@classmethod
def _eval_relation(cls, lhs, rhs):
return lhs > rhs
def __nonzero__(self):
return self.lhs.compare( self.rhs ) > 0
[docs]class StrictLessThan(_Less):
__doc__ = GreaterThan.__doc__
__slots__ = ()
rel_op = '<'
@classmethod
def _eval_relation(cls, lhs, rhs):
return lhs < rhs
def __nonzero__(self):
return self.lhs.compare( self.rhs ) < 0
# 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,
}
```