# Source code for diofant.core.relational

```
from ..logic.boolalg import Boolean, BooleanAtom
from .basic import S
from .compatibility import ordered
from .evalf import EvalfMixin
from .evaluate import global_evaluate
from .expr import Expr
from .function import _coeff_isneg
from .symbol import Dummy, Symbol
from .sympify import sympify
__all__ = (
'Rel', 'Eq', 'Ne', 'Lt', 'Le', 'Gt', 'Ge',
'Relational', 'Equality', 'Unequality', 'StrictLessThan', 'LessThan',
'StrictGreaterThan', 'GreaterThan',
)
# Note, see issue sympy/sympy#4986. Ideally, we wouldn't want to subclass both Boolean
# and Expr.
[docs]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
========
>>> Rel(y, x+x**2, '==')
Eq(y, x**2 + x)
"""
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:
new_cls = cls.ValidRelationOperator[rop]
return new_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
========
>>> 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.
The rules for the canonical form, in order of decreasing priority are:
1) Number on right if left is not a Number;
2) Symbol on the left;
3) Gt/Ge changed to Lt/Le;
4) Lt/Le are unchanged;
5) Eq and Ne get ordered args.
"""
r = self
if r.func in (Ge, Gt):
r = r.reversed
elif r.func in (Lt, Le):
pass
elif r.func in (Eq, Ne):
r = r.func(*ordered(r.args), evaluate=False)
else:
raise NotImplementedError
if r.lhs.is_Number and not r.rhs.is_Number:
r = r.reversed
elif r.rhs.is_Symbol and not r.lhs.is_Symbol:
r = r.reversed
if _coeff_isneg(r.lhs):
r = r.reversed.func(-r.lhs, -r.rhs, evaluate=False)
return r
[docs] 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 = a.lhs.equals(b.lhs, failing_expression=failing_expression)
r = a.rhs.equals(b.rhs, failing_expression=failing_expression)
if l is True:
return r
if r is True:
return l
lr = a.lhs.equals(b.rhs, failing_expression=failing_expression)
rl = a.rhs.equals(b.lhs, failing_expression=failing_expression)
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
if failing_expression:
return a.lhs - a.rhs - b.lhs + b.rhs
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.func(self.lhs.simplify(ratio=ratio, measure=measure),
self.rhs.simplify(ratio=ratio, measure=measure))
if r.is_Relational:
dif = r.lhs - r.rhs
# We want a Number to compare with zero and be sure to get a
# True/False answer. Check if we can deduce that dif is
# definitively zero or non-zero.
if not dif.has(Dummy, Symbol):
know = dif.equals(0)
if know is False:
if isinstance(r, Eq):
return False
elif isinstance(r, Ne):
return True
r = r.canonical
if measure(r) < ratio*measure(self):
return r
else:
return self
def __bool__(self):
raise TypeError("cannot determine truth value of Relational")
[docs] def as_set(self):
"""
Rewrites univariate inequality in terms of real sets
Examples
========
>>> x = Symbol('x', real=True)
>>> (x > 0).as_set()
(0, oo)
>>> Eq(x, 0).as_set()
{0}
"""
from ..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
========
>>> 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(E**x, sinh(x) + cosh(x))
>>> simplify(_)
true
See Also
========
diofant.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 = '=='
is_Equality = True
def __new__(cls, lhs, rhs=0, **options):
lhs = sympify(lhs, strict=True)
rhs = sympify(rhs, strict=True)
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 # equal args already evaluated
# If appropriate, check if the difference evaluates. Detect
# incompatibility such as lhs real and rhs not real.
if isinstance(lhs, Expr) and isinstance(rhs, Expr):
r = (lhs - rhs).is_zero
if r is not None:
return sympify(r, strict=True)
return Relational.__new__(cls, lhs, rhs, **options)
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
========
>>> 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 = '!='
def __new__(cls, lhs, rhs, **options):
lhs = sympify(lhs, strict=True)
rhs = sympify(rhs, strict=True)
evaluate = options.pop('evaluate', global_evaluate[0])
if evaluate:
is_equal = Equality(lhs, rhs)
if is_equal == S.true or is_equal == S.false:
return ~is_equal
return Relational.__new__(cls, lhs, rhs, **options)
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.
"""
def __new__(cls, lhs, rhs, **options):
lhs = sympify(lhs, strict=True)
rhs = sympify(rhs, strict=True)
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.
return cls._eval_relation(lhs, rhs)
# 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.
"""
@property
def gts(self):
"""Greater than side argument."""
return self.args[0]
@property
def lts(self):
"""Less than side argument."""
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.
"""
@property
def gts(self):
"""Greater than side argument."""
return self.args[1]
@property
def lts(self):
"""Less than side argument."""
return self.args[0]
[docs]class GreaterThan(_Greater):
r"""Class representations of inequalities.
The ``*Than`` classes represent unequal relationships, where the left-hand
side is generally bigger or smaller than the right-hand side. For example,
the GreaterThan class represents an unequal 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:
>>> 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 Diofant object
Python will try the reflective operation, (x > 1). Unfortunately, there is
no way available to Diofant 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 = Integer(1) > x
>>> e2 = Integer(1) >= x
>>> e3 = Integer(1) < x
>>> e4 = Integer(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:
>>> 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, to create a
chained inequality, the only method currently available is to make use of
And:
>>> And(x < y, y < z)
(x < y) & (y < z)
"""
rel_op = '>='
@classmethod
def _eval_relation(cls, lhs, rhs):
return sympify(lhs >= rhs, strict=True)
Ge = GreaterThan
[docs]class LessThan(_Less):
__doc__ = GreaterThan.__doc__
rel_op = '<='
@classmethod
def _eval_relation(cls, lhs, rhs):
return sympify(lhs <= rhs, strict=True)
Le = LessThan
[docs]class StrictGreaterThan(_Greater):
__doc__ = GreaterThan.__doc__
rel_op = '>'
@classmethod
def _eval_relation(cls, lhs, rhs):
return sympify(lhs > rhs, strict=True)
Gt = StrictGreaterThan
[docs]class StrictLessThan(_Less):
__doc__ = GreaterThan.__doc__
rel_op = '<'
@classmethod
def _eval_relation(cls, lhs, rhs):
return sympify(lhs < rhs, strict=True)
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,
}
```