Coverage for sympy/core/relational.py : 41%

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. 

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. 

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 NotImplemented 

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 

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() 

(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 

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.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: 

return S.true 

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 

# 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 

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] 

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/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 

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 

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 

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, 

}