```
"""
This module contains the machinery handling assumptions.
All symbolic objects have assumption attributes that can be accessed via
.is_<assumption name> attribute.
Assumptions determine certain properties of symbolic objects and can
have 3 possible values: True, False, None. True is returned if the
object has the property and False is returned if it doesn't or can't
(i.e. doesn't make sense):
>>> from sympy import I
>>> I.is_algebraic
True
>>> I.is_real
False
>>> I.is_prime
False
When the property cannot be determined (or when a method is not
implemented) None will be returned, e.g. a generic symbol, x, may or
may not be positive so a value of None is returned for x.is_positive.
By default, all symbolic values are in the largest set in the given context
without specifying the property. For example, a symbol that has a property
being integer, is also real, complex, etc.
Here follows a list of possible assumption names:
.. glossary::
commutative
object commutes with any other object with
respect to multiplication operation.
complex
object can have only values from the set
of complex numbers.
imaginary
object value is a number that can be written as a real
number multiplied by the imaginary unit ``I``. See
[3]_. Please note, that ``0`` is not considered to be an
imaginary number, see
`issue #7649 <https://github.com/sympy/sympy/issues/7649>`_.
real
object can have only values from the set
of real numbers.
integer
object can have only values from the set
of integers.
odd
even
object can have only values from the set of
odd (even) integers [2]_.
prime
object is a natural number greater than ``1`` that has
no positive divisors other than ``1`` and itself. See [6]_.
composite
object is a positive integer that has at least one positive
divisor other than ``1`` or the number itself. See [4]_.
zero
nonzero
object is zero (not zero).
rational
object can have only values from the set
of rationals.
algebraic
object can have only values from the set
of algebraic numbers [11]_.
transcendental
object can have only values from the set
of transcendental numbers [10]_.
irrational
object value cannot be represented exactly by Rational, see [5]_.
finite
infinite
object absolute value is bounded (is value is
arbitrarily large). See [7]_, [8]_, [9]_.
negative
nonnegative
object can have only negative (only
nonnegative) values [1]_.
positive
nonpositive
object can have only positive (only
nonpositive) values.
hermitian
antihermitian
object belongs to the field of hermitian
(antihermitian) operators.
Examples
========
>>> from sympy import Symbol
>>> x = Symbol('x', real=True); x
x
>>> x.is_real
True
>>> x.is_complex
True
See Also
========
.. seealso::
:py:class:`sympy.core.numbers.ImaginaryUnit`
:py:class:`sympy.core.numbers.Zero`
:py:class:`sympy.core.numbers.One`
Notes
=====
Assumption values are stored in obj._assumptions dictionary or
are returned by getter methods (with property decorators) or are
attributes of objects/classes.
References
==========
.. [1] http://en.wikipedia.org/wiki/Negative_number
.. [2] http://en.wikipedia.org/wiki/Parity_%28mathematics%29
.. [3] http://en.wikipedia.org/wiki/Imaginary_number
.. [4] http://en.wikipedia.org/wiki/Composite_number
.. [5] http://en.wikipedia.org/wiki/Irrational_number
.. [6] http://en.wikipedia.org/wiki/Prime_number
.. [7] http://en.wikipedia.org/wiki/Finite
.. [8] https://docs.python.org/3/library/math.html#math.isfinite
.. [9] http://docs.scipy.org/doc/numpy/reference/generated/numpy.isfinite.html
.. [10] http://en.wikipedia.org/wiki/Transcendental_number
.. [11] http://en.wikipedia.org/wiki/Algebraic_number
"""
from __future__ import print_function, division
from sympy.core.facts import FactRules, FactKB
from sympy.core.core import BasicMeta
from sympy.core.compatibility import integer_types
from random import shuffle
_assume_rules = FactRules([
'integer -> rational',
'rational -> real',
'rational -> algebraic',
'algebraic -> complex',
'real -> complex',
'real -> hermitian',
'imaginary -> complex',
'imaginary -> antihermitian',
'complex -> commutative',
'odd == integer & !even',
'even == integer & !odd',
'real == negative | zero | positive',
'transcendental == complex & !algebraic',
'negative == nonpositive & nonzero',
'positive == nonnegative & nonzero',
'zero == nonnegative & nonpositive',
'nonpositive == real & !positive',
'nonnegative == real & !negative',
'zero -> even & finite',
'prime -> integer & positive',
'composite -> integer & positive & !prime',
'irrational == real & !rational',
'imaginary -> !real',
'infinite -> !finite',
'noninteger == real & !integer',
'nonzero == real & !zero',
])
_assume_defined = _assume_rules.defined_facts.copy()
_assume_defined.add('polar')
_assume_defined = frozenset(_assume_defined)
class StdFactKB(FactKB):
"""A FactKB specialised for the built-in rules
This is the only kind of FactKB that Basic objects should use.
"""
rules = _assume_rules
def __init__(self, facts=None):
# save a copy of the facts dict
if not facts:
self._generator = {};
elif not isinstance(facts, FactKB):
self._generator = facts.copy()
else:
self._generator = facts.generator
if facts:
self.deduce_all_facts(facts)
def copy(self):
return self.__class__(self)
@property
def generator(self):
return self._generator.copy()
def as_property(fact):
"""Convert a fact name to the name of the corresponding property"""
return 'is_%s' % fact
def make_property(fact):
"""Create the automagic property corresponding to a fact."""
def getit(self):
try:
return self._assumptions[fact]
except KeyError:
if self._assumptions is self.default_assumptions:
self._assumptions = self.default_assumptions.copy()
return _ask(fact, self)
getit.func_name = as_property(fact)
return property(getit)
def _ask(fact, obj):
"""
Find the truth value for a property of an object.
This function is called when a request is made to see what a fact
value is.
For this we use several techniques:
First, the fact-evaluation function is tried, if it exists (for
example _eval_is_integer). Then we try related facts. For example
rational --> integer
another example is joined rule:
integer & !odd --> even
so in the latter case if we are looking at what 'even' value is,
'integer' and 'odd' facts will be asked.
In all cases, when we settle on some fact value, its implications are
deduced, and the result is cached in ._assumptions.
"""
assumptions = obj._assumptions
handler_map = obj._prop_handler
# Store None into the assumptions so that recursive attempts at
# evaluating the same fact don't trigger infinite recursion.
assumptions._tell(fact, None)
# First try the assumption evaluation function if it exists
try:
evaluate = handler_map[fact]
except KeyError:
pass
else:
a = evaluate(obj)
if a is not None:
assumptions.deduce_all_facts(((fact, a),))
return a
# Try assumption's prerequisites
prereq = list(_assume_rules.prereq[fact])
shuffle(prereq)
for pk in prereq:
if pk in assumptions:
continue
if pk in handler_map:
_ask(pk, obj)
# we might have found the value of fact
ret_val = assumptions.get(fact)
if ret_val is not None:
return ret_val
# Note: the result has already been cached
return None
class ManagedProperties(BasicMeta):
"""Metaclass for classes with old-style assumptions"""
def __init__(cls, *args, **kws):
BasicMeta.__init__(cls, *args, **kws)
local_defs = {}
for k in _assume_defined:
attrname = as_property(k)
v = cls.__dict__.get(attrname, '')
if isinstance(v, (bool, integer_types, type(None))):
if v is not None:
v = bool(v)
local_defs[k] = v
defs = {}
for base in reversed(cls.__bases__):
try:
defs.update(base._explicit_class_assumptions)
except AttributeError:
pass
defs.update(local_defs)
cls._explicit_class_assumptions = defs
cls.default_assumptions = StdFactKB(defs)
cls._prop_handler = {}
for k in _assume_defined:
try:
cls._prop_handler[k] = getattr(cls, '_eval_is_%s' % k)
except AttributeError:
pass
# Put definite results directly into the class dict, for speed
for k, v in cls.default_assumptions.items():
setattr(cls, as_property(k), v)
# protection e.g. for Integer.is_even=F <- (Rational.is_integer=F)
derived_from_bases = set()
for base in cls.__bases__:
try:
derived_from_bases |= set(base.default_assumptions)
except AttributeError:
continue # not an assumption-aware class
for fact in derived_from_bases - set(cls.default_assumptions):
pname = as_property(fact)
if pname not in cls.__dict__:
setattr(cls, pname, make_property(fact))
# Finally, add any missing automagic property (e.g. for Basic)
for fact in _assume_defined:
pname = as_property(fact)
if not hasattr(cls, pname):
setattr(cls, pname, make_property(fact))
```