from __future__ import print_function, division
from sympy.core.assumptions import StdFactKB
from sympy.core.compatibility import (string_types, range, is_sequence,
ordered)
from .basic import Basic
from .sympify import sympify
from .singleton import S
from .expr import Expr, AtomicExpr
from .cache import cacheit
from .function import FunctionClass
from sympy.core.logic import fuzzy_bool
from sympy.logic.boolalg import Boolean
from sympy.utilities.iterables import cartes
from sympy.core.containers import Tuple
import string
import re as _re
import random
def _symbol(s, matching_symbol=None, **assumptions):
"""Return s if s is a Symbol, else if s is a string, return either
the matching_symbol if the names are the same or else a new symbol
with the same assumptions as the matching symbol (or the
assumptions as provided).
Examples
========
>>> from sympy import Symbol, Dummy
>>> from sympy.core.symbol import _symbol
>>> _symbol('y')
y
>>> _.is_real is None
True
>>> _symbol('y', real=True).is_real
True
>>> x = Symbol('x')
>>> _symbol(x, real=True)
x
>>> _.is_real is None # ignore attribute if s is a Symbol
True
Below, the variable sym has the name 'foo':
>>> sym = Symbol('foo', real=True)
Since 'x' is not the same as sym's name, a new symbol is created:
>>> _symbol('x', sym).name
'x'
It will acquire any assumptions give:
>>> _symbol('x', sym, real=False).is_real
False
Since 'foo' is the same as sym's name, sym is returned
>>> _symbol('foo', sym)
foo
Any assumptions given are ignored:
>>> _symbol('foo', sym, real=False).is_real
True
NB: the symbol here may not be the same as a symbol with the same
name defined elsewhere as a result of different assumptions.
See Also
========
sympy.core.symbol.Symbol
"""
if isinstance(s, string_types):
if matching_symbol and matching_symbol.name == s:
return matching_symbol
return Symbol(s, **assumptions)
elif isinstance(s, Symbol):
return s
else:
raise ValueError('symbol must be string for symbol name or Symbol')
def _uniquely_named_symbol(xname, exprs=(), compare=str, modify=None, **assumptions):
"""Return a symbol which, when printed, will have a name unique
from any other already in the expressions given. The name is made
unique by prepending underscores (default) but this can be
customized with the keyword 'modify'.
Parameters
==========
xname : a string or a Symbol (when symbol xname <- str(xname))
compare : a single arg function that takes a symbol and returns
a string to be compared with xname (the default is the str
function which indicates how the name will look when it
is printed, e.g. this includes underscores that appear on
Dummy symbols)
modify : a single arg function that changes its string argument
in some way (the default is to preppend underscores)
Examples
========
>>> from sympy.core.symbol import _uniquely_named_symbol as usym, Dummy
>>> from sympy.abc import x
>>> usym('x', x)
_x
"""
default = None
if is_sequence(xname):
xname, default = xname
x = str(xname)
if not exprs:
return _symbol(x, default, **assumptions)
if not is_sequence(exprs):
exprs = [exprs]
syms = set().union(*[e.free_symbols for e in exprs])
if modify is None:
modify = lambda s: '_' + s
while any(x == compare(s) for s in syms):
x = modify(x)
return _symbol(x, default, **assumptions)
[docs]class Symbol(AtomicExpr, Boolean):
"""
Assumptions:
commutative = True
You can override the default assumptions in the constructor:
>>> from sympy import symbols
>>> A,B = symbols('A,B', commutative = False)
>>> bool(A*B != B*A)
True
>>> bool(A*B*2 == 2*A*B) == True # multiplication by scalars is commutative
True
"""
is_comparable = False
__slots__ = ['name']
is_Symbol = True
is_symbol = True
@property
def _diff_wrt(self):
"""Allow derivatives wrt Symbols.
Examples
========
>>> from sympy import Symbol
>>> x = Symbol('x')
>>> x._diff_wrt
True
"""
return True
@staticmethod
def _sanitize(assumptions, obj=None):
"""Remove None, covert values to bool, check commutativity *in place*.
"""
# be strict about commutativity: cannot be None
is_commutative = fuzzy_bool(assumptions.get('commutative', True))
if is_commutative is None:
whose = '%s ' % obj.__name__ if obj else ''
raise ValueError(
'%scommutativity must be True or False.' % whose)
# sanitize other assumptions so 1 -> True and 0 -> False
for key in list(assumptions.keys()):
from collections import defaultdict
from sympy.utilities.exceptions import SymPyDeprecationWarning
keymap = defaultdict(lambda: None)
keymap.update({'bounded': 'finite', 'unbounded': 'infinite', 'infinitesimal': 'zero'})
if keymap[key]:
SymPyDeprecationWarning(
feature="%s assumption" % key,
useinstead="%s" % keymap[key],
issue=8071,
deprecated_since_version="0.7.6").warn()
assumptions[keymap[key]] = assumptions[key]
assumptions.pop(key)
key = keymap[key]
v = assumptions[key]
if v is None:
assumptions.pop(key)
continue
assumptions[key] = bool(v)
def __new__(cls, name, **assumptions):
"""Symbols are identified by name and assumptions::
>>> from sympy import Symbol
>>> Symbol("x") == Symbol("x")
True
>>> Symbol("x", real=True) == Symbol("x", real=False)
False
"""
cls._sanitize(assumptions, cls)
return Symbol.__xnew_cached_(cls, name, **assumptions)
def __new_stage2__(cls, name, **assumptions):
if not isinstance(name, string_types):
raise TypeError("name should be a string, not %s" % repr(type(name)))
obj = Expr.__new__(cls)
obj.name = name
# TODO: Issue #8873: Forcing the commutative assumption here means
# later code such as ``srepr()`` cannot tell whether the user
# specified ``commutative=True`` or omitted it. To workaround this,
# we keep a copy of the assumptions dict, then create the StdFactKB,
# and finally overwrite its ``._generator`` with the dict copy. This
# is a bit of a hack because we assume StdFactKB merely copies the
# given dict as ``._generator``, but future modification might, e.g.,
# compute a minimal equivalent assumption set.
tmp_asm_copy = assumptions.copy()
# be strict about commutativity
is_commutative = fuzzy_bool(assumptions.get('commutative', True))
assumptions['commutative'] = is_commutative
obj._assumptions = StdFactKB(assumptions)
obj._assumptions._generator = tmp_asm_copy # Issue #8873
return obj
__xnew__ = staticmethod(
__new_stage2__) # never cached (e.g. dummy)
__xnew_cached_ = staticmethod(
cacheit(__new_stage2__)) # symbols are always cached
def __getnewargs__(self):
return (self.name,)
def __getstate__(self):
return {'_assumptions': self._assumptions}
def _hashable_content(self):
# Note: user-specified assumptions not hashed, just derived ones
return (self.name,) + tuple(sorted(self.assumptions0.items()))
def _eval_subs(self, old, new):
from sympy.core.power import Pow
if old.is_Pow:
return Pow(self, S.One, evaluate=False)._eval_subs(old, new)
@property
def assumptions0(self):
return dict((key, value) for key, value
in self._assumptions.items() if value is not None)
@cacheit
def sort_key(self, order=None):
return self.class_key(), (1, (str(self),)), S.One.sort_key(), S.One
def as_dummy(self):
return Dummy(self.name)
def as_real_imag(self, deep=True, **hints):
from sympy import im, re
if hints.get('ignore') == self:
return None
else:
return (re(self), im(self))
def _sage_(self):
import sage.all as sage
return sage.var(self.name)
def is_constant(self, *wrt, **flags):
if not wrt:
return False
return not self in wrt
@property
def free_symbols(self):
return {self}
binary_symbols = free_symbols # in this case, not always
def as_set(self):
return S.UniversalSet
[docs]class Dummy(Symbol):
"""Dummy symbols are each unique, even if they have the same name:
>>> from sympy import Dummy
>>> Dummy("x") == Dummy("x")
False
If a name is not supplied then a string value of an internal count will be
used. This is useful when a temporary variable is needed and the name
of the variable used in the expression is not important.
>>> Dummy() #doctest: +SKIP
_Dummy_10
"""
# In the rare event that a Dummy object needs to be recreated, both the
# `name` and `dummy_index` should be passed. This is used by `srepr` for
# example:
# >>> d1 = Dummy()
# >>> d2 = eval(srepr(d1))
# >>> d2 == d1
# True
#
# If a new session is started between `srepr` and `eval`, there is a very
# small chance that `d2` will be equal to a previously-created Dummy.
_count = 0
_prng = random.Random()
_base_dummy_index = _prng.randint(10**6, 9*10**6)
__slots__ = ['dummy_index']
is_Dummy = True
def __new__(cls, name=None, dummy_index=None, **assumptions):
if dummy_index is not None:
assert name is not None, "If you specify a dummy_index, you must also provide a name"
if name is None:
name = "Dummy_" + str(Dummy._count)
if dummy_index is None:
dummy_index = Dummy._base_dummy_index + Dummy._count
Dummy._count += 1
cls._sanitize(assumptions, cls)
obj = Symbol.__xnew__(cls, name, **assumptions)
obj.dummy_index = dummy_index
return obj
def __getstate__(self):
return {'_assumptions': self._assumptions, 'dummy_index': self.dummy_index}
@cacheit
def sort_key(self, order=None):
return self.class_key(), (
2, (str(self), self.dummy_index)), S.One.sort_key(), S.One
def _hashable_content(self):
return Symbol._hashable_content(self) + (self.dummy_index,)
[docs]class Wild(Symbol):
"""
A Wild symbol matches anything, or anything
without whatever is explicitly excluded.
Parameters
==========
name : str
Name of the Wild instance.
exclude : iterable, optional
Instances in ``exclude`` will not be matched.
properties : iterable of functions, optional
Functions, each taking an expressions as input
and returns a ``bool``. All functions in ``properties``
need to return ``True`` in order for the Wild instance
to match the expression.
Examples
========
>>> from sympy import Wild, WildFunction, cos, pi
>>> from sympy.abc import x, y, z
>>> a = Wild('a')
>>> x.match(a)
{a_: x}
>>> pi.match(a)
{a_: pi}
>>> (3*x**2).match(a*x)
{a_: 3*x}
>>> cos(x).match(a)
{a_: cos(x)}
>>> b = Wild('b', exclude=[x])
>>> (3*x**2).match(b*x)
>>> b.match(a)
{a_: b_}
>>> A = WildFunction('A')
>>> A.match(a)
{a_: A_}
Tips
====
When using Wild, be sure to use the exclude
keyword to make the pattern more precise.
Without the exclude pattern, you may get matches
that are technically correct, but not what you
wanted. For example, using the above without
exclude:
>>> from sympy import symbols
>>> a, b = symbols('a b', cls=Wild)
>>> (2 + 3*y).match(a*x + b*y)
{a_: 2/x, b_: 3}
This is technically correct, because
(2/x)*x + 3*y == 2 + 3*y, but you probably
wanted it to not match at all. The issue is that
you really didn't want a and b to include x and y,
and the exclude parameter lets you specify exactly
this. With the exclude parameter, the pattern will
not match.
>>> a = Wild('a', exclude=[x, y])
>>> b = Wild('b', exclude=[x, y])
>>> (2 + 3*y).match(a*x + b*y)
Exclude also helps remove ambiguity from matches.
>>> E = 2*x**3*y*z
>>> a, b = symbols('a b', cls=Wild)
>>> E.match(a*b)
{a_: 2*y*z, b_: x**3}
>>> a = Wild('a', exclude=[x, y])
>>> E.match(a*b)
{a_: z, b_: 2*x**3*y}
>>> a = Wild('a', exclude=[x, y, z])
>>> E.match(a*b)
{a_: 2, b_: x**3*y*z}
Wild also accepts a ``properties`` parameter:
>>> a = Wild('a', properties=[lambda k: k.is_Integer])
>>> E.match(a*b)
{a_: 2, b_: x**3*y*z}
"""
is_Wild = True
__slots__ = ['exclude', 'properties']
def __new__(cls, name, exclude=(), properties=(), **assumptions):
exclude = tuple([sympify(x) for x in exclude])
properties = tuple(properties)
cls._sanitize(assumptions, cls)
return Wild.__xnew__(cls, name, exclude, properties, **assumptions)
def __getnewargs__(self):
return (self.name, self.exclude, self.properties)
@staticmethod
@cacheit
def __xnew__(cls, name, exclude, properties, **assumptions):
obj = Symbol.__xnew__(cls, name, **assumptions)
obj.exclude = exclude
obj.properties = properties
return obj
def _hashable_content(self):
return super(Wild, self)._hashable_content() + (self.exclude, self.properties)
# TODO add check against another Wild
def matches(self, expr, repl_dict={}, old=False):
if any(expr.has(x) for x in self.exclude):
return None
if any(not f(expr) for f in self.properties):
return None
repl_dict = repl_dict.copy()
repl_dict[self] = expr
return repl_dict
_range = _re.compile('([0-9]*:[0-9]+|[a-zA-Z]?:[a-zA-Z])')
[docs]def symbols(names, **args):
r"""
Transform strings into instances of :class:`Symbol` class.
:func:`symbols` function returns a sequence of symbols with names taken
from ``names`` argument, which can be a comma or whitespace delimited
string, or a sequence of strings::
>>> from sympy import symbols, Function
>>> x, y, z = symbols('x,y,z')
>>> a, b, c = symbols('a b c')
The type of output is dependent on the properties of input arguments::
>>> symbols('x')
x
>>> symbols('x,')
(x,)
>>> symbols('x,y')
(x, y)
>>> symbols(('a', 'b', 'c'))
(a, b, c)
>>> symbols(['a', 'b', 'c'])
[a, b, c]
>>> symbols({'a', 'b', 'c'})
{a, b, c}
If an iterable container is needed for a single symbol, set the ``seq``
argument to ``True`` or terminate the symbol name with a comma::
>>> symbols('x', seq=True)
(x,)
To reduce typing, range syntax is supported to create indexed symbols.
Ranges are indicated by a colon and the type of range is determined by
the character to the right of the colon. If the character is a digit
then all contiguous digits to the left are taken as the nonnegative
starting value (or 0 if there is no digit left of the colon) and all
contiguous digits to the right are taken as 1 greater than the ending
value::
>>> symbols('x:10')
(x0, x1, x2, x3, x4, x5, x6, x7, x8, x9)
>>> symbols('x5:10')
(x5, x6, x7, x8, x9)
>>> symbols('x5(:2)')
(x50, x51)
>>> symbols('x5:10,y:5')
(x5, x6, x7, x8, x9, y0, y1, y2, y3, y4)
>>> symbols(('x5:10', 'y:5'))
((x5, x6, x7, x8, x9), (y0, y1, y2, y3, y4))
If the character to the right of the colon is a letter, then the single
letter to the left (or 'a' if there is none) is taken as the start
and all characters in the lexicographic range *through* the letter to
the right are used as the range::
>>> symbols('x:z')
(x, y, z)
>>> symbols('x:c') # null range
()
>>> symbols('x(:c)')
(xa, xb, xc)
>>> symbols(':c')
(a, b, c)
>>> symbols('a:d, x:z')
(a, b, c, d, x, y, z)
>>> symbols(('a:d', 'x:z'))
((a, b, c, d), (x, y, z))
Multiple ranges are supported; contiguous numerical ranges should be
separated by parentheses to disambiguate the ending number of one
range from the starting number of the next::
>>> symbols('x:2(1:3)')
(x01, x02, x11, x12)
>>> symbols(':3:2') # parsing is from left to right
(00, 01, 10, 11, 20, 21)
Only one pair of parentheses surrounding ranges are removed, so to
include parentheses around ranges, double them. And to include spaces,
commas, or colons, escape them with a backslash::
>>> symbols('x((a:b))')
(x(a), x(b))
>>> symbols(r'x(:1\,:2)') # or r'x((:1)\,(:2))'
(x(0,0), x(0,1))
All newly created symbols have assumptions set according to ``args``::
>>> a = symbols('a', integer=True)
>>> a.is_integer
True
>>> x, y, z = symbols('x,y,z', real=True)
>>> x.is_real and y.is_real and z.is_real
True
Despite its name, :func:`symbols` can create symbol-like objects like
instances of Function or Wild classes. To achieve this, set ``cls``
keyword argument to the desired type::
>>> symbols('f,g,h', cls=Function)
(f, g, h)
>>> type(_[0])
<class 'sympy.core.function.UndefinedFunction'>
"""
result = []
if isinstance(names, string_types):
marker = 0
literals = [r'\,', r'\:', r'\ ']
for i in range(len(literals)):
lit = literals.pop(0)
if lit in names:
while chr(marker) in names:
marker += 1
lit_char = chr(marker)
marker += 1
names = names.replace(lit, lit_char)
literals.append((lit_char, lit[1:]))
def literal(s):
if literals:
for c, l in literals:
s = s.replace(c, l)
return s
names = names.strip()
as_seq = names.endswith(',')
if as_seq:
names = names[:-1].rstrip()
if not names:
raise ValueError('no symbols given')
# split on commas
names = [n.strip() for n in names.split(',')]
if not all(n for n in names):
raise ValueError('missing symbol between commas')
# split on spaces
for i in range(len(names) - 1, -1, -1):
names[i: i + 1] = names[i].split()
cls = args.pop('cls', Symbol)
seq = args.pop('seq', as_seq)
for name in names:
if not name:
raise ValueError('missing symbol')
if ':' not in name:
symbol = cls(literal(name), **args)
result.append(symbol)
continue
split = _range.split(name)
# remove 1 layer of bounding parentheses around ranges
for i in range(len(split) - 1):
if i and ':' in split[i] and split[i] != ':' and \
split[i - 1].endswith('(') and \
split[i + 1].startswith(')'):
split[i - 1] = split[i - 1][:-1]
split[i + 1] = split[i + 1][1:]
for i, s in enumerate(split):
if ':' in s:
if s[-1].endswith(':'):
raise ValueError('missing end range')
a, b = s.split(':')
if b[-1] in string.digits:
a = 0 if not a else int(a)
b = int(b)
split[i] = [str(c) for c in range(a, b)]
else:
a = a or 'a'
split[i] = [string.ascii_letters[c] for c in range(
string.ascii_letters.index(a),
string.ascii_letters.index(b) + 1)] # inclusive
if not split[i]:
break
else:
split[i] = [s]
else:
seq = True
if len(split) == 1:
names = split[0]
else:
names = [''.join(s) for s in cartes(*split)]
if literals:
result.extend([cls(literal(s), **args) for s in names])
else:
result.extend([cls(s, **args) for s in names])
if not seq and len(result) <= 1:
if not result:
return ()
return result[0]
return tuple(result)
else:
for name in names:
result.append(symbols(name, **args))
return type(names)(result)
[docs]def var(names, **args):
"""
Create symbols and inject them into the global namespace.
This calls :func:`symbols` with the same arguments and puts the results
into the *global* namespace. It's recommended not to use :func:`var` in
library code, where :func:`symbols` has to be used::
Examples
========
>>> from sympy import var
>>> var('x')
x
>>> x
x
>>> var('a,ab,abc')
(a, ab, abc)
>>> abc
abc
>>> var('x,y', real=True)
(x, y)
>>> x.is_real and y.is_real
True
See :func:`symbol` documentation for more details on what kinds of
arguments can be passed to :func:`var`.
"""
def traverse(symbols, frame):
"""Recursively inject symbols to the global namespace. """
for symbol in symbols:
if isinstance(symbol, Basic):
frame.f_globals[symbol.name] = symbol
elif isinstance(symbol, FunctionClass):
frame.f_globals[symbol.__name__] = symbol
else:
traverse(symbol, frame)
from inspect import currentframe
frame = currentframe().f_back
try:
syms = symbols(names, **args)
if syms is not None:
if isinstance(syms, Basic):
frame.f_globals[syms.name] = syms
elif isinstance(syms, FunctionClass):
frame.f_globals[syms.__name__] = syms
else:
traverse(syms, frame)
finally:
del frame # break cyclic dependencies as stated in inspect docs
return syms
def disambiguate(*iter):
"""
Return a Tuple containing the passed expressions with symbols
that appear the same when printed replaced with numerically
subscripted symbols, and all Dummy symbols replaced with Symbols.
Parameters
==========
iter: list of symbols or expressions.
Examples
========
>>> from sympy.core.symbol import disambiguate
>>> from sympy import Dummy, Symbol, Tuple
>>> from sympy.abc import y
>>> tup = Symbol('_x'), Dummy('x'), Dummy('x')
>>> disambiguate(*tup)
(x_2, x, x_1)
>>> eqs = Tuple(Symbol('x')/y, Dummy('x')/y)
>>> disambiguate(*eqs)
(x_1/y, x/y)
>>> ix = Symbol('x', integer=True)
>>> vx = Symbol('x')
>>> disambiguate(vx + ix)
(x + x_1,)
To make your own mapping of symbols to use, pass only the free symbols
of the expressions and create a dictionary:
>>> free = eqs.free_symbols
>>> mapping = dict(zip(free, disambiguate(*free)))
>>> eqs.xreplace(mapping)
(x_1/y, x/y)
"""
new_iter = Tuple(*iter)
key = lambda x:tuple(sorted(x.assumptions0.items()))
syms = ordered(new_iter.free_symbols, keys=key)
mapping = {}
for s in syms:
mapping.setdefault(str(s).lstrip('_'), []).append(s)
reps = {}
for k in mapping:
# the first or only symbol doesn't get subscripted but make
# sure that it's a Symbol, not a Dummy
k0 = Symbol("%s" % (k), **mapping[k][0].assumptions0)
if k != k0:
reps[mapping[k][0]] = k0
# the others get subscripts (and are made into Symbols)
skip = 0
for i in range(1, len(mapping[k])):
while True:
name = "%s_%i" % (k, i + skip)
if name not in mapping:
break
skip += 1
ki = mapping[k][i]
reps[ki] = Symbol(name, **ki.assumptions0)
return new_iter.xreplace(reps)
