symbol.py

from __future__ import annotations

from .assumptions import StdFactKB, _assume_defined
from .basic import Basic, Atom
from .cache import cacheit
from .containers import Tuple
from .expr import Expr, AtomicExpr
from .function import AppliedUndef, FunctionClass
from .kind import NumberKind, UndefinedKind
from .logic import fuzzy_bool
from .singleton import S
from .sorting import ordered
from .sympify import sympify
from sympy.logic.boolalg import Boolean
from sympy.utilities.iterables import sift, is_sequence
from sympy.utilities.misc import filldedent

import string
import re as _re
import random
from itertools import product
from typing import Any


class Str(Atom):
    """
    Represents string in SymPy.

    Explanation
    ===========

    Previously, ``Symbol`` was used where string is needed in ``args`` of SymPy
    objects, e.g. denoting the name of the instance. However, since ``Symbol``
    represents mathematical scalar, this class should be used instead.

    """
    __slots__ = ('name',)

    def __new__(cls, name, **kwargs):
        if not isinstance(name, str):
            raise TypeError("name should be a string, not %s" % repr(type(name)))
        obj = Expr.__new__(cls, **kwargs)
        obj.name = name
        return obj

    def __getnewargs__(self):
        return (self.name,)

    def _hashable_content(self):
        return (self.name,)


def _filter_assumptions(kwargs):
    """Split the given dict into assumptions and non-assumptions.
    Keys are taken as assumptions if they correspond to an
    entry in ``_assume_defined``.
    """
    assumptions, nonassumptions = map(dict, sift(kwargs.items(),
        lambda i: i[0] in _assume_defined,
        binary=True))
    Symbol._sanitize(assumptions)
    return assumptions, nonassumptions

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
    >>> 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, str):
        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 whose name is derivated from *xname* but is unique
    from any other symbols in *exprs*.

    *xname* and symbol names in *exprs* are passed to *compare* to be
    converted to comparable forms. If ``compare(xname)`` is not unique,
    it is recursively passed to *modify* until unique name is acquired.

    Parameters
    ==========

    xname : str or Symbol
        Base name for the new symbol.

    exprs : Expr or iterable of Expr
        Expressions whose symbols are compared to *xname*.

    compare : function
        Unary function which transforms *xname* and symbol names from
        *exprs* to comparable form.

    modify : function
        Unary function which modifies the string. Default is appending
        the number, or increasing the number if exists.

    Examples
    ========

    By default, a number is appended to *xname* to generate unique name.
    If the number already exists, it is recursively increased.

    >>> from sympy.core.symbol import uniquely_named_symbol, Symbol
    >>> uniquely_named_symbol('x', Symbol('x'))
    x0
    >>> uniquely_named_symbol('x', (Symbol('x'), Symbol('x0')))
    x1
    >>> uniquely_named_symbol('x0', (Symbol('x1'), Symbol('x0')))
    x2

    Name generation can be controlled by passing *modify* parameter.

    >>> from sympy.abc import x
    >>> uniquely_named_symbol('x', x, modify=lambda s: 2*s)
    xx

    """
    def numbered_string_incr(s, start=0):
        if not s:
            return str(start)
        i = len(s) - 1
        while i != -1:
            if not s[i].isdigit():
                break
            i -= 1
        n = str(int(s[i + 1:] or start - 1) + 1)
        return s[:i + 1] + n

    default = None
    if is_sequence(xname):
        xname, default = xname
    x = compare(xname)
    if not exprs:
        return _symbol(x, default, **assumptions)
    if not is_sequence(exprs):
        exprs = [exprs]
    names = set().union(
        [i.name for e in exprs for i in e.atoms(Symbol)] +
        [i.func.name for e in exprs for i in e.atoms(AppliedUndef)])
    if modify is None:
        modify = numbered_string_incr
    while any(x == compare(s) for s in names):
        x = modify(x)
    return _symbol(x, default, **assumptions)
_uniquely_named_symbol = uniquely_named_symbol

class Symbol(AtomicExpr, Boolean):
    """
    Assumptions:
       commutative = True

    You can override the default assumptions in the constructor.

    Examples
    ========

    >>> 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',)

    name: str

    is_Symbol = True
    is_symbol = True

    @property
    def kind(self):
        if self.is_commutative:
            return NumberKind
        return UndefinedKind

    @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()):
            v = assumptions[key]
            if v is None:
                assumptions.pop(key)
                continue
            assumptions[key] = bool(v)

    def _merge(self, assumptions):
        base = self.assumptions0
        for k in set(assumptions) & set(base):
            if assumptions[k] != base[k]:
                raise ValueError(filldedent('''
                    non-matching assumptions for %s: existing value
                    is %s and new value is %s''' % (
                    k, base[k], assumptions[k])))
        base.update(assumptions)
        return base

    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)

    @staticmethod
    def __xnew__(cls, name, **assumptions):  # never cached (e.g. dummy)
        if not isinstance(name, str):
            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

    @staticmethod
    @cacheit
    def __xnew_cached_(cls, name, **assumptions):  # symbols are always cached
        return Symbol.__xnew__(cls, name, **assumptions)

    def __getnewargs_ex__(self):
        return ((self.name,), self.assumptions0)

    # NOTE: __setstate__ is not needed for pickles created by __getnewargs_ex__
    # but was used before Symbol was changed to use __getnewargs_ex__ in v1.9.
    # Pickles created in previous SymPy versions will still need __setstate__
    # so that they can be unpickled in SymPy > v1.9.

    def __setstate__(self, state):
        for name, value in state.items():
            setattr(self, name, value)

    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):
        if old.is_Pow:
            from sympy.core.power import Pow
            return Pow(self, S.One, evaluate=False)._eval_subs(old, new)

    def _eval_refine(self, assumptions):
        return self

    @property
    def assumptions0(self):
        return {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, (self.name,)), S.One.sort_key(), S.One

    def as_dummy(self):
        # only put commutativity in explicitly if it is False
        return Dummy(self.name) if self.is_commutative is not False \
            else Dummy(self.name, commutative=self.is_commutative)

    def as_real_imag(self, deep=True, **hints):
        if hints.get('ignore') == self:
            return None
        else:
            from sympy.functions.elementary.complexes import im, re
            return (re(self), im(self))

    def is_constant(self, *wrt, **flags):
        if not wrt:
            return False
        return self not 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


class Dummy(Symbol):
    """Dummy symbols are each unique, even if they have the same name:

    Examples
    ========

    >>> 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 __getnewargs_ex__(self):
        return ((self.name, self.dummy_index), self.assumptions0)

    @cacheit
    def sort_key(self, order=None):
        return self.class_key(), (
            2, (self.name, self.dummy_index)), S.One.sort_key(), S.One

    def _hashable_content(self):
        return Symbol._hashable_content(self) + (self.dummy_index,)


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 did not 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()._hashable_content() + (self.exclude, self.properties)

    # TODO add check against another Wild
    def matches(self, expr, repl_dict=None, old=False):
        if any(expr.has(x) for x in self.exclude):
            return None
        if not all(f(expr) for f in self.properties):
            return None
        if repl_dict is None:
            repl_dict = {}
        else:
            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])')


def symbols(names, *, cls=Symbol, **args) -> Any:
    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, str):
        marker = 0
        splitters = r'\,', r'\:', r'\ '
        literals: list[tuple[str, str]] = []
        for splitter in splitters:
            if splitter in names:
                while chr(marker) in names:
                    marker += 1
                lit_char = chr(marker)
                marker += 1
                names = names.replace(splitter, lit_char)
                literals.append((lit_char, splitter[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()

        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: list[str] = _range.split(name)
            split_list: list[list[str]] = []
            # 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 s in split:
                if ':' in s:
                    if s.endswith(':'):
                        raise ValueError('missing end range')
                    a, b = s.split(':')
                    if b[-1] in string.digits:
                        a_i = 0 if not a else int(a)
                        b_i = int(b)
                        split_list.append([str(c) for c in range(a_i, b_i)])
                    else:
                        a = a or 'a'
                        split_list.append([string.ascii_letters[c] for c in range(
                            string.ascii_letters.index(a),
                            string.ascii_letters.index(b) + 1)])  # inclusive
                    if not split_list[-1]:
                        break
                else:
                    split_list.append([s])
            else:
                seq = True
                if len(split_list) == 1:
                    names = split_list[0]
                else:
                    names = [''.join(s) for s in product(*split_list)]
                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, cls=cls, **args))

        return type(names)(result)


def var(names, **args):
    """
    Create symbols and inject them into the global namespace.

    Explanation
    ===========

    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 # noqa: F821
    x

    >>> var('a,ab,abc')
    (a, ab, abc)
    >>> abc # noqa: F821
    abc

    >>> var('x,y', real=True)
    (x, y)
    >>> x.is_real and y.is_real # noqa: F821
    True

    See :func:`symbols` 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
        mapk0 = Symbol("%s" % (k), **mapping[k][0].assumptions0)
        if mapping[k][0] != mapk0:
            reps[mapping[k][0]] = mapk0
        # 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)