def __init__(self, coordinates, metric = None):
        """
        An open subset of Euclidian space with a specific set of
        coordinates. See ``CoordinatePatch`` for details.

        INPUT::

        - ``coordinates`` -- a set of symbolic variables that serve
        as coordinates on this space.

        - ``metric`` (default: None) -- a metric tensor on this
        coordinate patch.  Providing anything other than ``None``
        is currently not defined.


        EXAMPLES::

            sage: x, y, z = var('x, y, z')
            sage: S = CoordinatePatch((x, y, z)); S
            Open subset of R^3 with coordinates x, y, z

        """

        from sage.symbolic.ring import is_SymbolicVariable

        if not all(is_SymbolicVariable(c) for c in coordinates):
            raise TypeError, "%s is not a valid vector of coordinates." % \
                coordinates

        self._coordinates = tuple(coordinates)
        dim = len(self._coordinates)

        if metric is not None:
            raise NotImplementedError, "Metric geometry not supported yet."

def preprocess_assumptions(args):
    """
    Turn a list of the form ``(var1, var2, ..., 'property')`` into a
    sequence of declarations ``(var1 is property), (var2 is property),
    ...``

    EXAMPLES::

        sage: from sage.symbolic.assumptions import preprocess_assumptions
        sage: preprocess_assumptions([x, 'integer', x > 4])
        [x is integer, x > 4]
        sage: var('x, y')
        (x, y)
        sage: preprocess_assumptions([x, y, 'integer', x > 4, y, 'even'])
        [x is integer, y is integer, x > 4, y is even]
    """
    args = list(args)
    last = None
    for i, x in reversed(list(enumerate(args))):
        if isinstance(x, str):
            del args[i]
            last = x
        elif ((not hasattr(x, 'assume') or is_SymbolicVariable(x))
              and last is not None):
            args[i] = GenericDeclaration(x, last)
        else:
            last = None
    return args

    def __init__(self, coordinates, metric = None):
        """
        An open subset of Euclidian space with a specific set of
        coordinates. See ``CoordinatePatch`` for details.

        INPUT:

        - ``coordinates`` -- a set of symbolic variables that serve
          as coordinates on this space.

        - ``metric`` (default: ``None``) -- a metric tensor on this
          coordinate patch.  Providing anything other than ``None``
          is currently not defined.

        EXAMPLES::

            sage: x, y, z = var('x, y, z')
            sage: S = CoordinatePatch((x, y, z)); S
            doctest:...: DeprecationWarning: Use Manifold instead.
            See http://trac.sagemath.org/24444 for details.
            Open subset of R^3 with coordinates x, y, z
        """
        from sage.symbolic.ring import is_SymbolicVariable
        from sage.misc.superseded import deprecation
        deprecation(24444, 'Use Manifold instead.')

        if not all(is_SymbolicVariable(c) for c in coordinates):
            raise TypeError("%s is not a valid vector of coordinates." % \
                coordinates)

        self._coordinates = tuple(coordinates)
        dim = len(self._coordinates)

        if metric is not None:
            raise NotImplementedError("Metric geometry not supported yet.")

    def create_key(self, args, check=True):
        """
        EXAMPLES::

            sage: x,y = var('x,y')
            sage: CallableSymbolicExpressionRing.create_key((x,y))
            (x, y)
        """
        if check:
            from sage.symbolic.ring import is_SymbolicVariable
            if len(args) == 1 and isinstance(args[0], (list, tuple)):
                args, = args
            for arg in args:
                if not is_SymbolicVariable(arg):
                    raise TypeError("Must construct a function with a tuple (or list) of variables.")
            args = tuple(args)
        return args

    def _print_latex_(self, f, x, a, b):
        r"""
        Convert this integral to LaTeX notation

        EXAMPLES::

            sage: from sage.symbolic.integration.integral import definite_integral
            sage: print_latex = definite_integral._print_latex_
            sage: var('x,a,b')
            (x, a, b)
            sage: f = function('f')
            sage: print_latex(f(x),x,0,1)
            '\\int_{0}^{1} f\\left(x\\right)\\,{d x}'
            sage: latex(integrate(tan(x)/x, x, 0, 1))
            \int_{0}^{1} \frac{\tan\left(x\right)}{x}\,{d x}
        """
        from sage.misc.latex import latex
        if not is_SymbolicVariable(x):
            dx_str = "{d \\left(%s\\right)}"%(latex(x))
        else:
            dx_str = "{d %s}"%(latex(x))
        return "\\int_{%s}^{%s} %s\\,%s"%(latex(a), latex(b), latex(f), dx_str)

    def __call__(self, *args):
        """
        EXAMPLES::

            sage: from sage.symbolic.operators import FDerivativeOperator
            sage: x,y = var('x,y')
            sage: f = function('foo')
            sage: op = FDerivativeOperator(f, [0,1])
            sage: op(x,y)
            diff(foo(x, y), x, y)
            sage: op(x,x^2)
            D[0, 1](foo)(x, x^2)

        TESTS:

        We should be able to operate on functions evaluated at a
        point, not just a symbolic variable, :trac:`12796`::

           sage: from sage.symbolic.operators import FDerivativeOperator
           sage: f = function('f')
           sage: op = FDerivativeOperator(f, [0])
           sage: op(1)
           D[0](f)(1)

        """
        if (not all(is_SymbolicVariable(x) for x in args) or
                len(args) != len(set(args))):
            # An evaluated derivative of the form f'(1) is not a
            # symbolic variable, yet we would like to treat it
            # like one. So, we replace the argument `1` with a
            # temporary variable e.g. `t0` and then evaluate the
            # derivative f'(t0) symbolically at t0=1. See trac
            # #12796.
            temp_args=[SR.var("t%s"%i) for i in range(len(args))]
            vars=[temp_args[i] for i in self._parameter_set]
            return self._f(*temp_args).diff(*vars).function(*temp_args)(*args)
        vars = [args[i] for i in self._parameter_set]
        return self._f(*args).diff(*vars)

def sr_to_max(expr):
    r"""
    Convert a symbolic expression into a Maxima object.

    INPUT:

    - ``expr`` - symbolic expression

    OUTPUT: ECL object

    EXAMPLES::

        sage: from sage.interfaces.maxima_lib import sr_to_max
        sage: var('x')
        x
        sage: sr_to_max(x)
        <ECL: $X>
        sage: sr_to_max(cos(x))
        <ECL: ((%COS) $X)>
        sage: f = function('f',x)
        sage: sr_to_max(f.diff())
        <ECL: ((%DERIVATIVE) (($F) $X) $X 1)>

    TESTS:

    We should be able to convert derivatives evaluated at a point,
    :trac:`12796`::

        sage: from sage.interfaces.maxima_lib import sr_to_max, max_to_sr
        sage: f = function('f')
        sage: f_prime = f(x).diff(x)
        sage: max_to_sr(sr_to_max(f_prime(x = 1)))
        D[0](f)(1)

    """
    global sage_op_dict, max_op_dict
    global sage_sym_dict, max_sym_dict
    if isinstance(expr,list) or isinstance(expr,tuple):
        return EclObject(([mlist],[sr_to_max(e) for e in expr]))
    op = expr.operator()
    if op:
        # Stolen from sage.symbolic.expression_conversion
        # Should be defined in a function and then put in special_sage_to_max
        # For that, we should change the API of the functions there
        # (we need to have access to op, not only to expr.operands()
        if isinstance(op, FDerivativeOperator):
            from sage.symbolic.ring import is_SymbolicVariable
            args = expr.operands()
            if (not all(is_SymbolicVariable(v) for v in args) or
                len(args) != len(set(args))):
                # An evaluated derivative of the form f'(1) is not a
                # symbolic variable, yet we would like to treat it
                # like one. So, we replace the argument `1` with a
                # temporary variable e.g. `t0` and then evaluate the
                # derivative f'(t0) symbolically at t0=1. See trac
                # #12796.
                temp_args=[var("t%s"%i) for i in range(len(args))]
                f =sr_to_max(op.function()(*temp_args))
                params = op.parameter_set()
                deriv_max = [[mdiff],f]
                for i in set(params):
                    deriv_max.extend([sr_to_max(temp_args[i]), EclObject(params.count(i))])
                at_eval=sr_to_max([temp_args[i]==args[i] for i in range(len(args))])
                return EclObject([[max_at],deriv_max,at_eval])

            f = sr_to_max(op.function()(*args))
            params = op.parameter_set()
            deriv_max = []
            [deriv_max.extend([sr_to_max(args[i]), EclObject(params.count(i))]) for i in set(params)]
            l = [[mdiff],f]
            l.extend(deriv_max)
            return EclObject(l)
        elif (op in special_sage_to_max):
            return EclObject(special_sage_to_max[op](*[sr_to_max(o) for o in expr.operands()]))
        elif not (op in sage_op_dict):
            # Maxima does some simplifications automatically by default
            # so calling maxima(expr) can change the structure of expr
            #op_max=caar(maxima(expr).ecl())
            # This should be safe if we treated all special operators above
            op_max=maxima(op).ecl()
            sage_op_dict[op]=op_max
            max_op_dict[op_max]=op
        return EclObject(([sage_op_dict[op]],
                     [sr_to_max(o) for o in expr.operands()]))
    elif expr.is_symbol() or expr.is_constant():
        if not expr in sage_sym_dict:
            sym_max=maxima(expr).ecl()
            sage_sym_dict[expr]=sym_max
            max_sym_dict[sym_max]=expr
        return sage_sym_dict[expr]
    else:
        try:
            return pyobject_to_max(expr.pyobject())
        except TypeError:
            return maxima(expr).ecl()

#.........这里部分代码省略.........
        ...
        TypeError: The first argument to solve() should be a symbolic expression or a list of symbolic expressions, cannot handle <type 'bool'>
        sage: solve([a, b], (1, a))
        Traceback (most recent call last):
        ...
        TypeError: 1 is not a valid variable.
        sage: solve([x == 1], (1, a))
        Traceback (most recent call last):
        ...
        TypeError: (1, a) are not valid variables.

    Test that the original version of a system in the French Sage book
    now works (:trac:`14306`)::

        sage: var('y,z')
        (y, z)
        sage: solve([x^2 * y * z == 18, x * y^3 * z == 24, x * y * z^4 == 6], x, y, z)
        [[x == 3, y == 2, z == 1], [x == (1.337215067... - 2.685489874...*I), y == (-1.700434271... + 1.052864325...*I), z == (0.9324722294... - 0.3612416661...*I)], ...]
    """
    from sage.symbolic.expression import is_Expression
    if is_Expression(f): # f is a single expression
        ans = f.solve(*args,**kwds)
        return ans

    if not isinstance(f, (list, tuple)):
        raise TypeError("The first argument must be a symbolic expression or a list of symbolic expressions.")

    if len(f)==1:
        # f is a list with a single element
        if is_Expression(f[0]):
            # if its a symbolic expression call solve method of this expression
            return f[0].solve(*args,**kwds)
        # otherwise complain
        raise TypeError("The first argument to solve() should be a symbolic "
                        "expression or a list of symbolic expressions, "
                        "cannot handle %s"%repr(type(f[0])))

    # f is a list of such expressions or equations
    from sage.symbolic.ring import is_SymbolicVariable

    if len(args)==0:
        raise TypeError("Please input variables to solve for.")
    if is_SymbolicVariable(args[0]):
        variables = args
    else:
        variables = tuple(args[0])

    for v in variables:
        if not is_SymbolicVariable(v):
            raise TypeError("%s is not a valid variable."%repr(v))

    try:
        f = [s for s in f if s is not True]
    except TypeError:
        raise ValueError("Unable to solve %s for %s"%(f, args))

    if any(s is False for s in f):
        return []

    from sage.calculus.calculus import maxima
    m = maxima(f)

    try:
        s = m.solve(variables)
    except Exception: # if Maxima gave an error, try its to_poly_solve
        try:
            s = m.to_poly_solve(variables)
        except TypeError as mess: # if that gives an error, raise an error.
            if "Error executing code in Maxima" in str(mess):
                raise ValueError("Sage is unable to determine whether the system %s can be solved for %s"%(f,args))
            else:
                raise

    if len(s)==0: # if Maxima's solve gave no solutions, try its to_poly_solve
        try:
            s = m.to_poly_solve(variables)
        except Exception: # if that gives an error, stick with no solutions
            s = []

    if len(s)==0: # if to_poly_solve gave no solutions, try use_grobner
        try:
            s = m.to_poly_solve(variables,'use_grobner=true')
        except Exception: # if that gives an error, stick with no solutions
            s = []

    sol_list = string_to_list_of_solutions(repr(s))

    # Relaxed form suggested by Mike Hansen (#8553):
    if kwds.get('solution_dict', False):
        if len(sol_list)==0: # fixes IndexError on empty solution list (#8553)
            return []
        if isinstance(sol_list[0], list):
            sol_dict=[dict([[eq.left(),eq.right()] for eq in solution])
                    for solution in sol_list]
        else:
            sol_dict=[{eq.left():eq.right()} for eq in sol_list]

        return sol_dict
    else:
        return sol_list

def sr_to_max(expr):
    r"""
    Convert a symbolic expression into a Maxima object.

    INPUT:

    - ``expr`` - symbolic expression

    OUTPUT: ECL object

    EXAMPLES::
        sage: from sage.interfaces.maxima_lib import sr_to_max
        sage: var('x')
        x
        sage: sr_to_max(x)
        <ECL: $X>
        sage: sr_to_max(cos(x))
        <ECL: ((%COS) $X)>
        sage: f = function('f',x)
        sage: sr_to_max(f.diff())
        <ECL: ((%DERIVATIVE) (($F) $X) $X 1)>
    """
    global sage_op_dict, max_op_dict
    global sage_sym_dict, max_sym_dict
    if isinstance(expr,list) or isinstance(expr,tuple):
        return EclObject(([mlist],[sr_to_max(e) for e in expr]))
    op = expr.operator()
    if op:
        # Stolen from sage.symbolic.expression_conversion
        # Should be defined in a function and then put in special_sage_to_max
        # For that, we should change the API of the functions there
        # (we need to have access to op, not only to expr.operands()
        if isinstance(op, FDerivativeOperator):
            from sage.symbolic.ring import is_SymbolicVariable
            args = expr.operands()
            if (not all(is_SymbolicVariable(v) for v in args) or
                len(args) != len(set(args))):
                raise NotImplementedError, "arguments must be distinct variables"
            f = sr_to_max(op.function()(*args))
            params = op.parameter_set()
            deriv_max = []
            [deriv_max.extend([sr_to_max(args[i]), EclObject(params.count(i))]) for i in set(params)]
            l = [[mdiff],f]
            l.extend(deriv_max)
            return EclObject(l)
        elif (op in special_sage_to_max):
            return EclObject(special_sage_to_max[op](*[sr_to_max(o) for o in expr.operands()]))
        elif not (op in sage_op_dict):
            # Maxima does some simplifications automatically by default
            # so calling maxima(expr) can change the structure of expr
            #op_max=caar(maxima(expr).ecl())
            # This should be safe if we treated all special operators above
            op_max=maxima(op).ecl()
            sage_op_dict[op]=op_max
            max_op_dict[op_max]=op
        return EclObject(([sage_op_dict[op]],
                     [sr_to_max(o) for o in expr.operands()]))
    elif expr.is_symbol() or expr.is_constant():
        if not expr in sage_sym_dict:
            sym_max=maxima(expr).ecl()
            sage_sym_dict[expr]=sym_max
            max_sym_dict[sym_max]=expr
        return sage_sym_dict[expr]
    else:
        try:
            return pyobject_to_max(expr.pyobject())
        except TypeError:
            return maxima(expr).ecl()

def wronskian(*args):
    """
    Returns the Wronskian of the provided functions, differentiating with
    respect to the given variable. If no variable is provided,
    diff(f) is called for each function f.

    wronskian(f1,...,fn, x) returns the Wronskian of f1,...,fn, with
    derivatives taken with respect to x.

    wronskian(f1,...,fn) returns the Wronskian of f1,...,fn where
    k'th derivatives are computed by doing ``.derivative(k)`` on each
    function.

    The Wronskian of a list of functions is a determinant of derivatives.
    The nth row (starting from 0) is a list of the nth derivatives of the
    given functions.

    For two functions::

                              | f   g  |
                 W(f, g) = det|        | = f*g' - g*f'.
                              | f'  g' |

    EXAMPLES::

        sage: wronskian(e^x, x^2)
        -x^2*e^x + 2*x*e^x

        sage: x,y = var('x, y')
        sage: wronskian(x*y, log(x), x)
        -y*log(x) + y

    If your functions are in a list, you can use `*' to turn them into
    arguments to :func:`wronskian`::

        sage: wronskian(*[x^k for k in range(1, 5)])
        12*x^4

    If you want to use 'x' as one of the functions in the Wronskian,
    you can't put it last or it will be interpreted as the variable
    with respect to which we differentiate. There are several ways to
    get around this.

    Two-by-two Wronskian of sin(x) and e^x::

        sage: wronskian(sin(x), e^x, x)
        -cos(x)*e^x + e^x*sin(x)

    Or don't put x last::

        sage: wronskian(x, sin(x), e^x)
        (cos(x)*e^x + e^x*sin(x))*x - 2*e^x*sin(x)

    Example where one of the functions is constant::

        sage: wronskian(1, e^(-x), e^(2*x))
        -6*e^x

    NOTES:

    - http://en.wikipedia.org/wiki/Wronskian
    - http://planetmath.org/encyclopedia/WronskianDeterminant.html

    AUTHORS:

    - Dan Drake (2008-03-12)
    """
    if len(args) == 0:
        raise TypeError('wronskian() takes at least one argument (0 given)')
    elif len(args) == 1:
        # a 1x1 Wronskian is just its argument
        return args[0]
    else:
        if is_SymbolicVariable(args[-1]):
            # if last argument is a variable, peel it off and
            # differentiate the other args
            v = args[-1]
            fs = args[0:-1]
            row = lambda n: map(lambda f: diff(f, v, n), fs)
        else:
            # if the last argument isn't a variable, just run
            # .derivative on everything
            fs = args
            row = lambda n: map(lambda f: diff(f, n), fs)
        # NOTE: I rewrote the below as two lines to avoid a possible subtle
        # memory management problem on some platforms (only VMware as far
        # as we know?).  See trac #2990.
        # There may still be a real problem that this is just hiding for now.
        A = matrix(map(row, range(len(fs))))
        return A.determinant()

#.........这里部分代码省略.........
        [[s == 1, j == 0], [s == g/b, j == (b - g)*m/(b*g)]]

    Inequalities can be also solved::

        sage: solve(x^2>8,x)
        [[x < -2*sqrt(2)], [x > 2*sqrt(2)]]

    Use use_grobner if no solution is obtained from to_poly_solve::

       sage: x,y=var('x y'); c1(x,y)=(x-5)^2+y^2-16; c2(x,y)=(y-3)^2+x^2-9
       sage: solve([c1(x,y),c2(x,y)],[x,y])                               
       [[x == -9/68*sqrt(55) + 135/68, y == -15/68*sqrt(5)*sqrt(11) + 123/68], [x == 9/68*sqrt(55) + 135/68, y == 15/68*sqrt(5)*sqrt(11) + 123/68]]
        
    TESTS::

        sage: solve([sin(x)==x,y^2==x],x,y)
        [sin(x) == x, y^2 == x]
        sage: solve(0==1,x)
        Traceback (most recent call last):
        ...
        TypeError:  The first argument must be a symbolic expression or a list of symbolic expressions.

    Test if the empty list is returned, too, when (a list of)
    dictionaries (is) are requested (#8553)::

        sage: solve([0==1],x)
        []
        sage: solve([0==1],x,solution_dict=True)
        []
        sage: solve([x==1,x==-1],x)
        []
        sage: solve([x==1,x==-1],x,solution_dict=True)
        []
        sage: solve((x==1,x==-1),x,solution_dict=0)
        []

    Relaxed form, suggested by Mike Hansen (#8553)::

        sage: solve([x^2-1],x,solution_dict=-1)
        [{x: -1}, {x: 1}]
        sage: solve([x^2-1],x,solution_dict=1)
        [{x: -1}, {x: 1}]
        sage: solve((x==1,x==-1),x,solution_dict=-1)
        []
        sage: solve((x==1,x==-1),x,solution_dict=1)
        []

    This inequality holds for any real ``x`` (trac #8078)::

        sage: solve(x^4+2>0,x)
        [x < +Infinity]

    """
    from sage.symbolic.expression import is_Expression
    if is_Expression(f): # f is a single expression
        ans = f.solve(*args,**kwds)
        return ans

    if not isinstance(f, (list, tuple)):
        raise TypeError("The first argument must be a symbolic expression or a list of symbolic expressions.")

    if len(f)==1 and is_Expression(f[0]):
        # f is a list with a single expression
        return f[0].solve(*args,**kwds)

    # f is a list of such expressions or equations
    from sage.symbolic.ring import is_SymbolicVariable

    if len(args)==0:
        raise TypeError, "Please input variables to solve for."
    if is_SymbolicVariable(args[0]):
        variables = args
    else:
        variables = tuple(args[0])
    
    for v in variables:
        if not is_SymbolicVariable(v):
            raise TypeError, "%s is not a valid variable."%v

    try:
        f = [s for s in f if s is not True]
    except TypeError:
        raise ValueError, "Unable to solve %s for %s"%(f, args)

    if any(s is False for s in f):
        return []

    from sage.calculus.calculus import maxima
    m = maxima(f)

    try:
        s = m.solve(variables)
    except: # if Maxima gave an error, try its to_poly_solve
        try:
            s = m.to_poly_solve(variables)
        except TypeError, mess: # if that gives an error, raise an error.
            if "Error executing code in Maxima" in str(mess):
                raise ValueError, "Sage is unable to determine whether the system %s can be solved for %s"%(f,args)
            else:
                raise