def _print_latex_matplotlib(o):
     """
     A function that returns a png rendered by mathtext
     """
     debug("_print_latex_matplotlib:", "called with %s" % o)
     if _can_print_latex(o):
         s = latex(o, mode='inline')
         return _matplotlib_wrapper(s)

 def _print_latex_text(o):
     debug("_print_latex_text:", "called with %s" % o)
     """
     A function to generate the latex representation of sympy expressions.
     """
     if _can_print_latex(o):
         s = latex(o, mode='plain')
         s = s.replace(r'\dag', r'\dagger')
         s = s.strip('$')
         debug("_print_latex_text:", "returns $$%s$$" % s)
         return '$$%s$$' % s

 def _preview_wrapper(o):
     exprbuffer = BytesIO()
     try:
         preview(
             o, output="png", viewer="BytesIO", outputbuffer=exprbuffer, preamble=preamble, dvioptions=dvioptions
         )
     except Exception as e:
         # IPython swallows exceptions
         debug("png printing:", "_preview_wrapper exception raised:", repr(e))
         raise
     return exprbuffer.getvalue()

 def _matplotlib_wrapper(o):
     # mathtext does not understand certain latex flags, so we try to
     # replace them with suitable subs
     o = o.replace(r'\operatorname', '')
     o = o.replace(r'\overline', r'\bar')
     # mathtext can't render some LaTeX commands. For example, it can't
     # render any LaTeX environments such as array or matrix. So here we
     # ensure that if mathtext fails to render, we return None.
     try:
         return latex_to_png(o)
     except ValueError as e:
         debug('matplotlib exception caught:', repr(e))
         return None

 def _matplotlib_wrapper(o):
     debug("_matplotlib_wrapper:", "called with %s" % o)
     # mathtext does not understand certain latex flags, so we try to
     # replace them with suitable subs
     o = o.replace(r'\operatorname', '')
     o = o.replace(r'\overline', r'\bar')
     try:
         return latex_to_png(o)
     except Exception:
         debug("_matplotlib_wrapper:", "exeption raised")
         # Matplotlib.mathtext cannot render some things (like
         # matrices)
         return None

def downvalues_rules(r, parsed):
    '''
    Function which generates parsed rules by substituting all possible
    combinations of default values.
    '''
    res = []
    index = 0
    for i in r:
        debug('parsing rule {}'.format(r.index(i) + 1))
        # Parse Pattern
        if i[1][1][0] == 'Condition':
            p = i[1][1][1].copy()
        else:
            p = i[1][1].copy()

        optional = get_default_values(p, {})
        pattern = generate_sympy_from_parsed(p.copy(), replace_Int=True)
        pattern, free_symbols = add_wildcards(pattern, optional=optional)
        free_symbols = list(set(free_symbols)) #remove common symbols

        # Parse Transformed Expression and Constraints
        if i[2][0] == 'Condition': # parse rules without constraints separately
            constriant = divide_constraint(i[2][2], free_symbols) # separate And constraints into individual constraints
            FreeQ_vars, FreeQ_x = seperate_freeq(i[2][2].copy()) # separate FreeQ into individual constraints
            transformed = generate_sympy_from_parsed(i[2][1].copy(), symbols=free_symbols)
        else:
            constriant = ''
            FreeQ_vars, FreeQ_x = [], []
            transformed = generate_sympy_from_parsed(i[2].copy(), symbols=free_symbols)

        FreeQ_constraint = parse_freeq(FreeQ_vars, FreeQ_x, free_symbols)
        pattern = sympify(pattern)
        pattern = rubi_printer(pattern, sympy_integers=True)
        pattern = setWC(pattern)
        transformed = sympify(transformed)

        index += 1
        if type(transformed) == Function('With') or type(transformed) == Function('Module'): # define separate function when With appears
            transformed, With_constraints = replaceWith(transformed, free_symbols, index)
            parsed += '    pattern' + str(index) +' = Pattern(' + pattern + '' + FreeQ_constraint + '' + constriant + With_constraints + ')'
            parsed += '\n{}'.format(transformed)
            parsed += '\n    ' + 'rule' + str(index) +' = ReplacementRule(' + 'pattern' + rubi_printer(index, sympy_integers=True) + ', lambda ' + ', '.join(free_symbols) + ' : ' + 'With{}({})'.format(index, ', '.join(free_symbols)) + ')\n    '
        else:
            transformed = rubi_printer(transformed, sympy_integers=True)
            parsed += '    pattern' + str(index) +' = Pattern(' + pattern + '' + FreeQ_constraint + '' + constriant + ')'
            parsed += '\n    ' + 'rule' + str(index) +' = ReplacementRule(' + 'pattern' + rubi_printer(index, sympy_integers=True) + ', lambda ' + ', '.join(free_symbols) + ' : ' + transformed + ')\n    '
        parsed += 'rubi.add(rule'+ str(index) +')\n\n'

    parsed += '    return rubi\n'

    return parsed

 def _print_latex_png(o):
     """
     A function that returns a png rendered by an external latex
     distribution, falling back to matplotlib rendering
     """
     if _can_print_latex(o):
         s = latex(o, mode=latex_mode)
         try:
             return _preview_wrapper(s)
         except RuntimeError as e:
             debug('preview failed with:', repr(e),
                   ' Falling back to matplotlib backend')
             if latex_mode != 'inline':
                 s = latex(o, mode='inline')
             return _matplotlib_wrapper(s)

 def _print_latex_png(o):
     debug("_print_latex_png:", "called with %s" % o)
     """
     A function that returns a png rendered by an external latex
     distribution, falling back to matplotlib rendering
     """
     if _can_print_latex(o):
         s = latex(o, mode=latex_mode)
         try:
             return _preview_wrapper(s)
         except RuntimeError:
             if latex_mode != 'inline':
                 s = latex(o, mode='inline')
             debug("_print_latex_png(o):",
                   "calling _matplotlib_wrapper")
             return _matplotlib_wrapper(s)

def param_rischDE(fa, fd, G, DE):
    """
    Solve a Parametric Risch Differential Equation: Dy + f*y == Sum(ci*Gi, (i, 1, m)).
    """
    _, (fa, fd) = weak_normalizer(fa, fd, DE)
    a, (ba, bd), G, hn = prde_normal_denom(ga, gd, G, DE)
    A, B, G, hs = prde_special_denom(a, ba, bd, G, DE)
    g = gcd(A, B)
    A, B, G = A.quo(g), B.quo(g), [gia.cancel(gid*g, include=True) for
        gia, gid in G]
    Q, M = prde_linear_constraints(A, B, G, DE)
    M, _ = constant_system(M, zeros(M.rows, 1), DE)
    # Reduce number of constants at this point
    try:
        # Similar to rischDE(), we try oo, even though it might lead to
        # non-termination when there is no solution.  At least for prde_spde,
        # it will always terminate no matter what n is.
        n = bound_degree(A, B, G, DE, parametric=True)
    except NotImplementedError:
        debug("param_rischDE: Proceeding with n = oo; may cause "
              "non-termination.")
        n = oo

    A, B, Q, R, n1 = prde_spde(A, B, Q, n, DE)

def build_parser(output_dir=dir_latex_antlr):
    check_antlr_version()

    debug("Updating ANTLR-generated code in {}".format(output_dir))

    if not os.path.exists(output_dir):
        os.makedirs(output_dir)

    with open(os.path.join(output_dir, "__init__.py"), "w+") as fp:
        fp.write(header)

    args = [
        "antlr4",
        grammar_file,
        "-o", output_dir,
        # for now, not generating these as latex2sympy did not use them
        "-no-visitor",
        "-no-listener",
    ]

    debug("Running code generation...\n\t$ {}".format(" ".join(args)))
    subprocess.check_output(args, cwd=output_dir)

    debug("Applying headers and renaming...")
    # Handle case insensitive file systems. If the files are already
    # generated, they will be written to latex* but LaTeX*.* won't match them.
    for path in (glob.glob(os.path.join(output_dir, "LaTeX*.*")) +
        glob.glob(os.path.join(output_dir, "latex*.*"))):
        offset = 0
        new_path = os.path.join(output_dir,
                                os.path.basename(path).lower())
        with open(path, 'r') as f:
            lines = [line.rstrip() + '\n' for line in f.readlines()]

        os.unlink(path)

        with open(new_path, "w") as out_file:
            if path.endswith(".py"):
                offset = 2
                out_file.write(header)
            out_file.writelines(lines[offset:])

        debug("\t{}".format(new_path))

    return True

def check_antlr_version():
    debug("Checking antlr4 version...")

    try:
        debug(subprocess.check_output(["antlr4"])
              .decode('utf-8').split("\n")[0])
        return True
    except (subprocess.CalledProcessError, FileNotFoundError):
        debug("The antlr4 command line tool is not installed, "
              "or not on your PATH\n"
              "> Please install it via your preferred package manager")
        return False

    def analyse_gens(gens, hints):
        """
        Analyse the generators ``gens``, using the hints ``hints``.

        The meaning of ``hints`` is described in the main docstring.
        Return a new list of generators, and also the ideal we should
        work with.
        """
        # First parse the hints
        n, funcs, iterables, extragens = parse_hints(hints)
        debug('n=%s' % n, 'funcs:', funcs, 'iterables:',
              iterables, 'extragens:', extragens)

        # We just add the extragens to gens and analyse them as before
        gens = list(gens)
        gens.extend(extragens)

        # remove duplicates
        funcs = list(set(funcs))
        iterables = list(set(iterables))
        gens = list(set(gens))

        # all the functions we can do anything with
        allfuncs = {sin, cos, tan, sinh, cosh, tanh}
        # sin(3*x) -> ((3, x), sin)
        trigterms = [(g.args[0].as_coeff_mul(), g.func) for g in gens
                     if g.func in allfuncs]
        # Our list of new generators - start with anything that we cannot
        # work with (i.e. is not a trigonometric term)
        freegens = [g for g in gens if g.func not in allfuncs]
        newgens = []
        trigdict = {}
        for (coeff, var), fn in trigterms:
            trigdict.setdefault(var, []).append((coeff, fn))
        res = [] # the ideal

        for key, val in trigdict.items():
            # We have now assembeled a dictionary. Its keys are common
            # arguments in trigonometric expressions, and values are lists of
            # pairs (fn, coeff). x0, (fn, coeff) in trigdict means that we
            # need to deal with fn(coeff*x0). We take the rational gcd of the
            # coeffs, call it ``gcd``. We then use x = x0/gcd as "base symbol",
            # all other arguments are integral multiples thereof.
            # We will build an ideal which works with sin(x), cos(x).
            # If hint tan is provided, also work with tan(x). Moreover, if
            # n > 1, also work with sin(k*x) for k <= n, and similarly for cos
            # (and tan if the hint is provided). Finally, any generators which
            # the ideal does not work with but we need to accommodate (either
            # because it was in expr or because it was provided as a hint)
            # we also build into the ideal.
            # This selection process is expressed in the list ``terms``.
            # build_ideal then generates the actual relations in our ideal,
            # from this list.
            fns = [x[1] for x in val]
            val = [x[0] for x in val]
            gcd = reduce(igcd, val)
            terms = [(fn, v/gcd) for (fn, v) in zip(fns, val)]
            fs = set(funcs + fns)
            for c, s, t in ([cos, sin, tan], [cosh, sinh, tanh]):
                if any(x in fs for x in (c, s, t)):
                    fs.add(c)
                    fs.add(s)
            for fn in fs:
                for k in range(1, n + 1):
                    terms.append((fn, k))
            extra = []
            for fn, v in terms:
                if fn == tan:
                    extra.append((sin, v))
                    extra.append((cos, v))
                if fn in [sin, cos] and tan in fs:
                    extra.append((tan, v))
                if fn == tanh:
                    extra.append((sinh, v))
                    extra.append((cosh, v))
                if fn in [sinh, cosh] and tanh in fs:
                    extra.append((tanh, v))
            terms.extend(extra)
            x = gcd*Mul(*key)
            r = build_ideal(x, terms)
            res.extend(r)
            newgens.extend(set(fn(v*x) for fn, v in terms))

        # Add generators for compound expressions from iterables
        for fn, args in iterables:
            if fn == tan:
                # Tan expressions are recovered from sin and cos.
                iterables.extend([(sin, args), (cos, args)])
            elif fn == tanh:
                # Tanh expressions are recovered from sihn and cosh.
                iterables.extend([(sinh, args), (cosh, args)])
            else:
                dummys = symbols('d:%i' % len(args), cls=Dummy)
                expr = fn( Add(*dummys)).expand(trig=True).subs(list(zip(dummys, args)))
                res.append(fn(Add(*args)) - expr)

        if myI in gens:
            res.append(myI**2 + 1)
            freegens.remove(myI)
            newgens.append(myI)
#.........这里部分代码省略.........

    def _ratsimpmodprime(a, b, allsol, N=0, D=0):
        r"""
        Computes a rational simplification of ``a/b`` which minimizes
        the sum of the total degrees of the numerator and the denominator.

        The algorithm proceeds by looking at ``a * d - b * c`` modulo
        the ideal generated by ``G`` for some ``c`` and ``d`` with degree
        less than ``a`` and ``b`` respectively.
        The coefficients of ``c`` and ``d`` are indeterminates and thus
        the coefficients of the normalform of ``a * d - b * c`` are
        linear polynomials in these indeterminates.
        If these linear polynomials, considered as system of
        equations, have a nontrivial solution, then `\frac{a}{b}
        \equiv \frac{c}{d}` modulo the ideal generated by ``G``. So,
        by construction, the degree of ``c`` and ``d`` is less than
        the degree of ``a`` and ``b``, so a simpler representation
        has been found.
        After a simpler representation has been found, the algorithm
        tries to reduce the degree of the numerator and denominator
        and returns the result afterwards.

        As an extension, if quick=False, we look at all possible degrees such
        that the total degree is less than *or equal to* the best current
        solution. We retain a list of all solutions of minimal degree, and try
        to find the best one at the end.
        """
        c, d = a, b
        steps = 0

        maxdeg = a.total_degree() + b.total_degree()
        if quick:
            bound = maxdeg - 1
        else:
            bound = maxdeg
        while N + D <= bound:
            if (N, D) in tested:
                break
            tested.add((N, D))

            M1 = staircase(N)
            M2 = staircase(D)
            debug('%s / %s: %s, %s' % (N, D, M1, M2))

            Cs = symbols("c:%d" % len(M1), cls=Dummy)
            Ds = symbols("d:%d" % len(M2), cls=Dummy)
            ng = Cs + Ds

            c_hat = Poly(
                sum([Cs[i] * M1[i] for i in range(len(M1))]), opt.gens + ng)
            d_hat = Poly(
                sum([Ds[i] * M2[i] for i in range(len(M2))]), opt.gens + ng)

            r = reduced(a * d_hat - b * c_hat, G, opt.gens + ng,
                        order=opt.order, polys=True)[1]

            S = Poly(r, gens=opt.gens).coeffs()
            sol = solve(S, Cs + Ds, particular=True, quick=True)

            if sol and not all([s == 0 for s in sol.values()]):
                c = c_hat.subs(sol)
                d = d_hat.subs(sol)

                # The "free" variables occurring before as parameters
                # might still be in the substituted c, d, so set them
                # to the value chosen before:
                c = c.subs(dict(list(zip(Cs + Ds, [1] * (len(Cs) + len(Ds))))))
                d = d.subs(dict(list(zip(Cs + Ds, [1] * (len(Cs) + len(Ds))))))

                c = Poly(c, opt.gens)
                d = Poly(d, opt.gens)
                if d == 0:
                    raise ValueError('Ideal not prime?')

                allsol.append((c_hat, d_hat, S, Cs + Ds))
                if N + D != maxdeg:
                    allsol = [allsol[-1]]

                break

            steps += 1
            N += 1
            D += 1

        if steps > 0:
            c, d, allsol = _ratsimpmodprime(c, d, allsol, N, D - steps)
            c, d, allsol = _ratsimpmodprime(c, d, allsol, N - steps, D)

        return c, d, allsol

#.........这里部分代码省略.........
        """
        Build generators for our ideal. Terms is an iterable with elements of
        the form (fn, coeff), indicating that we have a generator fn(coeff*x).

        If any of the terms is trigonometric, sin(x) and cos(x) are guaranteed
        to appear in terms. Similarly for hyperbolic functions. For tan(n*x),
        sin(n*x) and cos(n*x) are guaranteed.
        """
        I = []
        y = Dummy('y')
        for fn, coeff in terms:
            for c, s, t, rel in (
                    [cos, sin, tan, cos(x)**2 + sin(x)**2 - 1],
                    [cosh, sinh, tanh, cosh(x)**2 - sinh(x)**2 - 1]):
                if coeff == 1 and fn in [c, s]:
                    I.append(rel)
                elif fn == t:
                    I.append(t(coeff*x)*c(coeff*x) - s(coeff*x))
                elif fn in [c, s]:
                    cn = fn(coeff*y).expand(trig=True).subs(y, x)
                    I.append(fn(coeff*x) - cn)
        return list(set(I))

    def analyse_gens(gens, hints):
        """
        Analyse the generators ``gens``, using the hints ``hints``.

        The meaning of ``hints`` is described in the main docstring.
        Return a new list of generators, and also the ideal we should
        work with.
        """
        # First parse the hints
        n, funcs, iterables, extragens = parse_hints(hints)
        debug('n=%s' % n, 'funcs:', funcs, 'iterables:',
              iterables, 'extragens:', extragens)

        # We just add the extragens to gens and analyse them as before
        gens = list(gens)
        gens.extend(extragens)

        # remove duplicates
        funcs = list(set(funcs))
        iterables = list(set(iterables))
        gens = list(set(gens))

        # all the functions we can do anything with
        allfuncs = {sin, cos, tan, sinh, cosh, tanh}
        # sin(3*x) -> ((3, x), sin)
        trigterms = [(g.args[0].as_coeff_mul(), g.func) for g in gens
                     if g.func in allfuncs]
        # Our list of new generators - start with anything that we cannot
        # work with (i.e. is not a trigonometric term)
        freegens = [g for g in gens if g.func not in allfuncs]
        newgens = []
        trigdict = {}
        for (coeff, var), fn in trigterms:
            trigdict.setdefault(var, []).append((coeff, fn))
        res = [] # the ideal

        for key, val in trigdict.items():
            # We have now assembeled a dictionary. Its keys are common
            # arguments in trigonometric expressions, and values are lists of
            # pairs (fn, coeff). x0, (fn, coeff) in trigdict means that we
            # need to deal with fn(coeff*x0). We take the rational gcd of the
            # coeffs, call it ``gcd``. We then use x = x0/gcd as "base symbol",
            # all other arguments are integral multiples thereof.

#.........这里部分代码省略.........
    >>> init_printing(pretty_print=True) # doctest: +SKIP
    >>> sqrt(5) # doctest: +SKIP
      ___
    \/ 5
    >>> theta = Symbol('theta') # doctest: +SKIP
    >>> init_printing(use_unicode=True) # doctest: +SKIP
    >>> theta # doctest: +SKIP
    \u03b8
    >>> init_printing(use_unicode=False) # doctest: +SKIP
    >>> theta # doctest: +SKIP
    theta
    >>> init_printing(order='lex') # doctest: +SKIP
    >>> str(y + x + y**2 + x**2) # doctest: +SKIP
    x**2 + x + y**2 + y
    >>> init_printing(order='grlex') # doctest: +SKIP
    >>> str(y + x + y**2 + x**2) # doctest: +SKIP
    x**2 + x + y**2 + y
    >>> init_printing(order='grevlex') # doctest: +SKIP
    >>> str(y * x**2 + x * y**2) # doctest: +SKIP
    x**2*y + x*y**2
    >>> init_printing(order='old') # doctest: +SKIP
    >>> str(x**2 + y**2 + x + y) # doctest: +SKIP
    x**2 + x + y**2 + y
    >>> init_printing(num_columns=10) # doctest: +SKIP
    >>> x**2 + x + y**2 + y # doctest: +SKIP
    x + y +
    x**2 + y**2
    """
    import sys
    from sympy.printing.printer import Printer

    if pretty_print:
        if pretty_printer is not None:
            stringify_func = pretty_printer
        else:
            from sympy.printing import pretty as stringify_func
    else:
        if str_printer is not None:
            stringify_func = str_printer
        else:
            from sympy.printing import sstrrepr as stringify_func

    # Even if ip is not passed, double check that not in IPython shell
    in_ipython = False
    if ip is None:
        try:
            ip = get_ipython()
        except NameError:
            pass
        else:
            in_ipython = (ip is not None)

    if ip and not in_ipython:
        in_ipython = _is_ipython(ip)

    if in_ipython and pretty_print:
        try:
            import IPython
            # IPython 1.0 deprecates the frontend module, so we import directly
            # from the terminal module to prevent a deprecation message from being
            # shown.
            if V(IPython.__version__) >= '1.0':
                from IPython.terminal.interactiveshell import TerminalInteractiveShell
            else:
                from IPython.frontend.terminal.interactiveshell import TerminalInteractiveShell
            from code import InteractiveConsole
        except ImportError:
            pass
        else:
            # This will be True if we are in the qtconsole or notebook
            if not isinstance(ip, (InteractiveConsole, TerminalInteractiveShell)) \
                    and 'ipython-console' not in ''.join(sys.argv):
                if use_unicode is None:
                    debug("init_printing: Setting use_unicode to True")
                    use_unicode = True
                if use_latex is None:
                    debug("init_printing: Setting use_latex to True")
                    use_latex = True

    if not no_global:
        Printer.set_global_settings(order=order, use_unicode=use_unicode,
                                    wrap_line=wrap_line, num_columns=num_columns)
    else:
        _stringify_func = stringify_func

        if pretty_print:
            stringify_func = lambda expr: \
                             _stringify_func(expr, order=order,
                                             use_unicode=use_unicode,
                                             wrap_line=wrap_line,
                                             num_columns=num_columns)
        else:
            stringify_func = lambda expr: _stringify_func(expr, order=order)

    if in_ipython:
        _init_ipython_printing(ip, stringify_func, use_latex, euler,
                               forecolor, backcolor, fontsize, latex_mode,
                               print_builtin, latex_printer)
    else:
        _init_python_printing(stringify_func)

def _init_ipython_printing(ip, stringify_func, use_latex, euler, forecolor,
                           backcolor, fontsize, latex_mode, print_builtin,
                           latex_printer):
    """Setup printing in IPython interactive session. """
    try:
        from IPython.lib.latextools import latex_to_png
    except ImportError:
        pass

    preamble = "\\documentclass[%s]{article}\n" \
               "\\pagestyle{empty}\n" \
               "\\usepackage{amsmath,amsfonts}%s\\begin{document}"
    if euler:
        addpackages = '\\usepackage{euler}'
    else:
        addpackages = ''
    preamble = preamble % (fontsize, addpackages)

    imagesize = 'tight'
    offset = "0cm,0cm"
    resolution = 150
    dvi = r"-T %s -D %d -bg %s -fg %s -O %s" % (
        imagesize, resolution, backcolor, forecolor, offset)
    dvioptions = dvi.split()
    debug("init_printing: DVIOPTIONS:", dvioptions)
    debug("init_printing: PREAMBLE:", preamble)

    latex = latex_printer or default_latex

    def _print_plain(arg, p, cycle):
        """caller for pretty, for use in IPython 0.11"""
        if _can_print_latex(arg):
            p.text(stringify_func(arg))
        else:
            p.text(IPython.lib.pretty.pretty(arg))

    def _preview_wrapper(o):
        exprbuffer = BytesIO()
        try:
            preview(o, output='png', viewer='BytesIO',
                    outputbuffer=exprbuffer, preamble=preamble,
                    dvioptions=dvioptions)
        except Exception as e:
            # IPython swallows exceptions
            debug("png printing:", "_preview_wrapper exception raised:",
                  repr(e))
            raise
        return exprbuffer.getvalue()

    def _matplotlib_wrapper(o):
        # mathtext does not understand certain latex flags, so we try to
        # replace them with suitable subs
        o = o.replace(r'\operatorname', '')
        o = o.replace(r'\overline', r'\bar')
        # mathtext can't render some LaTeX commands. For example, it can't
        # render any LaTeX environments such as array or matrix. So here we
        # ensure that if mathtext fails to render, we return None.
        try:
            return latex_to_png(o)
        except ValueError as e:
            debug('matplotlib exception caught:', repr(e))
            return None

    def _can_print_latex(o):
        """Return True if type o can be printed with LaTeX.

        If o is a container type, this is True if and only if every element of
        o can be printed with LaTeX.
        """
        from sympy import Basic
        from sympy.matrices import MatrixBase
        from sympy.physics.vector import Vector, Dyadic
        if isinstance(o, (list, tuple, set, frozenset)):
            return all(_can_print_latex(i) for i in o)
        elif isinstance(o, dict):
            return all(_can_print_latex(i) and _can_print_latex(o[i]) for i in o)
        elif isinstance(o, bool):
            return False
        # TODO : Investigate if "elif hasattr(o, '_latex')" is more useful
        # to use here, than these explicit imports.
        elif isinstance(o, (Basic, MatrixBase, Vector, Dyadic)):
            return True
        elif isinstance(o, (float, integer_types)) and print_builtin:
            return True
        return False

    def _print_latex_png(o):
        """
        A function that returns a png rendered by an external latex
        distribution, falling back to matplotlib rendering
        """
        if _can_print_latex(o):
            s = latex(o, mode=latex_mode)
            try:
                return _preview_wrapper(s)
            except RuntimeError as e:
                debug('preview failed with:', repr(e),
                      ' Falling back to matplotlib backend')
                if latex_mode != 'inline':
                    s = latex(o, mode='inline')
#.........这里部分代码省略.........

def ratsimpmodprime(expr, G, *gens, **args):
    """
    Simplifies a rational expression ``expr`` modulo the prime ideal
    generated by ``G``.  ``G`` should be a Groebner basis of the
    ideal.

    >>> from sympy.simplify.ratsimp import ratsimpmodprime
    >>> from sympy.abc import x, y
    >>> eq = (x + y**5 + y)/(x - y)
    >>> ratsimpmodprime(eq, [x*y**5 - x - y], x, y, order='lex')
    (x**2 + x*y + x + y)/(x**2 - x*y)

    If ``polynomial`` is False, the algorithm computes a rational
    simplification which minimizes the sum of the total degrees of
    the numerator and the denominator.

    If ``polynomial`` is True, this function just brings numerator and
    denominator into a canonical form. This is much faster, but has
    potentially worse results.

    References
    ==========

    .. [1] M. Monagan, R. Pearce, Rational Simplification Modulo a Polynomial
    Ideal,
    http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.163.6984
    (specifically, the second algorithm)
    """
    from sympy import solve

    quick = args.pop('quick', True)
    polynomial = args.pop('polynomial', False)
    debug('ratsimpmodprime', expr)

    # usual preparation of polynomials:

    num, denom = cancel(expr).as_numer_denom()

    try:
        polys, opt = parallel_poly_from_expr([num, denom] + G, *gens, **args)
    except PolificationFailed:
        return expr

    domain = opt.domain

    if domain.has_assoc_Field:
        opt.domain = domain.get_field()
    else:
        raise DomainError(
            "can't compute rational simplification over %s" % domain)

    # compute only once
    leading_monomials = [g.LM(opt.order) for g in polys[2:]]
    tested = set()

    def staircase(n):
        """
        Compute all monomials with degree less than ``n`` that are
        not divisible by any element of ``leading_monomials``.
        """
        if n == 0:
            return [1]
        S = []
        for mi in combinations_with_replacement(range(len(opt.gens)), n):
            m = [0]*len(opt.gens)
            for i in mi:
                m[i] += 1
            if all([monomial_div(m, lmg) is None for lmg in
                    leading_monomials]):
                S.append(m)

        return [Monomial(s).as_expr(*opt.gens) for s in S] + staircase(n - 1)

    def _ratsimpmodprime(a, b, allsol, N=0, D=0):
        r"""
        Computes a rational simplification of ``a/b`` which minimizes
        the sum of the total degrees of the numerator and the denominator.

        The algorithm proceeds by looking at ``a * d - b * c`` modulo
        the ideal generated by ``G`` for some ``c`` and ``d`` with degree
        less than ``a`` and ``b`` respectively.
        The coefficients of ``c`` and ``d`` are indeterminates and thus
        the coefficients of the normalform of ``a * d - b * c`` are
        linear polynomials in these indeterminates.
        If these linear polynomials, considered as system of
        equations, have a nontrivial solution, then `\frac{a}{b}
        \equiv \frac{c}{d}` modulo the ideal generated by ``G``. So,
        by construction, the degree of ``c`` and ``d`` is less than
        the degree of ``a`` and ``b``, so a simpler representation
        has been found.
        After a simpler representation has been found, the algorithm
        tries to reduce the degree of the numerator and denominator
        and returns the result afterwards.

        As an extension, if quick=False, we look at all possible degrees such
        that the total degree is less than *or equal to* the best current
        solution. We retain a list of all solutions of minimal degree, and try
        to find the best one at the end.
        """
        c, d = a, b
#.........这里部分代码省略.........

def downvalues_rules(r, header, cons_dict, cons_index, index):
    '''
    Function which generates parsed rules by substituting all possible
    combinations of default values.
    '''
    rules = '['
    parsed = '\n\n'
    cons = ''
    cons_import = [] # it contains name of constraints that need to be imported for rules.
    for i in r:
        debug('parsing rule {}'.format(r.index(i) + 1))
        # Parse Pattern
        if i[1][1][0] == 'Condition':
            p = i[1][1][1].copy()
        else:
            p = i[1][1].copy()

        optional = get_default_values(p, {})
        pattern = generate_sympy_from_parsed(p.copy(), replace_Int=True)
        pattern, free_symbols = add_wildcards(pattern, optional=optional)
        free_symbols = list(set(free_symbols)) #remove common symbols
        # Parse Transformed Expression and Constraints
        if i[2][0] == 'Condition': # parse rules without constraints separately
            constriant, constraint_def, cons_index = divide_constraint(i[2][2], free_symbols, cons_index, cons_dict, cons_import) # separate And constraints into individual constraints
            FreeQ_vars, FreeQ_x = seperate_freeq(i[2][2].copy()) # separate FreeQ into individual constraints
            transformed = generate_sympy_from_parsed(i[2][1].copy(), symbols=free_symbols)
        else:
            constriant = ''
            constraint_def = ''
            FreeQ_vars, FreeQ_x = [], []
            transformed = generate_sympy_from_parsed(i[2].copy(), symbols=free_symbols)
        FreeQ_constraint, free_cons_def, cons_index = parse_freeq(FreeQ_vars, FreeQ_x, cons_index, cons_dict, cons_import, free_symbols)
        pattern = sympify(pattern, locals={"Or": Function("Or"), "And": Function("And"), "Not":Function("Not") })
        pattern = rubi_printer(pattern, sympy_integers=True)

        pattern = setWC(pattern)
        transformed = sympify(transformed, locals={"Or": Function("Or"), "And": Function("And"), "Not":Function("Not") })
        constraint_def = constraint_def + free_cons_def
        cons+=constraint_def
        index += 1

        # below are certain if - else condition depending on various situation that may be encountered
        if type(transformed) == Function('With') or type(transformed) == Function('Module'): # define separate function when With appears
            transformed, With_constraints, return_type = replaceWith(transformed, free_symbols, index)
            if return_type is None:
                parsed += '{}'.format(transformed)
                parsed += '\n    pattern' + str(index) +' = Pattern(' + pattern + '' + FreeQ_constraint + '' + constriant + ')'
                parsed += '\n    ' + 'rule' + str(index) +' = ReplacementRule(' + 'pattern' + rubi_printer(index, sympy_integers=True) + ', With{}'.format(index) + ')\n'
            else:

                parsed += '{}'.format(transformed)
                parsed += '\n    pattern' + str(index) +' = Pattern(' + pattern + '' + FreeQ_constraint + '' + constriant + With_constraints + ')'
                parsed += '\n    def replacement{}({}):\n'.format(index, ', '.join(free_symbols)) + return_type[0] + '\n        rubi.append({})\n        return '.format(index) + return_type[1]
                parsed += '\n    ' + 'rule' + str(index) +' = ReplacementRule(' + 'pattern' + rubi_printer(index, sympy_integers=True) + ', replacement{}'.format(index) + ')\n'

        else:
            transformed = rubi_printer(transformed, sympy_integers=True)
            parsed += '    pattern' + str(index) +' = Pattern(' + pattern + '' + FreeQ_constraint + '' + constriant + ')'
            parsed += '\n    def replacement{}({}):\n        rubi.append({})\n        return '.format(index, ', '.join(free_symbols), index) + transformed
            parsed += '\n    ' + 'rule' + str(index) +' = ReplacementRule(' + 'pattern' + rubi_printer(index, sympy_integers=True) + ', replacement{}'.format(index) + ')\n'
        rules += 'rule{}, '.format(index)
    rules += ']'
    parsed += '    return ' + rules +'\n'

    header += '    from sympy.integrals.rubi.constraints import ' + ', '.join(word for word in cons_import)
    parsed = header + parsed
    return parsed, cons_index, cons, index

def _init_ipython_printing(ip, stringify_func, use_latex, euler,
                           forecolor, backcolor, fontsize, latex_mode):
    """Setup printing in IPython interactive session. """
    try:
        from IPython.lib.latextools import latex_to_png
    except ImportError:
        pass

    preamble = "\\documentclass[%s]{article}\n" \
               "\\pagestyle{empty}\n" \
               "\\usepackage{amsmath,amsfonts}%s\\begin{document}"
    if euler:
        addpackages = '\\usepackage{euler}'
    else:
        addpackages = ''
    preamble = preamble % (fontsize, addpackages)

    imagesize = 'tight'
    offset = "0cm,0cm"
    resolution = 150
    dvi = r"-T %s -D %d -bg %s -fg %s -O %s" % (
        imagesize, resolution, backcolor, forecolor, offset)
    dvioptions = dvi.split()
    debug("init_printing: DVIOPTIONS:", dvioptions)
    debug("init_printing: PREAMBLE:", preamble)

    def _print_plain(arg, p, cycle):
        """caller for pretty, for use in IPython 0.11""&