def trigsimp_groebner(expr, hints=[], quick=False, order="grlex",
                      polynomial=False):
    """
    Simplify trigonometric expressions using a groebner basis algorithm.

    This routine takes a fraction involving trigonometric or hyperbolic
    expressions, and tries to simplify it. The primary metric is the
    total degree. Some attempts are made to choose the simplest possible
    expression of the minimal degree, but this is non-rigorous, and also
    very slow (see the ``quick=True`` option).

    If ``polynomial`` is set to True, instead of simplifying numerator and
    denominator together, this function just brings numerator and denominator
    into a canonical form. This is much faster, but has potentially worse
    results. However, if the input is a polynomial, then the result is
    guaranteed to be an equivalent polynomial of minimal degree.

    The most important option is hints. Its entries can be any of the
    following:

    - a natural number
    - a function
    - an iterable of the form (func, var1, var2, ...)
    - anything else, interpreted as a generator

    A number is used to indicate that the search space should be increased.
    A function is used to indicate that said function is likely to occur in a
    simplified expression.
    An iterable is used indicate that func(var1 + var2 + ...) is likely to
    occur in a simplified .
    An additional generator also indicates that it is likely to occur.
    (See examples below).

    This routine carries out various computationally intensive algorithms.
    The option ``quick=True`` can be used to suppress one particularly slow
    step (at the expense of potentially more complicated results, but never at
    the expense of increased total degree).

    Examples
    ========

    >>> from sympy.abc import x, y
    >>> from sympy import sin, tan, cos, sinh, cosh, tanh
    >>> from sympy.simplify.trigsimp import trigsimp_groebner

    Suppose you want to simplify ``sin(x)*cos(x)``. Naively, nothing happens:

    >>> ex = sin(x)*cos(x)
    >>> trigsimp_groebner(ex)
    sin(x)*cos(x)

    This is because ``trigsimp_groebner`` only looks for a simplification
    involving just ``sin(x)`` and ``cos(x)``. You can tell it to also try
    ``2*x`` by passing ``hints=[2]``:

    >>> trigsimp_groebner(ex, hints=[2])
    sin(2*x)/2
    >>> trigsimp_groebner(sin(x)**2 - cos(x)**2, hints=[2])
    -cos(2*x)

    Increasing the search space this way can quickly become expensive. A much
    faster way is to give a specific expression that is likely to occur:

    >>> trigsimp_groebner(ex, hints=[sin(2*x)])
    sin(2*x)/2

    Hyperbolic expressions are similarly supported:

    >>> trigsimp_groebner(sinh(2*x)/sinh(x))
    2*cosh(x)

    Note how no hints had to be passed, since the expression already involved
    ``2*x``.

    The tangent function is also supported. You can either pass ``tan`` in the
    hints, to indicate that tan should be tried whenever cosine or sine are,
    or you can pass a specific generator:

    >>> trigsimp_groebner(sin(x)/cos(x), hints=[tan])
    tan(x)
    >>> trigsimp_groebner(sinh(x)/cosh(x), hints=[tanh(x)])
    tanh(x)

    Finally, you can use the iterable form to suggest that angle sum formulae
    should be tried:

    >>> ex = (tan(x) + tan(y))/(1 - tan(x)*tan(y))
    >>> trigsimp_groebner(ex, hints=[(tan, x, y)])
    tan(x + y)
    """
    # TODO
    #  - preprocess by replacing everything by funcs we can handle
    # - optionally use cot instead of tan
    # - more intelligent hinting.
    #     For example, if the ideal is small, and we have sin(x), sin(y),
    #     add sin(x + y) automatically... ?
    # - algebraic numbers ...
    # - expressions of lowest degree are not distinguished properly
    #   e.g. 1 - sin(x)**2
    # - we could try to order the generators intelligently, so as to influence
#.........这里部分代码省略.........

def primitive_element(extension, x=None, **args):
    """Construct a common number field for all extensions. """
    if not extension:
        raise ValueError("can't compute primitive element for empty extension")

    if x is not None:
        x, cls = sympify(x), Poly
    else:
        x, cls = Dummy('x'), PurePoly
    if not args.get('ex', False):
        extension = [ AlgebraicNumber(ext, gen=x) for ext in extension ]

        g, coeffs = extension[0].minpoly.replace(x), [1]

        for ext in extension[1:]:
            s, _, g = sqf_norm(g, x, extension=ext)
            coeffs = [ s*c for c in coeffs ] + [1]

        if not args.get('polys', False):
            return g.as_expr(), coeffs
        else:
            return cls(g), coeffs

    generator = numbered_symbols('y', cls=Dummy)

    F, Y = [], []

    for ext in extension:
        y = next(generator)

        if ext.is_Poly:
            if ext.is_univariate:
                f = ext.as_expr(y)
            else:
                raise ValueError("expected minimal polynomial, got %s" % ext)
        else:
            f = minpoly(ext, y)

        F.append(f)
        Y.append(y)

    coeffs_generator = args.get('coeffs', _coeffs_generator)

    for coeffs in coeffs_generator(len(Y)):
        f = x - sum([ c*y for c, y in zip(coeffs, Y)])
        G = groebner(F + [f], Y + [x], order='lex', field=True)

        H, g = G[:-1], cls(G[-1], x, domain='QQ')

        for i, (h, y) in enumerate(zip(H, Y)):
            try:
                H[i] = Poly(y - h, x,
                            domain='QQ').all_coeffs()  # XXX: composite=False
            except CoercionFailed:  # pragma: no cover
                break  # G is not a triangular set
        else:
            break
    else:  # pragma: no cover
        raise RuntimeError("run out of coefficient configurations")

    _, g = g.clear_denoms()

    if not args.get('polys', False):
        return g.as_expr(), coeffs, H
    else:
        return g, coeffs, H

def _minpoly_groebner(ex, x, cls):
    """
    Computes the minimal polynomial of an algebraic number
    using Groebner bases

    Examples
    ========

    >>> from sympy import minimal_polynomial, sqrt, Rational
    >>> from sympy.abc import x
    >>> minimal_polynomial(sqrt(2) + 3*Rational(1, 3), x, compose=False)
    x**2 - 2*x - 1

    """
    from sympy.polys.polytools import degree
    from sympy.core.function import expand_multinomial

    generator = numbered_symbols('a', cls=Dummy)
    mapping, symbols, replace = {}, {}, []

    def update_mapping(ex, exp, base=None):
        a = next(generator)
        symbols[ex] = a

        if base is not None:
            mapping[ex] = a**exp + base
        else:
            mapping[ex] = exp.as_expr(a)

        return a

    def bottom_up_scan(ex):
        if ex.is_Atom:
            if ex is S.ImaginaryUnit:
                if ex not in mapping:
                    return update_mapping(ex, 2, 1)
                else:
                    return symbols[ex]
            elif ex.is_Rational:
                return ex
        elif ex.is_Add:
            return Add(*[ bottom_up_scan(g) for g in ex.args ])
        elif ex.is_Mul:
            return Mul(*[ bottom_up_scan(g) for g in ex.args ])
        elif ex.is_Pow:
            if ex.exp.is_Rational:
                if ex.exp < 0 and ex.base.is_Add:
                    coeff, terms = ex.base.as_coeff_add()
                    elt, _ = primitive_element(terms, polys=True)

                    alg = ex.base - coeff

                    # XXX: turn this into eval()
                    inverse = invert(elt.gen + coeff, elt).as_expr()
                    base = inverse.subs(elt.gen, alg).expand()

                    if ex.exp == -1:
                        return bottom_up_scan(base)
                    else:
                        ex = base**(-ex.exp)
                if not ex.exp.is_Integer:
                    base, exp = (
                        ex.base**ex.exp.p).expand(), Rational(1, ex.exp.q)
                else:
                    base, exp = ex.base, ex.exp
                base = bottom_up_scan(base)
                expr = base**exp

                if expr not in mapping:
                    return update_mapping(expr, 1/exp, -base)
                else:
                    return symbols[expr]
        elif ex.is_AlgebraicNumber:
            if ex.root not in mapping:
                return update_mapping(ex.root, ex.minpoly)
            else:
                return symbols[ex.root]

        raise NotAlgebraic("%s doesn't seem to be an algebraic number" % ex)

    def simpler_inverse(ex):
        """
        Returns True if it is more likely that the minimal polynomial
        algorithm works better with the inverse
        """
        if ex.is_Pow:
            if (1/ex.exp).is_integer and ex.exp < 0:
                if ex.base.is_Add:
                    return True
        if ex.is_Mul:
            hit = True
            a = []
            for p in ex.args:
                if p.is_Add:
                    return False
                if p.is_Pow:
                    if p.base.is_Add and p.exp > 0:
                        return False

            if hit:
#.........这里部分代码省略.........

def minimal_polynomial(ex, x=None, **args):
    """
    Computes the minimal polynomial of an algebraic number.

    **Example**

    >>> from sympy import minimal_polynomial, sqrt
    >>> from sympy.abc import x

    >>> minimal_polynomial(sqrt(2), x)
    x**2 - 2
    >>> minimal_polynomial(sqrt(2) + sqrt(3), x)
    x**4 - 10*x**2 + 1

    """
    generator = numbered_symbols('a', cls=Dummy)
    mapping, symbols, replace = {}, {}, []

    ex = sympify(ex)

    if x is not None:
        x, cls = sympify(x), Poly
    else:
        x, cls = Dummy('x'), PurePoly

    def update_mapping(ex, exp, base=None):
        a = generator.next()
        symbols[ex] = a

        if base is not None:
            mapping[ex] = a**exp + base
        else:
            mapping[ex] = exp.as_expr(a)

        return a

    def bottom_up_scan(ex):
        if ex.is_Atom:
            if ex is S.ImaginaryUnit:
                if ex not in mapping:
                    return update_mapping(ex, 2, 1)
                else:
                    return symbols[ex]
            elif ex.is_Rational and ex.q != 0:
                return ex
        elif ex.is_Add:
            return Add(*[ bottom_up_scan(g) for g in ex.args ])
        elif ex.is_Mul:
            return Mul(*[ bottom_up_scan(g) for g in ex.args ])
        elif ex.is_Pow:
            if ex.exp.is_Rational:
                if ex.exp < 0 and ex.base.is_Add:
                    coeff, terms = ex.base.as_coeff_add()
                    elt, _ = primitive_element(terms, polys=True)

                    alg = ex.base - coeff

                    # XXX: turn this into eval()
                    inverse = invert(elt.gen + coeff, elt).as_expr()
                    base = inverse.subs(elt.gen, alg).expand()

                    if ex.exp == -1:
                        return bottom_up_scan(base)
                    else:
                        ex = base**(-ex.exp)

                if not ex.exp.is_Integer:
                    base, exp = (ex.base**ex.exp.p).expand(), Rational(1, ex.exp.q)
                else:
                    base, exp = ex.base, ex.exp

                base = bottom_up_scan(base)
                expr = base**exp

                if expr not in mapping:
                    return update_mapping(expr, 1/exp, -base)
                else:
                    return symbols[expr]
        elif ex.is_AlgebraicNumber:
            if ex.root not in mapping:
                return update_mapping(ex.root, ex.minpoly)
            else:
                return symbols[ex.root]

        raise NotAlgebraic("%s doesn't seem to be an algebraic number" % ex)

    polys = args.get('polys', False)

    if ex.is_AlgebraicNumber:
        if not polys:
            return ex.minpoly.as_expr(x)
        else:
            return ex.minpoly.replace(x)
    elif ex.is_Rational and ex.q != 0:
        result = ex.q*x - ex.p
    else:
        F = [x - bottom_up_scan(ex)] + mapping.values()
        G = groebner(F, symbols.values() + [x], order='lex')

        _, factors = factor_list(G[-1])
#.........这里部分代码省略.........

def minimal_polynomial(ex, x=None, **args):
    """
    Computes the minimal polynomial of an algebraic number.

    Parameters
    ==========

    ex : algebraic number expression

    x : indipendent variable of the minimal polynomial

    Options
    =======

    compose : if ``True`` _minpoly1`` is used, else the ``groebner`` algorithm

    polys : if ``True`` returns a ``Poly`` object

    Notes
    =====

    By default ``compose=True``, the minimal polynomial of the subexpressions of ``ex``
    are computed, then the arithmetic operations on them are performed using the resultant
    and factorization.
    If ``compose=False``, a bottom-up algorithm is used with ``groebner``.
    The default algorithm stalls less frequently.

    Examples
    ========

    >>> from sympy import minimal_polynomial, sqrt, solve
    >>> from sympy.abc import x

    >>> minimal_polynomial(sqrt(2), x)
    x**2 - 2
    >>> minimal_polynomial(sqrt(2) + sqrt(3), x)
    x**4 - 10*x**2 + 1
    >>> minimal_polynomial(solve(x**3 + x + 3)[0], x)
    x**3 + x + 3

    """
    from sympy.polys.polytools import degree
    from sympy.core.function import expand_multinomial
    from sympy.core.basic import preorder_traversal

    compose = args.get('compose', True)
    polys = args.get('polys', False)
    ex = sympify(ex)
    for expr in preorder_traversal(ex):
        if expr.is_AlgebraicNumber:
            compose = False
            break

    if ex.is_AlgebraicNumber:
        compose = False

    if x is not None:
        x, cls = sympify(x), Poly
    else:
        x, cls = Dummy('x'), PurePoly

    if compose:
        result = _minpoly1(ex, x)
        result = result.primitive()[1]
        c = result.coeff(x**degree(result, x))
        if c < 0:
            result = expand_mul(-result)
            c = -c
        return cls(result, x, field=True) if polys else result

    generator = numbered_symbols('a', cls=Dummy)
    mapping, symbols, replace = {}, {}, []

    def update_mapping(ex, exp, base=None):
        a = generator.next()
        symbols[ex] = a

        if base is not None:
            mapping[ex] = a**exp + base
        else:
            mapping[ex] = exp.as_expr(a)

        return a

    def bottom_up_scan(ex):
        if ex.is_Atom:
            if ex is S.ImaginaryUnit:
                if ex not in mapping:
                    return update_mapping(ex, 2, 1)
                else:
                    return symbols[ex]
            elif ex.is_Rational:
                return ex
        elif ex.is_Add:
            return Add(*[ bottom_up_scan(g) for g in ex.args ])
        elif ex.is_Mul:
            return Mul(*[ bottom_up_scan(g) for g in ex.args ])
        elif ex.is_Pow:
            if ex.exp.is_Rational:
                if ex.exp < 0 and ex.base.is_Add:
#.........这里部分代码省略.........

#.........这里部分代码省略.........
                    coeff, terms = ex.base.as_coeff_add()
                    elt, _ = primitive_element(terms, polys=True)

                    alg = ex.base - coeff

                    # XXX: turn this into eval()
                    inverse = invert(elt.gen + coeff, elt).as_expr()
                    base = inverse.subs(elt.gen, alg).expand()

                    if ex.exp == -1:
                        return bottom_up_scan(base)
                    else:
                        ex = base ** (-ex.exp)
                if not ex.exp.is_Integer:
                    base, exp = (ex.base ** ex.exp.p).expand(), Rational(1, ex.exp.q)
                else:
                    base, exp = ex.base, ex.exp
                base = bottom_up_scan(base)
                expr = base ** exp

                if expr not in mapping:
                    return update_mapping(expr, 1 / exp, -base)
                else:
                    return symbols[expr]
        elif ex.is_AlgebraicNumber:
            if ex.root not in mapping:
                return update_mapping(ex.root, ex.minpoly)
            else:
                return symbols[ex.root]

        raise NotAlgebraic("%s doesn't seem to be an algebraic number" % ex)

    def simpler_inverse(ex):
        """
        Returns True if it is more likely that the minimal polynomial
        algorithm works better with the inverse
        """
        if ex.is_Pow:
            if (1 / ex.exp).is_integer and ex.exp < 0:
                if ex.base.is_Add:
                    return True
        if ex.is_Mul:
            hit = True
            a = []
            for p in ex.args:
                if p.is_Add:
                    return False
                if p.is_Pow:
                    if p.base.is_Add and p.exp > 0:
                        return False

            if hit:
                return True
        return False

    polys = args.get("polys", False)
    prec = args.pop("prec", 10)

    inverted = False
    ex = expand_multinomial(ex)
    if ex.is_AlgebraicNumber:
        if not polys:
            return ex.minpoly.as_expr(x)
        else:
            return ex.minpoly.replace(x)
    elif ex.is_Rational:
        result = ex.q * x - ex.p
    else:
        inverted = simpler_inverse(ex)
        if inverted:
            ex = ex ** -1
        res = None
        if ex.is_Pow and (1 / ex.exp).is_Integer:
            n = 1 / ex.exp
            res = _minimal_polynomial_sq(ex.base, n, x, prec)

        elif _is_sum_surds(ex):
            res = _minimal_polynomial_sq(ex, S.One, x, prec)

        if res is not None:
            result = res

        if res is None:
            bus = bottom_up_scan(ex)
            F = [x - bus] + mapping.values()
            G = groebner(F, symbols.values() + [x], order="lex")

            _, factors = factor_list(G[-1])
            # by construction G[-1] has root `ex`
            result = _choose_factor(factors, x, ex, prec)
            if result is None:
                raise NotImplementedError("multiple candidates for the minimal polynomial of %s" % ex)
    if inverted:
        result = expand_mul(x ** degree(result) * result.subs(x, 1 / x))
        if result.coeff(x ** degree(result)) < 0:
            result = expand_mul(-result)
    if polys:
        return cls(result, x, field=True)
    else:
        return result