def _dmp_inner_gcd(f, g, u, K):
    """Helper function for `dmp_inner_gcd()`. """
    if not K.is_Exact:
        try:
            exact = K.get_exact()
        except DomainError:
            return dmp_one(u, K), f, g

        f = dmp_convert(f, u, K, exact)
        g = dmp_convert(g, u, K, exact)

        h, cff, cfg = _dmp_inner_gcd(f, g, u, exact)

        h = dmp_convert(h, u, exact, K)
        cff = dmp_convert(cff, u, exact, K)
        cfg = dmp_convert(cfg, u, exact, K)

        return h, cff, cfg
    elif K.has_Field:
        if K.is_QQ and query('USE_HEU_GCD'):
            try:
                return dmp_qq_heu_gcd(f, g, u, K)
            except HeuristicGCDFailed:
                pass

        return dmp_ff_prs_gcd(f, g, u, K)
    else:
        if K.is_ZZ and query('USE_HEU_GCD'):
            try:
                return dmp_zz_heu_gcd(f, g, u, K)
            except HeuristicGCDFailed:
                pass

        return dmp_rr_prs_gcd(f, g, u, K)

    def poly_unify(f, g):
        """Unify a multivariate fraction and a polynomial. """
        if not isinstance(g, DMP) or f.lev != g.lev:
            raise UnificationFailed("can't unify %s with %s" % (f, g))

        if f.dom == g.dom:
            return (f.lev, f.dom, f.per, (f.num, f.den), g.rep)
        else:
            lev, dom = f.lev, f.dom.unify(g.dom)

            F = (dmp_convert(f.num, lev, f.dom, dom),
                 dmp_convert(f.den, lev, f.dom, dom))

            G = dmp_convert(g.rep, lev, g.dom, dom)

            def per(num, den, cancel=True, kill=False):
                if kill:
                    if not lev:
                        return num/den
                    else:
                        lev = lev - 1

                if cancel:
                    num, den = dmp_cancel(num, den, lev, dom)

                return f.__class__.new((num, den), dom, lev)

            return lev, dom, per, F, G

    def unify(f, g):
        """Unify representations of two multivariate polynomials. """
        return f.lev, f.dom, f.per, f.rep, g.rep

        if not isinstance(g, DMP) or f.lev != g.lev:
            raise UnificationFailed("can't unify %s with %s" % (f, g))

        if f.dom == g.dom:
            return f.lev, f.dom, f.per, f.rep, g.rep
        else:
            lev, dom = f.lev, f.dom.unify(g.dom)

            F = dmp_convert(f.rep, lev, f.dom, dom)
            G = dmp_convert(g.rep, lev, g.dom, dom)

            def per(rep, dom=dom, lev=lev, kill=False):
                if kill:
                    if not lev:
                        return rep
                    else:
                        lev -= 1

                return DMP(rep, dom, lev)

            return lev, dom, per, F, G

def dmp_cancel(f, g, u, K, multout=True):
    """
    Cancel common factors in a rational function ``f/g``.

    **Examples**

    >>> from sympy.polys.domains import ZZ
    >>> from sympy.polys.euclidtools import dmp_cancel

    >>> f = ZZ.map([[2], [0], [-2]])
    >>> g = ZZ.map([[1], [-2], [1]])

    >>> dmp_cancel(f, g, 1, ZZ)
    ([[2], [2]], [[1], [-1]])

    """
    if dmp_zero_p(f, u) or dmp_zero_p(g, u):
        if multout:
            return f, g
        else:
            return K.one, K.one, f, g

    K0 = None

    if K.has_Field and K.has_assoc_Ring:
        K0, K = K, K.get_ring()

        cq, f = dmp_clear_denoms(f, u, K0, K, convert=True)
        cp, g = dmp_clear_denoms(g, u, K0, K, convert=True)
    else:
        cp, cq = K.one, K.one

    _, p, q = dmp_inner_gcd(f, g, u, K)

    if K0 is not None:
        p = dmp_convert(p, u, K, K0)
        q = dmp_convert(q, u, K, K0)

        K = K0

    p_neg = K.is_negative(dmp_ground_LC(p, u, K))
    q_neg = K.is_negative(dmp_ground_LC(q, u, K))

    if p_neg and q_neg:
        p, q = dmp_neg(p, u, K), dmp_neg(q, u, K)
    elif p_neg:
        cp, p = -cp, dmp_neg(p, u, K)
    elif q_neg:
        cp, q = -cp, dmp_neg(q, u, K)

    if not multout:
        return cp, cq, p, q

    p = dmp_mul_ground(p, cp, u, K)
    q = dmp_mul_ground(q, cq, u, K)

    return p, q

def dmp_cancel(f, g, u, K, include=True):
    """
    Cancel common factors in a rational function `f/g`.

    Examples
    ========

    >>> from sympy.polys import ring, ZZ
    >>> R, x,y = ring("x,y", ZZ)

    >>> R.dmp_cancel(2*x**2 - 2, x**2 - 2*x + 1)
    (2*x + 2, x - 1)

    """
    K0 = None

    if K.has_Field and K.has_assoc_Ring:
        K0, K = K, K.get_ring()

        cq, f = dmp_clear_denoms(f, u, K0, K, convert=True)
        cp, g = dmp_clear_denoms(g, u, K0, K, convert=True)
    else:
        cp, cq = K.one, K.one

    _, p, q = dmp_inner_gcd(f, g, u, K)

    if K0 is not None:
        _, cp, cq = K.cofactors(cp, cq)

        p = dmp_convert(p, u, K, K0)
        q = dmp_convert(q, u, K, K0)

        K = K0

    p_neg = K.is_negative(dmp_ground_LC(p, u, K))
    q_neg = K.is_negative(dmp_ground_LC(q, u, K))

    if p_neg and q_neg:
        p, q = dmp_neg(p, u, K), dmp_neg(q, u, K)
    elif p_neg:
        cp, p = -cp, dmp_neg(p, u, K)
    elif q_neg:
        cp, q = -cp, dmp_neg(q, u, K)

    if not include:
        return cp, cq, p, q

    p = dmp_mul_ground(p, cp, u, K)
    q = dmp_mul_ground(q, cq, u, K)

    return p, q

def dmp_ext_factor(f, u, K):
    """Factor multivariate polynomials over algebraic number fields. """
    if not u:
        return dup_ext_factor(f, K)

    lc = dmp_ground_LC(f, u, K)
    f = dmp_ground_monic(f, u, K)

    if all([ d <= 0 for d in dmp_degree_list(f, u) ]):
        return lc, []

    f, F = dmp_sqf_part(f, u, K), f
    s, g, r = dmp_sqf_norm(f, u, K)

    factors = dmp_factor_list_include(r, u, K.dom)

    if len(factors) == 1:
        coeff, factors = lc, [f]
    else:
        H = dmp_raise([K.one, s*K.unit], u, 0, K)

        for i, (factor, _) in enumerate(factors):
            h = dmp_convert(factor, u, K.dom, K)
            h, _, g = dmp_inner_gcd(h, g, u, K)
            h = dmp_compose(h, H, u, K)
            factors[i] = h

    return lc, dmp_trial_division(F, factors, u, K)

def dmp_clear_denoms(f, u, K0, K1=None, convert=False):
    """
    Clear denominators, i.e. transform ``K_0`` to ``K_1``.

    Examples
    ========

    >>> from sympy.polys.domains import QQ, ZZ
    >>> from sympy.polys.densetools import dmp_clear_denoms

    >>> f = [[QQ(1,2)], [QQ(1,3), QQ(1)]]
    >>> dmp_clear_denoms(f, 1, QQ, convert=False)
    (6, [[3/1], [2/1, 6/1]])

    >>> f = [[QQ(1,2)], [QQ(1,3), QQ(1)]]
    >>> dmp_clear_denoms(f, 1, QQ, convert=True)
    (6, [[3], [2, 6]])

    """
    if not u:
        return dup_clear_denoms(f, K0, K1, convert=convert)

    if K1 is None:
        K1 = K0.get_ring()

    common = _rec_clear_denoms(f, u, K0, K1)

    if not K1.is_one(common):
        f = dmp_mul_ground(f, common, u, K0)

    if not convert:
        return common, f
    else:
        return common, dmp_convert(f, u, K0, K1)

def test_dmp_convert():
    K0, K1 = ZZ['x'], ZZ

    f = [[K0(1)], [K0(2)], [], [K0(3)]]

    assert dmp_convert(f, 1, K0, K1) == \
        [[ZZ(1)], [ZZ(2)], [], [ZZ(3)]]

def test_dmp_convert():
    K0, K1 = ZZ['x'], ZZ

    f = [[DMP([1], ZZ)],[DMP([2], ZZ)],[],[DMP([3], ZZ)]]

    assert dmp_convert(f, 1, K0, K1) == \
        [[ZZ(1)],[ZZ(2)],[],[ZZ(3)]]

def test_dmp_integrate_in():
    f = dmp_convert(f_6, 3, ZZ, QQ)

    assert dmp_integrate_in(f, 2, 1, 3, QQ) == dmp_swap(dmp_integrate(dmp_swap(f, 0, 1, 3, QQ), 2, 3, QQ), 0, 1, 3, QQ)
    assert dmp_integrate_in(f, 3, 1, 3, QQ) == dmp_swap(dmp_integrate(dmp_swap(f, 0, 1, 3, QQ), 3, 3, QQ), 0, 1, 3, QQ)
    assert dmp_integrate_in(f, 2, 2, 3, QQ) == dmp_swap(dmp_integrate(dmp_swap(f, 0, 2, 3, QQ), 2, 3, QQ), 0, 2, 3, QQ)
    assert dmp_integrate_in(f, 3, 2, 3, QQ) == dmp_swap(dmp_integrate(dmp_swap(f, 0, 2, 3, QQ), 3, 3, QQ), 0, 2, 3, QQ)

def dmp_qq_heu_gcd(f, g, u, K0):
    """
    Heuristic polynomial GCD in `Q[X]`.

    Returns ``(h, cff, cfg)`` such that ``a = gcd(f, g)``,
    ``cff = quo(f, h)``, and ``cfg = quo(g, h)``.

    Examples
    ========

    >>> from sympy.polys import ring, QQ
    >>> R, x,y, = ring("x,y", QQ)

    >>> f = QQ(1,4)*x**2 + x*y + y**2
    >>> g = QQ(1,2)*x**2 + x*y

    >>> R.dmp_qq_heu_gcd(f, g)
    (x + 2*y, 1/4*x + 1/2*y, 1/2*x)

    """
    result = _dmp_ff_trivial_gcd(f, g, u, K0)

    if result is not None:
        return result

    K1 = K0.get_ring()

    cf, f = dmp_clear_denoms(f, u, K0, K1)
    cg, g = dmp_clear_denoms(g, u, K0, K1)

    f = dmp_convert(f, u, K0, K1)
    g = dmp_convert(g, u, K0, K1)

    h, cff, cfg = dmp_zz_heu_gcd(f, g, u, K1)

    h = dmp_convert(h, u, K1, K0)

    c = dmp_ground_LC(h, u, K0)
    h = dmp_ground_monic(h, u, K0)

    cff = dmp_convert(cff, u, K1, K0)
    cfg = dmp_convert(cfg, u, K1, K0)

    cff = dmp_mul_ground(cff, K0.quo(c, cf), u, K0)
    cfg = dmp_mul_ground(cfg, K0.quo(c, cg), u, K0)

    return h, cff, cfg

def dmp_qq_heu_gcd(f, g, u, K0):
    """
    Heuristic polynomial GCD in `Q[X]`.

    Returns ``(h, cff, cfg)`` such that ``a = gcd(f, g)``,
    ``cff = quo(f, h)``, and ``cfg = quo(g, h)``.

    Examples
    ========

    >>> from sympy.polys.domains import QQ
    >>> from sympy.polys.euclidtools import dmp_qq_heu_gcd

    >>> f = [[QQ(1,4)], [QQ(1), QQ(0)], [QQ(1), QQ(0), QQ(0)]]
    >>> g = [[QQ(1,2)], [QQ(1), QQ(0)], []]

    >>> dmp_qq_heu_gcd(f, g, 1, QQ)
    ([[1/1], [2/1, 0/1]], [[1/4], [1/2, 0/1]], [[1/2], []])

    """
    result = _dmp_ff_trivial_gcd(f, g, u, K0)

    if result is not None:
        return result

    K1 = K0.get_ring()

    cf, f = dmp_clear_denoms(f, u, K0, K1)
    cg, g = dmp_clear_denoms(g, u, K0, K1)

    f = dmp_convert(f, u, K0, K1)
    g = dmp_convert(g, u, K0, K1)

    h, cff, cfg = dmp_zz_heu_gcd(f, g, u, K1)

    h = dmp_convert(h, u, K1, K0)

    c = dmp_ground_LC(h, u, K0)
    h = dmp_ground_monic(h, u, K0)

    cff = dmp_convert(cff, u, K1, K0)
    cfg = dmp_convert(cfg, u, K1, K0)

    cff = dmp_mul_ground(cff, K0.quo(c, cf), u, K0)
    cfg = dmp_mul_ground(cfg, K0.quo(c, cg), u, K0)

    return h, cff, cfg

def dmp_qq_collins_resultant(f, g, u, K0):
    """
    Collins's modular resultant algorithm in `Q[X]`.

    Examples
    ========

    >>> from sympy.polys import ring, QQ
    >>> R, x,y = ring("x,y", QQ)

    >>> f = QQ(1,2)*x + y + QQ(2,3)
    >>> g = 2*x*y + x + 3

    >>> R.dmp_qq_collins_resultant(f, g)
    -2*y**2 - 7/3*y + 5/6

    """
    n = dmp_degree(f, u)
    m = dmp_degree(g, u)

    if n < 0 or m < 0:
        return dmp_zero(u - 1)

    K1 = K0.get_ring()

    cf, f = dmp_clear_denoms(f, u, K0, K1)
    cg, g = dmp_clear_denoms(g, u, K0, K1)

    f = dmp_convert(f, u, K0, K1)
    g = dmp_convert(g, u, K0, K1)

    r = dmp_zz_collins_resultant(f, g, u, K1)
    r = dmp_convert(r, u - 1, K1, K0)

    c = K0.convert(cf**m * cg**n, K1)

    return dmp_quo_ground(r, c, u - 1, K0)

def dmp_qq_collins_resultant(f, g, u, K0):
    """
    Collins's modular resultant algorithm in `Q[X]`.

    Examples
    ========

    >>> from sympy.polys.domains import QQ
    >>> from sympy.polys.euclidtools import dmp_qq_collins_resultant

    >>> f = [[QQ(1,2)], [QQ(1), QQ(2,3)]]
    >>> g = [[QQ(2), QQ(1)], [QQ(3)]]

    >>> dmp_qq_collins_resultant(f, g, 1, QQ)
    [-2/1, -7/3, 5/6]

    """
    n = dmp_degree(f, u)
    m = dmp_degree(g, u)

    if n < 0 or m < 0:
        return dmp_zero(u-1)

    K1 = K0.get_ring()

    cf, f = dmp_clear_denoms(f, u, K0, K1)
    cg, g = dmp_clear_denoms(g, u, K0, K1)

    f = dmp_convert(f, u, K0, K1)
    g = dmp_convert(g, u, K0, K1)

    r = dmp_zz_collins_resultant(f, g, u, K1)
    r = dmp_convert(r, u-1, K1, K0)

    c = K0.convert(cf**m * cg**n, K1)

    return dmp_quo_ground(r, c, u-1, K0)

    def frac_unify(f, g):
        """Unify representations of two multivariate fractions. """
        if not isinstance(g, DMF) or f.lev != g.lev:
            raise UnificationFailed("can't unify %s with %s" % (f, g))

        if f.dom == g.dom and f.ring == g.ring:
            return (f.lev, f.dom, f.per, (f.num, f.den),
                                         (g.num, g.den))
        else:
            lev, dom = f.lev, f.dom.unify(g.dom)
            ring = f.ring
            if g.ring is not None:
                if ring is not None:
                    ring = ring.unify(g.ring)
                else:
                    ring = g.ring

            F = (dmp_convert(f.num, lev, f.dom, dom),
                 dmp_convert(f.den, lev, f.dom, dom))

            G = (dmp_convert(g.num, lev, g.dom, dom),
                 dmp_convert(g.den, lev, g.dom, dom))

            def per(num, den, cancel=True, kill=False, lev=lev):
                if kill:
                    if not lev:
                        return num/den
                    else:
                        lev = lev - 1

                if cancel:
                    num, den = dmp_cancel(num, den, lev, dom)

                return f.__class__.new((num, den), dom, lev, ring=ring)

            return lev, dom, per, F, G

def dmp_lift(f, u, K):
    """
    Convert algebraic coefficients to integers in ``K[X]``.

    Examples
    ========

    >>> from sympy.polys import ring, QQ
    >>> from sympy import I

    >>> K = QQ.algebraic_field(I)
    >>> R, x = ring("x", K)

    >>> f = x**2 + K([QQ(1), QQ(0)])*x + K([QQ(2), QQ(0)])

    >>> R.dmp_lift(f)
    x**8 + 2*x**6 + 9*x**4 - 8*x**2 + 16

    """
    if not K.is_Algebraic:
        raise DomainError(
            'computation can be done only in an algebraic domain')

    F, monoms, polys = dmp_to_dict(f, u), [], []

    for monom, coeff in F.items():
        if not coeff.is_ground:
            monoms.append(monom)

    perms = variations([-1, 1], len(monoms), repetition=True)

    for perm in perms:
        G = dict(F)

        for sign, monom in zip(perm, monoms):
            if sign == -1:
                G[monom] = -G[monom]

        polys.append(dmp_from_dict(G, u, K))

    return dmp_convert(dmp_expand(polys, u, K), u, K, K.dom)

def dmp_lift(f, u, K):
    """
    Convert algebraic coefficients to integers in ``K[X]``.

    Examples
    ========

    >>> from sympy import I
    >>> from sympy.polys.domains import QQ
    >>> from sympy.polys.densetools import dmp_lift

    >>> K = QQ.algebraic_field(I)
    >>> f = [K(1), K([QQ(1), QQ(0)]), K([QQ(2), QQ(0)])]

    >>> dmp_lift(f, 0, K)
    [1/1, 0/1, 2/1, 0/1, 9/1, 0/1, -8/1, 0/1, 16/1]

    """
    if not K.is_Algebraic:
        raise DomainError(
            'computation can be done only in an algebraic domain')

    F, monoms, polys = dmp_to_dict(f, u), [], []

    for monom, coeff in F.iteritems():
        if not coeff.is_ground:
            monoms.append(monom)

    perms = variations([-1, 1], len(monoms), repetition=True)

    for perm in perms:
        G = dict(F)

        for sign, monom in zip(perm, monoms):
            if sign == -1:
                G[monom] = -G[monom]

        polys.append(dmp_from_dict(G, u, K))

    return dmp_convert(dmp_expand(polys, u, K), u, K, K.dom)

def dmp_clear_denoms(f, u, K0, K1=None, convert=False):
    """
    Clear denominators, i.e. transform ``K_0`` to ``K_1``.

    Examples
    ========

    >>> from sympy.polys import ring, QQ
    >>> R, x,y = ring("x,y", QQ)

    >>> f = QQ(1,2)*x + QQ(1,3)*y + 1

    >>> R.dmp_clear_denoms(f, convert=False)
    (6, 3*x + 2*y + 6)
    >>> R.dmp_clear_denoms(f, convert=True)
    (6, 3*x + 2*y + 6)

    """
    if not u:
        return dup_clear_denoms(f, K0, K1, convert=convert)

    if K1 is None:
        if K0.has_assoc_Ring:
            K1 = K0.get_ring()
        else:
            K1 = K0

    common = _rec_clear_denoms(f, u, K0, K1)

    if not K1.is_one(common):
        f = dmp_mul_ground(f, common, u, K0)

    if not convert:
        return common, f
    else:
        return common, dmp_convert(f, u, K0, K1)

def dmp_factor_list(f, u, K0, **args):
    """Factor polynomials into irreducibles in `K[X]`. """
    if not u:
        return dup_factor_list(f, K0, **args)

    if not K0.has_CharacteristicZero: # pragma: no cover
        raise DomainError('only characteristic zero allowed')

    if K0.is_Algebraic:
        coeff, factors = dmp_ext_factor(f, u, K0)
    else:
        if not K0.is_Exact:
            K0_inexact, K0 = K0, K0.get_exact()
            f = dmp_convert(f, u, K0_inexact, K0)
        else:
            K0_inexact = None

        if K0.has_Field:
            K = K0.get_ring()

            denom, f = dmp_ground_to_ring(f, u, K0, K)
            f = dmp_convert(f, u, K0, K)
        else:
            K = K0

        if K.is_ZZ:
            coeff, factors = _dmp_inner_factor(f, u, K)
        elif K.is_Poly:
            f, v = dmp_inject(f, u, K)

            coeff, factors = dmp_factor_list(f, v, K.dom, **args)

            for i, (f, k) in enumerate(factors):
                factors[i] = (dmp_eject(f, v, K), k)

            coeff = K.convert(coeff, K.dom)
        else: # pragma: no cover
            raise DomainError('factorization not supported over %s' % K0)

        if K0.has_Field:
            for i, (f, k) in enumerate(factors):
                factors[i] = (dmp_convert(f, u, K, K0), k)

            coeff = K0.convert(coeff, K)
            denom = K0.convert(denom, K)

            coeff = K0.quo(coeff, denom)

        if K0_inexact is not None:
            for i, (f, k) in enumerate(factors):
                factors[i] = (dmp_convert(f, u, K0, K0_inexact), k)

            coeff = K0_inexact.convert(coeff, K0)

    if not args.get('include', False):
        return coeff, factors
    else:
        if not factors:
            return [(dmp_ground(coeff, u), 1)]
        else:
            g = dmp_mul_ground(factors[0][0], coeff, u, K)
            return [(g, factors[0][1])] + factors[1:]

def dmp_factor_list(f, u, K0):
    """Factor polynomials into irreducibles in `K[X]`. """
    if not u:
        return dup_factor_list(f, K0)

    J, f = dmp_terms_gcd(f, u, K0)

    if not K0.has_CharacteristicZero: # pragma: no cover
        coeff, factors = dmp_gf_factor(f, u, K0)
    elif K0.is_Algebraic:
        coeff, factors = dmp_ext_factor(f, u, K0)
    else:
        if not K0.is_Exact:
            K0_inexact, K0 = K0, K0.get_exact()
            f = dmp_convert(f, u, K0_inexact, K0)
        else:
            K0_inexact = None

        if K0.has_Field:
            K = K0.get_ring()

            denom, f = dmp_clear_denoms(f, u, K0, K)
            f = dmp_convert(f, u, K0, K)
        else:
            K = K0

        if K.is_ZZ:
            levels, f, v = dmp_exclude(f, u, K)
            coeff, factors = dmp_zz_factor(f, v, K)

            for i, (f, k) in enumerate(factors):
                factors[i] = (dmp_include(f, levels, v, K), k)
        elif K.is_Poly:
            f, v = dmp_inject(f, u, K)

            coeff, factors = dmp_factor_list(f, v, K.dom)

            for i, (f, k) in enumerate(factors):
                factors[i] = (dmp_eject(f, v, K), k)

            coeff = K.convert(coeff, K.dom)
        else: # pragma: no cover
            raise DomainError('factorization not supported over %s' % K0)

        if K0.has_Field:
            for i, (f, k) in enumerate(factors):
                factors[i] = (dmp_convert(f, u, K, K0), k)

            coeff = K0.convert(coeff, K)
            denom = K0.convert(denom, K)

            coeff = K0.quo(coeff, denom)

        if K0_inexact is not None:
            for i, (f, k) in enumerate(factors):
                factors[i] = (dmp_convert(f, u, K0, K0_inexact), k)

            coeff = K0_inexact.convert(coeff, K0)

    for i, j in enumerate(reversed(J)):
        if not j:
            continue

        term = {(0,)*(u-i) + (1,) + (0,)*i: K0.one}
        factors.insert(0, (dmp_from_dict(term, u, K0), j))

    return coeff, _sort_factors(factors)