def test_dmp_degree_list():
    assert dmp_degree_list([[[[ ]]]], 3) == (-1,-1,-1,-1)
    assert dmp_degree_list([[[[1]]]], 3) == ( 0, 0, 0, 0)

    assert dmp_degree_list(f_0, 2) == (2, 2, 2)
    assert dmp_degree_list(f_1, 2) == (3, 3, 3)
    assert dmp_degree_list(f_2, 2) == (5, 3, 3)
    assert dmp_degree_list(f_3, 2) == (5, 4, 7)
    assert dmp_degree_list(f_4, 2) == (9, 12, 8)
    assert dmp_degree_list(f_5, 2) == (3, 3, 3)
    assert dmp_degree_list(f_6, 3) == (4, 4, 6, 3)

def dmp_zz_mignotte_bound(f, u, K):
    """Mignotte bound for multivariate polynomials in `K[X]`. """
    a = dmp_max_norm(f, u, K)
    b = abs(dmp_ground_LC(f, u, K))
    n = sum(dmp_degree_list(f, u))

    return K.sqrt(K(n+1))*2**n*a*b

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_zz_wang_hensel_lifting(f, H, LC, A, p, u, K):
    """Wang/EEZ: Parallel Hensel lifting algorithm. """
    S, n, v = [f], len(A), u-1

    H = list(H)

    for i, a in enumerate(reversed(A[1:])):
        s = dmp_eval_in(S[0], a, n-i, u-i, K)
        S.insert(0, dmp_ground_trunc(s, p, v-i, K))

    d = max(dmp_degree_list(f, u)[1:])

    for j, s, a in zip(xrange(2, n+2), S, A):
        G, w = list(H), j-1

        I, J = A[:j-2], A[j-1:]

        for i, (h, lc) in enumerate(zip(H, LC)):
            lc = dmp_ground_trunc(dmp_eval_tail(lc, J, v, K), p, w-1, K)
            H[i] = [lc] + dmp_raise(h[1:], 1, w-1, K)

        m = dmp_nest([K.one, -a], w, K)
        M = dmp_one(w, K)

        c = dmp_sub(s, dmp_expand(H, w, K), w, K)

        dj = dmp_degree_in(s, w, w)

        for k in xrange(0, dj):
            if dmp_zero_p(c, w):
                break

            M = dmp_mul(M, m, w, K)
            C = dmp_diff_eval_in(c, k+1, a, w, w, K)

            if not dmp_zero_p(C, w-1):
                C = dmp_quo_ground(C, K.factorial(k+1), w-1, K)
                T = dmp_zz_diophantine(G, C, I, d, p, w-1, K)

                for i, (h, t) in enumerate(zip(H, T)):
                    h = dmp_add_mul(h, dmp_raise(t, 1, w-1, K), M, w, K)
                    H[i] = dmp_ground_trunc(h, p, w, K)

                h = dmp_sub(s, dmp_expand(H, w, K), w, K)
                c = dmp_ground_trunc(h, p, w, K)

    if dmp_expand(H, u, K) != f:
        raise ExtraneousFactors # pragma: no cover
    else:
        return H

def dmp_zz_factor(f, u, K):
    """
    Factor (non square-free) polynomials in `Z[X]`.

    Given a multivariate polynomial `f` in `Z[x]` computes its complete
    factorization `f_1, ..., f_n` into irreducibles over integers::

                 f = content(f) f_1**k_1 ... f_n**k_n

    The factorization is computed by reducing the input polynomial
    into a primitive square-free polynomial and factoring it using
    Enhanced Extended Zassenhaus (EEZ) algorithm. Trial division
    is used to recover the multiplicities of factors.

    The result is returned as a tuple consisting of::

             (content(f), [(f_1, k_1), ..., (f_n, k_n))

    Consider polynomial `f = 2*(x**2 - y**2)`::

        >>> from sympy.polys.factortools import dmp_zz_factor
        >>> from sympy.polys.domains import ZZ

        >>> dmp_zz_factor([[2], [], [-2, 0, 0]], 1, ZZ)
        (2, [([[1], [-1, 0]], 1), ([[1], [1, 0]], 1)])

    In result we got the following factorization::

                    f = 2 (x - y) (x + y)

    **References**

    1. [Gathen99]_

    """
    if not u:
        return dup_zz_factor(f, K)

    if dmp_zero_p(f, u):
        return K.zero, []

    cont, g = dmp_ground_primitive(f, u, K)

    if dmp_ground_LC(g, u, K) < 0:
        cont, g = -cont, dmp_neg(g, u, K)

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

    G, g = dmp_primitive(g, u, K)

    factors = []

    if dmp_degree(g, u) > 0:
        g = dmp_sqf_part(g, u, K)
        H = dmp_zz_wang(g, u, K)

        for h in H:
            k = 0

            while True:
                q, r = dmp_div(f, h, u, K)

                if dmp_zero_p(r, u):
                    f, k = q, k+1
                else:
                    break

            factors.append((h, k))

    for g, k in dmp_zz_factor(G, u-1, K)[1]:
        factors.insert(0, ([g], k))

    return cont, _sort_factors(factors)

 def total_degree(f):
     """Returns the total degree of `f`. """
     return sum(dmp_degree_list(f.rep, f.lev))

 def degree_list(f):
     """Returns a list of degrees of `f`. """
     return dmp_degree_list(f.rep, f.lev)

 def is_ground(f):
     """Returns `True` if `f` is an element of the ground domain. """
     return all(d <= 0 for d in dmp_degree_list(f.rep, f.lev))

def dmp_zz_factor(f, u, K):
    """
    Factor (non square-free) polynomials in `Z[X]`.

    Given a multivariate polynomial `f` in `Z[x]` computes its complete
    factorization `f_1, ..., f_n` into irreducibles over integers::

                 f = content(f) f_1**k_1 ... f_n**k_n

    The factorization is computed by reducing the input polynomial
    into a primitive square-free polynomial and factoring it using
    Enhanced Extended Zassenhaus (EEZ) algorithm. Trial division
    is used to recover the multiplicities of factors.

    The result is returned as a tuple consisting of::

             (content(f), [(f_1, k_1), ..., (f_n, k_n))

    Consider polynomial `f = 2*(x**2 - y**2)`::

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

        >>> R.dmp_zz_factor(2*x**2 - 2*y**2)
        (2, [(x - y, 1), (x + y, 1)])

    In result we got the following factorization::

                    f = 2 (x - y) (x + y)

    References
    ==========

    1. [Gathen99]_

    """
    if not u:
        return dup_zz_factor(f, K)

    if dmp_zero_p(f, u):
        return K.zero, []

    cont, g = dmp_ground_primitive(f, u, K)

    if dmp_ground_LC(g, u, K) < 0:
        cont, g = -cont, dmp_neg(g, u, K)

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

    G, g = dmp_primitive(g, u, K)

    factors = []

    if dmp_degree(g, u) > 0:
        g = dmp_sqf_part(g, u, K)
        H = dmp_zz_wang(g, u, K)
        factors = dmp_trial_division(f, H, u, K)

    for g, k in dmp_zz_factor(G, u - 1, K)[1]:
        factors.insert(0, ([g], k))

    return cont, _sort_factors(factors)