human
I wrote the below code

    order = ring.order
    domain = ring.domain
    monomial_mul = ring.monomial_mul
    monomial_div = ring.monomial_div
    monomial_lcm = ring.monomial_lcm
    def select(P):
        pr = min(P, key=(lambda pair: order(monomial_lcm(f[pair[0]].LM, f[pair[1]].LM))))
        return pr
    def normal(g, J):
        h = g.rem([f[j] for j in J])
        if (not h):
            return None
        else:
            h = h.monic()
            if (not (h in I)):
                I[h] = len(f)
                f.append(h)
            return (h.LM, I[h])
    def update(G, B, ih):
        h = f[ih]
        mh = h.LM
        C = G.copy()
        D = set()
        while C:
            ig = C.pop()
            g = f[ig]
            mg = g.LM
            LCMhg = monomial_lcm(mh, mg)
            def lcm_divides(ip):
                m = monomial_lcm(mh, f[ip].LM)
                return monomial_div(LCMhg, m)
            if ((monomial_mul(mh, mg) == LCMhg) or ((not any((lcm_divides(ipx) for ipx in C))) and (not any((lcm_divides(pr[1]) for pr in D))))):
                D.add((ih, ig))
        E = set()
        while D:
            (ih, ig) = D.pop()
            mg = f[ig].LM
            LCMhg = monomial_lcm(mh, mg)
            if (not (monomial_mul(mh, mg) == LCMhg)):
                E.add((ih, ig))
        B_new = set()
        while B:
            (ig1, ig2) = B.pop()
            mg1 = f[ig1].LM
            mg2 = f[ig2].LM
            LCM12 = monomial_lcm(mg1, mg2)
            if ((not monomial_div(LCM12, mh)) or (monomial_lcm(mg1, mh) == LCM12) or (monomial_lcm(mg2, mh) == LCM12)):
                B_new.add((ig1, ig2))
        B_new |= E
        G_new = set()
        while G:
            ig = G.pop()
            mg = f[ig].LM
            if (not monomial_div(mg, mh)):
                G_new.add(ig)
        G_new.add(ih)
        return (G_new, B_new)
    if (not f):
        return []
    f1 = f[:]
    while True:
        f = f1[:]
        f1 = []
        for i in range(len(f)):
            p = f[i]
            r = p.rem(f[:i])
...

What's a good function header?

=====
Here's a good function header?
assistant
#Here's a good function header.

def _buchberger(f, ring):