def is_tolerance(t, L):
    # reflexive:
    if not all((e, e) in t for e in L):
        return False
    # symmetric:
    if not all((f, e) in t for e, f in t):
        return False
    # compatible:
    for (x, y), (z, u) in Subsets(t, 2):
        # join:
        if not (L.join(x, z), L.join(y, u)) in t:
            return False
        if not (L.meet(x, z), L.meet(y, u)) in t:
            return False
    return True

# extremely stupid code
def all_tolerances_iter(L):
    n = Subsets(Subsets(L, 2)).cardinality()
    if n > 2^15:
        raise ValueError("%s too large (%s)" % (L, n))
    for t0 in Subsets(Subsets(L, 2)):
        t = [tuple(x) for x in t0]
        t.extend((e, e) for e in L)
        t.extend((f, e) for e, f in t0)
        assert len(set(t)) == len(t)
        if is_tolerance(t, L):
            yield t

def all_tolerances(L):
    """
    sage: from sage.databases.findstat import _finite_lattices
    sage: [len(all_tolerances(L)) for L in _finite_lattices(6)]
    [2, 5, 13, 42, 13]
    """
    return list(all_tolerances_iter(L))

@cached_function
def statistic(L):
    return len(all_tolerances(L))