human
What is a the Python code you requested given the following:
Complete the below
def TR10i(rv):

    ''''Sum of products to function of sum.
    Examples
    >>> from sympy.simplify.fu import TR10i
    >>> from sympy import cos, sin, pi, Add, Mul, sqrt, Symbol
    >>> from sympy.abc import x, y
    >>> TR10i(cos(1)*cos(3) + sin(1)*sin(3))
    cos(2)
    >>> TR10i(cos(1)*sin(3) + sin(1)*cos(3) + cos(3))
    cos(3) + sin(4)
    >>> TR10i(sqrt(2)*cos(x)*x + sqrt(6)*sin(x)*x)
    2*sqrt(2)*x*sin(x + pi/6)'
    '''
assistant
#Here is the Python code you requested.
    global _ROOT2, _ROOT3, _invROOT3
    if (_ROOT2 is None):
        _roots()
    def f(rv):
        if (not rv.is_Add):
            return rv
        def do(rv, first=True):
            if (not rv.is_Add):
                return rv
            args = list(ordered(rv.args))
            if (len(args) != 2):
                hit = False
                for i in range(len(args)):
                    ai = args[i]
                    if (ai is None):
                        continue
                    for j in range((i + 1), len(args)):
                        aj = args[j]
                        if (aj is None):
                            continue
                        was = (ai + aj)
                        new = do(was)
                        if (new != was):
                            args[i] = new
                            args[j] = None
                            hit = True
                            break
                if hit:
                    rv = Add(*[_f for _f in args if _f])
                    if rv.is_Add:
                        rv = do(rv)
                return rv
            split = trig_split(two=True, *args)
            if (not split):
                return rv
            (gcd, n1, n2, a, b, same) = split
            if same:
                gcd = (n1 * gcd)
                if (n1 == n2):
                    return (gcd * cos((a - b)))
                return (gcd * cos((a + b)))
            else:
                gcd = (n1 * gcd)
                if (n1 == n2):
                    return (gcd * sin((a + b)))
                return (gcd * sin((b - a)))
        rv = process_common_addends(rv, do, (lambda x: tuple(ordered(x.free_symbols))))
        while rv.is_Add:
            byrad = defaultdict(list)
            for a in rv.args:
                hit = 0
                if a.is_Mul:
                    for ai in a.args:
                        if (ai.is_Pow and (ai.exp is S.Half) and ai.base.is_Integer):
                            byrad[ai].append(a)
                            hit...