human
Complete the below
@timethis('combsimp')
def combsimp(expr):

    ''''Simplify combinatorial expressions.
    This function takes as input an expression containing factorials,
    binomials, Pochhammer symbol and other "combinatorial" functions,
    and tries to minimize the number of those functions and reduce
    the size of their arguments.
    The algorithm works by rewriting all combinatorial functions as
    expressions involving rising factorials (Pochhammer symbols) and
    applies recurrence relations and other transformations applicable
    to rising factorials, to reduce their arguments, possibly letting
    the resulting rising factorial to cancel. Rising factorials with
    the second argument being an integer are expanded into polynomial
    forms and finally all other rising factorial are rewritten in terms
    of more familiar functions. If the initial expression consisted of
    gamma functions alone, the result is expressed in terms of gamma
    functions. If the initial expression consists of gamma function
    with some other combinatorial, the result is expressed in terms of
    gamma functions.
    If the result is expressed using gamma functions, the following three
    additional steps are performed:
    1. Reduce the number of gammas by applying the reflection theorem
    gamma(x)*gamma(1-x) == pi/sin(pi*x).
    2. Reduce the number of gammas by applying the multiplication theorem
    gamma(x)*gamma(x+1/n)*...*gamma(x+(n-1)/n) == C*gamma(n*x).
    3. Reduce the number of prefactors by absorbing them into gammas, where
    possible.
    All transformation rules can be found (or was derived from) here:
    1. http://functions.wolfram.com/GammaBetaErf/Pochhammer/17/01/02/
    2. http://functions.wolfram.com/GammaBetaErf/Pochhammer/27/01/0005/
    Examples
    >>> from sympy.simplify import combsimp
    >>> from sympy import factorial, binomial
    >>> from sympy.abc import n, k
    >>> combsimp(factorial(n)/factorial(n - 3))
    n*(n - 2)*(n - 1)
    >>> combsimp(binomial(n+1, k+1)/binomial(n, k))
    (n + 1)/(k + 1)'
    '''

=====
Write me a the Python program you requested?
assistant
#Here is the Python program you requested.
    as_gamma = expr.has(gamma)
    as_factorial = expr.has(factorial)
    as_binomial = expr.has(binomial)
    expr = expr.replace(binomial, (lambda n, k: (_rf(((n - k) + 1).expand(), k.expand()) / _rf(1, k.expand()))))
    expr = expr.replace(factorial, (lambda n: _rf(1, n.expand())))
    expr = expr.rewrite(gamma)
    expr = expr.replace(gamma, (lambda n: _rf(1, (n - 1).expand())))
    if as_gamma:
        expr = expr.replace(_rf, (lambda a, b: (gamma((a + b)) / gamma(a))))
    else:
        expr = expr.replace(_rf, (lambda a, b: (binomial(((a + b) - 1), b) * gamma((b + 1)))))
    def rule(n, k):
        (coeff, rewrite) = (S.One, False)
        (cn, _n) = n.as_coeff_Add()
        if (_n and cn.is_Integer and cn):
            coeff *= (_rf((_n + 1), cn) / _rf(((_n - k) + 1), cn))
            rewrite = True
            n = _n
        if k.is_Add:
            (ck, _k) = k.as_coeff_Add()
            if (_k and ck.is_Integer and ck):
                coeff *= (_rf((((n - ck) - _k) + 1), ck) / _rf((_k + 1), ck))
                rewrite = True
                k = _k
        if rewrite:
            return (coeff * binomial(n, k))
    expr = expr.replace(binomial, rule)
    def rule_gamma(expr, level=0):
        '    Simplify    products    of    gamma    functions    further.    '
        if expr.is_Atom:
            return expr
        def gamma_rat(x):
            was = x.count(gamma)
            xx = x.replace(gamma, (lambda n: _rf(1, (n - 1).expand()).replace(_rf, (lambda a, b: (gamma((a + b)) / gamma(a))))))
            if (xx.count(gamma) < was):
                x = xx
            return x
        def gamma_factor(x):
            if (x.func is gamma):
                return True
            if (x.is_Add or x.is_Mul):
                return any((gamma_factor(xi) for xi in x.args))
            if (x.is_Pow and (x.exp.is_integer or x.base.is_positive)):
                return gamma_factor(x.base)
            return False
        if (level == 0):
            expr = expr.func(*[rule_gamma(x, (level + 1)) for x in...