Given the following equation, how do I prove that $T(i)$ is a decreasing function? Given that $i, j, n, k, d_i, g_k, T(i)$ are all positive integers.
$$T(i) = d_i + \min_{i < j < n}(\max(T(j), \max_{i<k<j} ((\sum_{s=i+1}^{k} d_s) + g_{k-1} + g_{k})))$$
My current attempt is that since $\max(T(j), \max_{i<k<j} ((\sum_{s=i+1}^{k} d_s) + g_{k-1} + g_{k})) \ge T(j)$, then we can conclude that $T(i)$ is $d_i$ plus the minimal among values all greater than $T(j)$.
It's stated to be a decreasing function of $i$, but since I'm not the original writer of this formula so I don't know whether or not this is true. And unfortunately, the original writer only stated: by direct observation we know that it is a decreasing function...