Given a vector $\mathbf{a} \in \mathbb{Z}^7_{> 0}$, I have the following inequalities:
$$\begin{array}{l} a_1 < a_2 < a_3 < a_4 < a_5 < a_6 < a_7\\ a_7 < a_1 + a_2\\ a_6 + a_7 < a_1 + a_2 + a_3\\ a_5 + a_6 + a_7 < a_1 + a_2 + a_3 + a_4 \end{array}$$
$[10, 11, 12, 13, 14, 15, 16]$ satisfies the constraints.
Someone said all $\mathbf{a}$ that satisfy the above can be found by adding non-negative weighted sums of the following row vectors to this vector:
$$ \left[\begin{array}{lllllll} 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 0 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 2 & 2 & 2 & 2 & 2\\ 2 & 2 & 2 & 3 & 3 & 3 & 3\\ 3 & 3 & 3 & 3 & 4 & 4 & 4\\ 2 & 2 & 2 & 2 & 2 & 3 & 3\\ 1 & 1 & 1 & 1 & 1 & 1 & 2 \end{array}\right] $$
Is this correct, and why is this? How were these row vectors obtained?