Last edit: I used these Matlab scripts to test the circulation condition for every set in the power set of $V$.
I am trying to understand Hoffman's circulation theorem. For this, I have read different sources, and what I interpreted from them is that the theorem states that if there is a directed graph $D=(V,A)$, and there are functions $l,u:A\rightarrow \mathbb{R_{\geq 0}}$, such that $l(e)\leq u(e)$, there exists a circulation/function $\phi: A\rightarrow \mathbb{R}$, $l(e) \leq \phi(e)\leq u(e)$ such that:
$\sum_{e\in \delta^+(X)}\phi(e) = \sum_{e\in \delta^-(X)}\phi(e)$ for every $X \subseteq V$
where $\delta^-(X)$ is the set of arcs going into the nodes of $X$, and $\delta^+(X)$ is the set of arcs going out of the nodes of $X$.
if and only if
$\sum_{e\in \delta^+(X)}u(e) \geq \sum_{e\in \delta^-(X)}l(e)$, for every $X\subseteq V$
I defined a graph with 8 nodes and 12 arcs, with the value of the functions $l,u$ for every arc in the graph. In the image you can identify these values as follows $[l(e), u(e)]$. I checked that the circulation condition is met for the power set of $V$, but a circulation does not exists, since the only way to have $\phi((1,4)) + \phi((2,4)) = \phi((4,6) + \phi((4,7))$ is by setting $\phi((4,6)) = 0.1$, and this makes $\phi((3,6)) + \phi((4,6)) \neq \phi((6,8))$.
I guess I am missing something and may not be fully understanding the theorem, which makes me arrive to this conclusion. Can you help me to determine if the graph in the image meets the circulation condition or not?
I apologize in advance for any omission or mistake.