How does the RCPSP's precedence constraint work?

Question

In [1] the authors define the RCPSP (resource-constrained project scheduling problem) as follows:

minimize $$ \sum_{t} t x_{n t} $$ subject to $$ \begin{array}{c} \sum_{t} x_{j t}=1, \quad j \in J, \\ \sum_{s=t}^{T} x_{i s}+\sum_{s=0}^{t+d_{i j}-1} x_{j s} \leq 1, \quad(i, j) \in L, t=0, \ldots, T, \\ x_{j t} \geq 0, \quad j \in J, t=0, \ldots, T, \\ x_{j t} \text { integer, } \quad j \in J, t=0, \ldots, T \\ \sum_{j} r_{j k}\left(\sum_{s=t-p_{j}+1}^{t} x_{j s}\right) \leq R_{k}, k \in R, t=0, \ldots, T . \end{array} $$

What I do not understand here is the second constraint, which is the precedence constraint. Why do we need the sum $\sum_{s=t}^{T} x_{i s} $ and cannot just use $x_{i t} $ instead? Would'nt $$ x_{i t}+\sum_{s=0}^{t+d_{i j}-1} x_{j s} \leq 1, \quad(i, j) \in L, t=0, \ldots, T \\ $$ be enough to be sure that if the predecessor starts at time t the successor j cannot start at any time before $t+d_{i j}$, when $d_{i j}$ is the enforced time lag between the predecessor i and succesor j (which equals the duration of the predecessor if we have strict finish-start precedence constraints)?

[1] Möhring, Rolf H.; Schulz, Andreas S.; Stork, Frederik; Uetz, Marc (2003): Solving Project Scheduling Problems by Minimum Cut Computations. In: Management Science 49 (3), S. 330–350. DOI: 10.1287/mnsc.49.3.330.12737.

Answer 1

Yes, your proposal suffices. But the published second constraint is stronger, yielding a tighter formulation. You can think of it as a lifting obtained by using the first constraint.