# How to model this chain of logical implication II

I would like to seek some advice on modeling the following (chain of) logical implication:

For instance $$\omega_{xz}$$ might indicate precedence, i.e., $$x$$, $$z$$ being the nodes $$x$$ and $$z$$, respectively. Thus if $$\omega_{xz}=1$$, it implies that the departure time of a vehicle from node $$x$$ is to be less than or equal to the arrival time at node $$y$$.

Say for instance, there is an additional node $$y_1$$, which must visited after $$x$$ temporally, and for other nodes that are also temporally behind $$y_1$$, such as $$y_1...y_n$$, i.e $$x$$ is the first node to be visited followed by $$y_1...y_n$$.

If $$z$$ is before $$x$$, $$z$$ must definitely be visited before $$y_1$$, followed by $$y_2$$ and so on

Hence, I would like to force the logical implication such that

$$\omega_{zx}\ge$$ $$\omega_{zy_1}$$ $$\implies$$ $$\omega_{zx}\ge$$ $$\omega_{zy_2}$$

$$\omega_{zx}\ge$$ $$\omega_{zy_1}$$ $$\implies$$ $$\omega_{zx}\ge$$ $$\omega_{zy_3}$$

...

$$\omega_{zx}\ge$$ $$\omega_{zy_1}$$ $$\implies$$ $$\omega_{zx}\ge$$ $$\omega_{zy_n}$$

on the other hand the converse might also be true such If $$y_n$$ is before $$z$$, $$y_1$$ be visited before $$z$$. As the chain of nodes could either be on the extreme left or extreme right of $$z$$.

$$\omega_{y_nz}\le$$ $$\omega_{y_{n-1}z}$$ $$\implies$$ $$\omega_{y_nz}\le$$ $$\omega_{y_{n-2}z}$$

$$\omega_{y_nz}\le$$ $$\omega_{y_{n-1}z}$$ $$\implies$$ $$\omega_{y_nz}\le$$ $$\omega_{y_{n-3}z}$$

...

$$\omega_{y_nz}\le$$ $$\omega_{y_{n-1}z}$$ $$\implies$$ $$\omega_{y_nz}\le$$ $$\omega_{xz}$$

You can use conjunctive normal form to derive the desired constraints. The first one is: $$a \ge b \implies a\ge c\\ (b \implies a) \implies (c \implies a)\\ \lnot(\lnot b \lor a) \lor (\lnot c \lor a)\\ (b \land \lnot a) \lor (\lnot c \lor a)\\ (b \lor \lnot c \lor a) \land (\lnot a \lor \lnot c \lor a)\\ (b \lor \lnot c \lor a)\\ b+ 1- c +a \ge 1\\ a+b \ge c$$
The second one is: $$a \le b \implies a\le c\\ (a \implies b) \implies (a \implies c)\\ \lnot(\lnot a \lor b) \lor (\lnot a \lor c)\\ (a \land \lnot b) \lor (\lnot a \lor c)\\ (a \lor \lnot a \lor c) \land (\lnot b \lor \lnot a \lor c)\\ \lnot b \lor \lnot a \lor c\\ 1-b + 1-a + c \ge 1\\ a+b \le c+1$$