Global Optimization Principle for Internet IP routing by Hop-Dijkstra algorithm proof, route-dijkstra

Global Optimization Principle for Internet IP routing by Hop-Dijkstra algorithm proof, route-dijkstra
I made up half of what I wrote over the weekend and achieved a perfect day.
One fact we know is that there are too many IP addresses and it is impossible to manage them in a unified manner, either on the data plane or on the control/management plane. Therefore, the IP protocol is designed to be scalable. For IP routing, route computing is skip-by-hop. Of course, the "Source Route" option is also supported. The source route means that the route has been planned before departure, hop-by-hop routing is a standard form of IP routing, that is, IP packets plan routes on the road in real time.
I like IP routing because it is also my way of traveling. I like traveling, but I don't like booking a hotel, planning a route, navigation, etc, my approach is to look at the road signs on the road. When I arrive at a temporary stop, I carry a bag to find a place to live, and then where to go, this is a kind of wandering with no destination... of course, IP data packets are destination.
The global optimization of IP routing on a hop-by-hop mode is performed on each vro, which leads to a problem where the Internet is larger, how can I believe that a complete path combined by so many hop-by-hop routes is indeed the most optimal? The answer is clearly definite, and the question is how to prove it.
In the routing algorithm book, the routing algorithms are basically divided into distance vector algorithms and link state algorithms. Their respective protocol representatives are RIP and OSPF (I am the first job I found by these two ), this is indeed the case, but from the proof of the correctness of these algorithms, you will find that "the hop-by-hop optimal route is actually the global optimal route ". In this article, I only provide proof of the Dijkstra Algorithm Based on the Link Status routing protocol, because the link status database of each device on the entire network is the same, so it is easy to understand.
To prove the correctness of the Dijkstra algorithm, we must first prove the correctness of the Dijkstra algorithm. After all, the Dijkstra algorithm only guides the step by step operation steps and does not prove that every path in the shortest path tree obtained by such a round is indeed the shortest. This fact is required to prove the skip-by-hop global optimization principle.
The following provides a simple proof of the correctness of the Dijkstra algorithm. For detailed and complete mathematical proof, refer to this idea:

The question of skip-by-Skip global optimization is as follows:

The following is a simple proof of global optimization for a hop-by-hop model. There are many ways to prove the global optimization. Here I only provide one of them:

Appendix: The greedy model of Dijkstra algorithm if we pour a cup of water on the ground and observe the traces of water penetration, we will understand Dijkstra algorithm, which is indeed self-evident. Nature is lazy and always acts in the most effort-saving way. water molecules start to land at that point, on uneven ground due to gravity (not considering other molecular forces for the time being) the path must be the shortest path. We can regard the rugged ground as the path weight. Isn't this exactly the same as the Dijkstra Algorithm Model?