Dijkstra's Algorithm Proof Explanation I read over this link from a previous question regarding Dijkstra's algorithm but I'm confused as to some of the conclusions drawn.
In the graphic within the question, there are a bunch of inequalities set up.
I'd like to explain how I currently see/understand it, but I know I'm misssing something.
$l(p)$ in the algorithm is the total length of the path $P$, which is some s-v path.
$l(p')$ is some sub-path of $P$ that  does not leave $P$;
$l(x, y)$ is the first edge that leaves $P$;
and $d(x)$ is the length of the shortest s-x path.
So I understand that that $l(p)$ would be the largest value since it's just SOME s-v path not necessarily the minimal s-v path (which would be $\pi(v)$). 
I understand that $l(p') \geq d(x)$ since $d(x)$ is the minimal $l(p')$, but what is the intuition behind $l(p') + l(x, y) \geq d(x) + l(x, y)$?
And furthermore, I don't understand the intuition behind $\pi(y) \geq \pi(v)$ in the example. 
If anybody could explain it, that would be great. Thank you.
 A: 
I understand that $l(p') \geq d(x)$ since $d(x)$ is the minimal $l(p')$, but what is the intuition behind $l(p') + l(x, y) \geq d(x) + l(x, y)$?

Well, this is just a basic algebra rule. If you agree that $a \geq b$, then it follows that $a + c \geq b + c$. Here's a reference if you need it.

And furthermore, I don't understand the intuition behind $\pi(y) \geq \pi(v)$ in the example. 

This is exactly what the question that you linked is asking, it seems. To paraphrase the answer, at this point, we are assuming that $v$ is the next vertex that the algorithm adds, which means precisely that $\pi(v) \leq \pi(y)$ for all other vertices $y$ that are not yet added. If this weren't true, then this wouldn't be the Djikstra algorithm.
To see why this isn't circular reasoning, it helps to step back and look at what the algorithm is doing, and what we're trying to prove about it. I don't know what reference you have for how the algorithm runs, but the one on Wikipedia is fine. That's how it runs, and we want to prove that this ultimately gives the shortest distance between the starting vertex and the destination. $\pi(y) \geq \pi(v)$ is a consequence of step 6 in Wikipedia's description of the algorithm. We picked $v$ because its distance is shorter than anything else at that point.
