I recently read this blog, about segment tree beats, but it took me a lot of time to understand this. Thus I have written some blogs on my own stating my own understanding of the topic, so that it is closely linked to the original blog, yet so that its a bit easier to understand. I have explained some of the key concepts that I feel were only touched upon in the original blog but was actually important. However, whatever I have written is based on the original article.
This only includes the explanations for the general concept and the proofs of task 1 and task 2 only. However I hope this explains the proof method itself in a better fashion.
I have also tried to prove the first task using the method in task 2. However I dont yet know whether it is actually correct, and if someone points out the flaw then I would remove it from my post.
Also, any improvements can be suggested in the comments and I would try to update them as soon as possible.