I'm pretty sure a large amount of people have at the very least heard of Kadane's algorithm. It is a way to find the maximum subarray sum (of an array). But why does it work? Besides the normal proof/explanation, is there another way to explain why it is correct?
Let's fix the right boundary of the subarray. From now on, it is called $$$r$$$. Now we want to find the optimal left boundary, $$$l$$$, for every $$$r$$$. For the answer we can just find the maximum of all fixed $$$r$$$. How could we do this? Well for a given $$$r$$$ let's assume we already found the $$$l$$$. How does the answer change for $$$r+1$$$? For $$$r+1$$$, all we did was just add $$$a_{r+1}$$$ to every subarray we had so far. For every integer value $$$a,b,c$$$ it holds true that
This means that the greatest subarray sum will still be greater than every subarray whose left boundary was contained in the
subarray, so we only need to check if the subarray sum $a_{{r+1}..{r+1}}$ (the element $$$a_{r+1}$$$) is greater than the sum of $$$a_{l..r+1}$$$. Because this is true for the $$$a_{1..2}$$$ subarray it holds true for every subarray, completing our proof by induction. Why does this prove Kadane's algorithm? Because when we consider the $$$a_{{r+1}..{r+1}}$$$ subarray, in the perspective of the $$$r$$$ before, we are considering the neutral element of addition, 0. Following the property stated beforehand, this means the answer doesn't change based on whether we evaluate $$$0 > a_{l..r}$$$ or $$$a_{r+1} > a_{l..r+1}$$$
Does this only work for subarray sums? It turns out that every function for which
holds true this works (the function takes array elements as parameters). We in turn get a very simple general Kadane's algorithm. A considerable number of practical functions have this property, but I don't know of much problems that can be solved by reduction to such functions. If you want to prove that the function has the property you can also use something like I used in the proof of Kadane's subarray sum algorithm.
Besides the proof by induction I presented here is a proof by AC (if you don't believe me):
CSES Kadane's: https://cses.fi/problemset/result/7030713/
Some other problems should also be solvable like this. 1872G - Replace With Product is a good example, as well as 1796D - Maximum Subarray