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Complete mathematical proof for Leetcode Leetcode 2193…

Complete mathematical proof for Leetcode Leetcode 2193. Minimum Number of Moves to Make Palindrome

I always try to prove a solution before I code it. I took one night and wrote a paper on Leetcode 2193 https://leetcode.com/problems/minimum-number-of-moves-to-make-palindrome/description/ .

Overleaf paper link here https://www.overleaf.com/read/qmsfhdsgbvqq#6fc040

What do you guys think? AI says it can't really be proved easier if you want to be 100% rigorous and not make an intuition-based proof.

I might make edits to the paper if I re-read it and find I don't like something. Please tell me if you find mistakes!
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