Create a dp array where dp[i] is the maximum number of books you can take if you can only take books from bookshelves 0 to i and you must take books from bookshelf i.

Keep taking as many books as you can (i.e. starting from bookshelf i and going backwards, you take arr[i], arr[i] - 1, arr[i] - 2, … books).

You may reach an index j where arr[j] < arr[i] - (i - j). Have we already found the maximum number of books you can take from bookshelves 0 to j? How do we quickly find such an index j?

Keep a stack of possible indices for j. If x is the number at the top of the stack, keep popping from the stack while arr[x] ≥ arr[i] - (i - x). This is because if the inequality mentioned before is true, x will never be an index j as index i will run out of items first.