The minimum number of “insertions” to sort an array

Question

Suppose there is an unordered list. The only operation we can do is to move an element and insert it back to any place. How many moves does it take to sort the whole list?

Answer 1

It is easy to prove that size of the list - size of longest ordered sequence is enough always to sort any sequence, e.g. with mathematical induction.

You can also easily prove that for some unordered sequences, it is the best what you can do by simply finding such trivial sequence. E.g. to sort the sequence 3, 1, 2 you need one move of one item (3) and it's trivial to see that it cannot be made faster, and obviously size of the list - size of longest ordered sequence is equal to 1.

However, proving that it is the best for all sequences is impossible because that statement is not actually true: A counter example is a sequence with multiple sorted sub-sequences S[i], where max(S[i]) < min(S[i+1]) for every i. For example see the sequence 1, 2, 3, 1000, 4, 5, 6.