[jojo3000]

But your "intermediate specifications" are not part of your specification! The specification says that the result is a sorted version of the input. It does not care about the induction hypothesis in your program. It might be technically necessary depending on the verification condition generator you are using. When you verify a functional program using Isabelle, Lean, or Coq your statement is that the output is a sorted version of the input.

[xgk]

Yes. Mostly.

But in practise, you need to specify what sortedness means, and in a generic sorting algorithm that means axiomatising the comparison predicate, including its side-effects. Note for example the C standard library implementation of Quicksort does take an arbitrary predicate, which may not be transitive, may have arbitrary side effects etc. The behaviour of sorting becomes pretty complicated in this scenario.