The proof of a program property can be reduced to the proof of satisfiability of a set of constrained Horn clauses (CHCs) which can be automatically generated from the pro- gram and the property. In this paper we have conducted a case study in Horn clause verification by considering several sorting programs with the aim of exploring the ef- fectiveness of a transformation technique which allows us to eliminate inductive data structures such as lists or trees. If this technique is successful, we derive a set of CHCs with constraints over the integers and booleans only, and the satisfiability check can often be performed in an effective way by using state-of-the-art CHC solvers, such as Eldarica or Z3. In this case study we have also illustrated the usefulness of a companion tech- nique based on the introduction of the so-called difference predicates, whose definitions correspond to lemmata required during the verification. We have considered functional programs which implement the following kinds of sorting algorithms acting on lists of integers: (i) linear recursive sorting algorithms, such as insertion sort and selection sort, and (ii) non-linear recursive sorting algorithms, such as quicksort and mergesort, and we have considered the following properties: (i) the partial correctness properties, that is, the orderedness of the output lists, and the equality of the input and output lists when viewed as multisets, and (ii) some arithmetic properties, such as the equality of the sum of the elements before and after sorting.