Formalising Bitonic Sort using Dependent Types

Rapport, 2004

We present a complete formalisation of bitonic sort and its correctness proof in constructive type theory. Bitonic sort is one of the fastest sorting algorithms where the sequence of comparisons is not data-dependent. In addition, it is a general recursive algorithm that works on sequences of length 2^n. In the formalisation we face two main problems: only structural recursion is allowed in type theory, and a formal proof of the correctness of the algorithm needs to consider quite a number of cases. We define the bitonic sort algorithm over dependently-typed binary trees with information in the leaves. In proving that the algorithm sorts its input we make use of the 0-1-principle. To support the use of that principle we also prove a parametricity theorem derived from the type of our bitonic sort from which the 0-1-principle can be proved.