Refinement with Time - Refining the Run-Time of Algorithms in Isabelle/HOL

Separation Logic with Time Credits is a well established method to formally verify the correctness and run-time of algorithms, which has been applied to various medium-sized use-cases. Refinement is a technique in program verification that makes software projects of larger scale manageable. Combining these two techniques for the first time, we present a methodology for verifying the functional correctness and the run-time analysis of algorithms in a modular way. We use it to verify Kruskal\u27s minimum spanning tree algorithm and the Edmonds - Karp algorithm for network flow. An adaptation of the Isabelle Refinement Framework [Lammich and Tuerk, 2012] enables us to specify the functional result and the run-time behaviour of abstract algorithms which can be refined to more concrete algorithms. From these, executable imperative code can be synthesized by an extension of the Sepref tool [Lammich, 2015], preserving correctness and the run-time bounds of the abstract algorithm