Coinductive Verification of Program Optimizations Using Similarity Relations

Formal verification methods have gained increased importance due to their ability to guarantee system correctness and improve reliability. Nevertheless, the question how proofs are to be formalized in theorem provers is far from being trivial, yet very important as one needs to spend much more time on verification if the formalization was not cleverly chosen. In this paper, we develop and compare two different possibilities to express coinductive proofs in the theorem prover Isabelle/HOL. Coinduction is a proof method that allows for the verification of properties of also non-terminating state-transition systems. Since coinduction is not as widely used as other proof techniques as e.g. induction, there are much fewer “recipes” available how to formalize corresponding proofs and there are also fewer proof strategies implemented in theorem provers for coinduction. In this paper, we investigate formalizations for coinductive proofs of properties on state transition sequences. In particular, we compare two different possibilities for their formalization and show their equivalence. The first of these two formalizations captures the mathematical intuition, while the second can be used more easily in a theorem prover. We have formally verified the equivalence of these criteria in Isabelle/HOL, thus establishing a coalgebraic verification framework. To demonstrate that our verification framework is suitable for the verification of compiler optimizations, we have introduced three different, rather simple transformations that capture typical problems in the verification of optimizing compilers, even for non-terminating source programs.