Realizability in Cyclic Proof: Extracting Ordering Information for Infinite Descent

In program veri_cation, measures for proving the termination of programs are typically constructed using (notions of size for) the data manipulated by the program. Such data are often described by means of logical formulas. For example, the cyclic proof technique makes use of semantic approximations of inductively de_ned predicates to construct Fermat-style in_nite descent arguments. However, logical formulas must often incorporate explicit size information (e.g. a list length parameter) in order to support inter-procedural analysis. In this paper, we show that information relating the sizes of inductively de_ned data can be automatically extracted from cyclic proofs of logical entailments.We characterise this information in terms of a graph-theoretic condition on proofs, and show that this condition can be encoded as a containment between weighted automata. We also show that under certain conditions this containment falls within known decidability results. Our results can be viewed as a form of realizability for cyclic proof theory.