The Transitive Composability of Relation Transition Systems

Relation Transition Systems (RTSs) have recently been proposed as a foundation for reasoning effectively about program equivalence in higher-order imperative languages like ML. RTSs fruitfully synthesize the coinductive style of bisimulation-based methods with the treatment of local state in recent work on step-indexed Kripke logical relations (SKLRs). Like SKLRs, RTSs are designed to have the potential to scale to inter-language reasoning; but unlike SKLRs, RTS proofs are also transitively composable, which is of critical importance for applications such as multi-stage verified compilation. In a POPL’12 paper [6], we presented the first RTS model for an ML-like core language, F, supporting higher-order functions, recursive types, abstract types, and general mutable references, and we proved soundness of the model w.r.t. contextual equivalence. In addition, we briefly sketched the proof that RTSs are transitively composable, but our proof only covered a restricted fragment of the language/model omitting abstract types and mutable state. Here, we present the transitivity proof for the full RTS model of the full F language. The proof is highly intricate, requiring a number of technical innovations. We have mechanized all our results in Coq.