Types and verification for infinite state systems

Server-like or non-terminating programs are central to modern computing. It is a common requirement for these programs that they always be available to produce a behaviour. One method of showing such availability is by endowing a type-theory with constraints that demonstrate that a program will always produce some behaviour or halt. Such a constraint is often called productivity. We introduce a type theory which can be used to type-check a polymorphic functional programming language similar to a fragment of the Haskell programming language. This allows placing constraints on program terms such that they will not type-check unless they are productive. We show that using program transformation techniques, one can restructure some programs which are not provably productive in our type theory into programs which are manifestly productive. This allows greater programmer flexibility in the specification of such programs. We have demonstrated a mechanisation of some of these important results in the proof-assistant Coq. We have also written a program transformation system for this term-language in the programming language Haskell