Locally Nameless Sigma Calculus

We present a Theory of Objects based on the original functional ςcalculus by Abadi and Cardelli [1] but with an additional parameter to methods. We prove confluence of the operational semantics following the outline of Nipkow’s proof of confluence for the λ-calculus reusing his general Commutation.thy [4] a generic diamond lemma reduction. We furthermore formalize a simple type system for our ς-calculus including a proof of type safety. The entire development uses the concept of Locally Nameless representation for binders [2]. We reuse an earlier proof of confluence [3] for a simpler ς-calculus based on de Bruijn indices and lists to represent objects.