A Formalization of a Concurrent Object Calculus up to α-Conversion

Abstract

We experiment a method for representing a concurrent object calculus in the Calculus of Inductive Constructions. Terms are first defined in de Bruijn style, then names are re-introduced in binders. The terms of the calculus are formalized in the mechanized logic by suitable subsets of the de Bruijn terms; namely those whose de Bruijn indices are relayed beyond the scene. The α-equivalence relation is the Leibnitz equality and the substitution functions can be defined as sets of partial rewriting rules on these terms. We prove induction schemes for both the terms and some properties of the calculus which internalize the re-naming of bound variables. We show that despite the fact the terms which formalize the calculus are not generated by a last fixed point relation, we can prove the desired inversion lemmas. We formalize the computational part of the semantic and a simple type system of the calculus. Finally, we prove a subject reduction theorem and see that the specifications and proofs have the nice feature of not mixing de Bruijn technical manipulations with real proofs.