Formalizing the Meta-Theory of Q0 in Rogue-Sigma-Pi

Introduced by Peter Andrews in the 1960’s, Q0 is a classical higher-order logic based on simply-typed lambda calculus. This paper presents our work in progress on the formalizing of Q0 in a programming language, Rogue-Sigma-Pi (RSP), with the aim of validating its meta-theory. The main challenge of this project arises from the fact that while all logical derivations are carried out in much detail in Andrews’ formalism, many of the syntactic derivations have been kept implicit. Therefore, most of our work has been devoted to setting up a framework that allows us to formalize low-level syntactic notions of Q0, such as variable occurrences, bindings and replacement. This formalization also includes proving meta-theoretic properties of these various syntactic notions. Building on the the ability to prove syntactic derivations assumed in Andrews’ formalism, recent progress has led to the proving of basic meta-theorems of Q0, such as the equality rules, alpha-equivalence, betaand eta-conversions, as well as capture-avoiding substitution. This paper will discuss the theoretical and engineering challenges faced in our formalizing of Q0 in RSP that is guided by a faithful adherence to Andrews’ presentation on paper.