Encoding of Predicate Subtyping with Proof Irrelevance in the λΠ-Calculus Modulo Theory

Abstract

The λΠ-calculus modulo theory is a logical framework in which various logics and type systems can be encoded, thus helping the cross-verification and interoperability of proof systems based on those logics and type systems. In this paper, we show how to encode predicate subtyping and proof irrelevance, two important features of the PVS proof assistant. We prove that this encoding is correct and that encoded proofs can be mechanically checked by Dedukti, a type checker for the λΠ-calculus modulo theory using rewriting.

Adequate and computational encodings in the logical framework Dedukti

“…The logical framework Dedukti [3] has the syntax of the λ-calculus with dependent types [16] (λΠ-calculus). Like works such as [17], we consider here a version with arities, with the following syntax.…”

“…The logical framework Dedukti [3] addresses this point by extending the LF with rewriting rules, thus allowing for the representation of both deduction and computation. This framework was already proven to be as a very expressive system, and has been used to encode the logics of many proof assistants, such as Coq [13], Agda [14], PVS [17] and others.…”

“…They also both rely on intricate properties of the encoded systems, which is unnatural given that logical frameworks should ideally only require the encoded systems to satisfy some basic properties, and be agnostic with respect to more deep ones -for instance, one should not be obliged to show that a given system is consistent in order to encode it in a logical framework. This reason, coupled with the technicality of these proofs, may explain why recent works such as [22], [17] and [14] have left conservativity as conjecture. Moreover, none of these works have addressed the lack of an adequacy theorem, which until now has remained an overlooked problem in the Dedukti community.…”