Reasoning Using Higher-Order Abstract Syntax in a Higher-Order Logic Proof Environment: Improvements to Hybrid and a Case Study

Abstract

We present a series of improvements to the Hybrid system, a formal theory implemented in Isabelle/HOL to support specifying and reasoning about formal systems using higher-order abstract syntax (HOAS). We modify Hybrid's type of terms, which is built definitionally in terms of de Bruijn indices, to exclude at the type level terms with `dangling' indices. We strengthen the injectivity property for Hybrid's variable-binding operator, and develop rules for compositional proof of its side condition, avoiding conversion from HOAS to de Bruijn indices. We prove representational adequacy of Hybrid (with these improvements) for a lambda-calculus-like subset of Isabelle/HOL syntax, at the level of set-theoretic semantics and without unfolding Hybrid's definition in terms of de Bruijn indices. In further work, we prove an induction principle that maintains some of the benefits of HOAS even for open terms. We also present a case study of the formalization in Hybrid of a small programming language, Mini-ML with mutable references, including its operational semantics and a type-safety property. This is the largest case study in Hybrid to date, and the first to formalize a language with mutable references. We compare four variants of this formalization based on the two-level approach adopted by Felty and Momigliano in other recent work on Hybrid, with various specification logics (SLs), including substructural logics, formalized in Isabelle/HOL and used in turn to encode judgments of the object language. We also compare these with a variant that does not use an intermediate SL layer. In the course of the case study, we explore and develop new proof techniques, particularly in connection with context invariants and induction on SL statements.