Abstract. Combining higher-order abstract syntax and (co)-induction in a logical framework is well known to be problematic. Previous work [ACM02] described the implementation of a tool called Hybrid, within Isabelle HOL, syntax, and reasoned about using tactical theorem proving and principles of (co)induction. Moreover, it is definitional, which guarantees consistency within a classical type theory. The idea is to have a de Bruijn representation of syntax, while offering tools for reasoning about them at the higher level. In this paper we describe how to use it in a multi-level reasoning fashion, similar in spirit to other meta-logics such as Linc and Twelf. By explicitly referencing provability in a middle layer called a specification logic, we solve the problem of reasoning by (co)induction in the presence of non-stratifiable hypothetical judgments, which allow very elegant and succinct specifications of object logic inference rules. We first demonstrate the method on a simple example, formally proving type soundness (subject reduction) for a fragment of a pure functional language, using a minimal intuitionistic logic as the specification logic. We then prove an analogous result for a continuation-machine presentation of the operational semantics of the same language, encoded this time in an ordered linear logic that serves as the specification layer. This example demonstrates the ease with which we can incorporate new specification logics, and also illustrates a significantly

Abstract. Theorem provers can be used to reason formally about programming languages and there are various general methods for the formalization of variable binding operators. Hence there are choices for the style of formalization of such languages, even within a single theorem prover. The choice of formalization can affect how easy or difficult it is to do automated reasoning. The aim of this paper is to compare and contrast three formalizations (termed de Bruijn, weak HOAS and full HOAS) of a typical functional programming language. Our contribution is a detailed report on our formalizations, a survey of related work, and a final comparative summary, in which we mention a novel approach to a hybrid de Bruijn/HOAS syntax. 1

The Hybrid system (Ambler et al., 2002b), implemented within Isabelle/HOL, allows object logics to be represented using higher order abstract syntax (HOAS), and reasoned about using tactical theorem proving in general and principles of (co)induction in particular. The form of HOAS provided by Hybrid is essentially a lambda calculus with constants. Of fundamental interest is the form of the lambda abstractions provided by Hybrid. The user has the convenience of writing lambda abstractions using names for the binding variables. However each abstraction is actually a definition of a de Bruijn expression, and Hybrid can unwind the user’s abstractions (written with names) to machine friendly de Bruijn expressions (without names). In this sense the formal system contains a hybrid of named and nameless bound variable notation. In this paper, we present a formal theory in a logical framework which can be viewed as a model of core Hybrid, and state and prove that the model is representationally adequate for HOAS. In particular, it is the canonical translation function from λ-expressions to Hybrid that witnesses adequacy. We also prove two results that characterise how Hybrid represents certain classes of λ-expressions. The Hybrid system contains a number of different syntactic classes of expression, and associated abstraction mechanisms. Hence this paper also aims to provide a self-contained theoretical introduction to both the syntax and key ideas of the system; background in automated theorem proving is not essential, although this paper will be of considerable interest to those who wish to work with Hybrid in Isabelle/HOL.