Syntax S. J. Ambler (S.Ambler@mcs.le.ac.uk) R. L. Crole (R.Crole@mcs.le.ac.uk) & A. Momigliano (A.Momigliano@mcs.le.ac.uk) Department of Mathematics and Computer Science, University of Leicester, Leicester, LE1 7RH, U.K.

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.

It’s blatantly clear You stupid machine, that what I tell you is true — Michael Norrish 1 This paper compares three models for formal reasoning about programming languages with binding. Higher order abstract syntax (hoas) uses meta-level binding to represent object-level binding [PE88]. Nominal Logic couples a concrete representation of bound variables with a formal apparatus for safely manipulating bound variables [Pit03]. The locally named binding representation places bound and free variables in different syntactic sorts [MP99]. This paper surveys each binding model, and compares it to the others and to Gordon and Melham’s axiomatization of the untyped lambda calculus [GM97]. Comparisons are made based on expressive power, transparency to human readers, and suitability for mechanized reasoning of each binding model. Each system excels in one area; hoas is most expressive, Nominal Logic most transparent, and locally named most mechanizable. 1

Abstract—We present a point of view concerning HOAS (Higher-Order Abstract Syntax) and an extensive exercise in HOAS along this point of view. The point of view is that HOAS can be soundly and fruitfully regarded as a definitional extension on top of FOAS (First-Order Abstract Syntax). As such, HOAS is not only an encoding technique, but also a higher-order view of a first-order reality. A rich collection of concepts and proof principles is developed inside the standard mathematical universe to give technical life to this point of view. The exercise consists of a new proof of Strong Normalization for System F. HOAS makes our proof considerably more direct than previous proofs. The concepts and results presented here have been formalized in the theorem prover Isabelle/HOL.