The representational adequacy of Hybrid

Abstract

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, andHybrid 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-expression.We provide the first detailed proof to be published that proper locally nameless de Bruijn expressions and α-equivalence classes of λ-expressions are in bijective correspondence. This result is presented as a form of de Bruijn representational adequacy, and is a key component of the proof of Hybrid adequacy.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. Although this paper will be of considerable interest to those who wish to work with Hybrid in Isabelle/HOL, a background in automated theorem proving is not essential.

A focused linear logical framework and its application to metatheory of object logics

“…Adequacy for inference rules is stated and proved in Section 5.2. Adequacy of syntax encoding, also called representational adequacy, is discussed for the lambda calculus as an OL in Hybrid in Ambler et al (2002) and proved in detail in Crole (2011), adequacy for a fragment of a functional programming language known as Mini-ML is proved in Felty and Momigliano (2012), and adequacy for a quantum programming language whose core is a linear lambda calculus is discussed in detail in Mahmoud and Felty (2019). The OLs described here are similar to these others in the sense that the quantifiers in this paper are represented in the same manner as the lambda operator of the lambda calculus in the cited papers, and thus such adequacy results can be adapted fairly directly.…”

Formalizing Adequacy: A Case Study for Higher-order Abstract Syntax

“…There are some treatments of correctness properties for other techniques besides LF, notably Crole's [7] development of adequacy for the Hybrid system, Norrish and Vestergaard's [17] formalization of isomorphisms among nominal and de Bruijn representations of the untyped λ-calculus, and Urban's isomorphism between raw and nominal representations of the λ-calculus [21,Section 3]. Cheney and Urban [3,Section 5] also discuss adequacy in the context of nominal logic programming.…”

“…However, none of this work has provided a clear, general definition of adequacy that is applicable to any representation technique; rather, particular instances of correspondences between object-languages and higher-order abstract syntax representations are typically called adequacy theorems. This has led to confusion between researchers familiar with different techniques, since properties such as "compositional bijection" that are called adequacy theorems in (for example) LF [10][11][12]18] are couched in terms of LF or higher-order abstract syntax and bear little superficial resemblance to correctness properties established for other techniques [7,8,17,20].…”