Strong Normalization for System F by HOAS on Top of FOAS

Abstract

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. [49] and has ever since been extensively developed in frameworks with a wide variety of features and flavors. We distinguish two main (overlapping) directions in these developments. Keywords--(I) First, the employment of a chosen meta logic as a pure logical framework, used for defining object systems for the purpose of reasoning inside those systems. A standard example is higher-order logic (HOL) as the meta logic and firstorder logic (FOL) as the object system. Thanks to affinities between the mechanisms of these two logics, one obtains an adequate encoding of FOL in HOL by merely declaring in HOL types and constants and stating the FOL axioms and rules as HOL axioms -then the mechanisms for building FOL deductions (including substitution, instantiation, etc.) are already present in the meta logic, HOL.-(II) Second, the employment of the meta-logic to reason about the represented object systems, i.e., to represent not only the object systems, but also (some of) their metatheory. (E.g., cut elimination is a property about Gentzenstyle FOL, not expressible in a standard HOAS-encoding of FOL into HOL.) While direction (I) has been quasi-saturated by the achievement of quasi-maximally convenient logical frameworks (such Edinburgh LF [31] and generic Isabelle [49]), this second direction undergoes these days a period of active research. We distinguish two main approaches here:

An Improved Implementation and Abstract Interface for Hybrid

“…al. have developed an approach motivated by a new proof of strong normalization for System F that takes advantage of HOAS techniques [21]. It is also definitional, implements full HOAS, and is implemented in Isabelle/HOL, though the details of the formalizations as well as the case studies carried out in each system are quite different.…”

Cardinals in Isabelle/HOL

“…In his Ph.D. thesis [17][18][19], Popescu has formalized a general theory of syntax with bindings, parameterized over a binding signature with possibly infinitary operation symbols. For handling infinitary syntax, cardinal support was crucially needed.…”

GMeta: A Generic Formal Metatheory Framework for First-Order Representations

“…It is well-known that in Coq it is not possible to use the usual HOAS encodings, although Despeyroux et al (1995) and Chlipala (2008) have shown how weaker variations of HOAS can be encoded in Coq. Popescu et al (2010) investigate how formalizations using HOAS can avoid standard problems by being encoded on top of first-order representations. Approaches like GMeta or LNgen are aimed at recovering many of the properties that one expects from a logical framework for free.…”