Formalized Metatheory with Terms Represented by an Indexed Family of Types

Abstract

It is possible to represent the terms of a syntax with binding constructors by a family of types, indexed by the free variables that may occur. This approach has been used several times for the study of syntax and substitution, but never for the formalization of the metatheory of a typing system. We describe a recent formalization of the metatheory of Pure Type Systems in Coq as an example of such a formalization. In general, careful thought is required as to how each definition and theorem should be stated, usually in an unfamiliar 'big-step' form; but, once the correct form has been found, the proofs are very elegant and direct.

Strongly Typed Term Representations in Coq

“…Some researchers have used syntax definitions that are well-scoped (i.e. whose types express an upper bound on their free de Bruijn indices) by construction, but are still actually typed by an extrinsic typing relation [1,7,17]. This seems particularly natural if one is trying to formalize type theory within type theory, as it is common to treat dependency by working with 'pre-terms' in the first instance.…”

“…In the functional programming community, 'strongly typed' encodings are a popular motivating example for generalized algebraic datatypes (GADTs) [10,24,27], and similar techniques can also be applied in modern object oriented languages [19]. In the type theory and automated reasoning community, such 'internal' representations have been described by a number of authors [1,2,8,12,13,16,23]. So the idea is well-known, and it is intuitively clear that dependently-typed calculi such as CiC have the power to express intrinsic encodings of, for example, simply typed languages.…”

“…This paper is a tutorial account of one way of working with intrinsic encodings in Coq, intended to show programming language metatheorists that this really is a viable option, rather than at telling hard-core type-theorists something new. We make no great claims of originality: most of the basic ideas are drawn from papers by Goguen and McKinna [15], Altenkirch and Reus [2] and Adams [1], and a note on Fig. 1 The simply-typed lambda calculus with naturals representing simply typed terms and substitutions in Epigram by the fourth author [21].…”

Completeness and Decidability of de Bruijn Substitution Algebra in Coq

“…One then defines composition based on instantiation (corresponding to step (5)). Such a two-level definition of instantiation is used by Adams [2]. Notationally, we give instantiation higher precedence than composition, and composition higher precedence than cons.…”

Implementing a normalizer using sized heterogeneous types

“…Adams [7] has carried out metatheory of pure type systems using de Bruijn representations as a heterogeneous type. He found that he had to pass from small-step statements, which speak about a single variable, e.g., a single substitution, to big-step statements, that take all variables into account at the same time, like simultaneous substitution.…”