Combining de Bruijn Indices and Higher-Order Abstract Syntax in Coq

Abstract

Abstract. The use of higher-order abstract syntax is an important approach for the representation of binding constructs in encodings of languages and logics in a logical framework. Formal meta-reasoning about such object languages is a particular challenge. We present a mechanism for such reasoning, formalized in Coq, inspired by the Hybrid tool in Isabelle. At the base level, we define a de Bruijn representation of terms with basic operations and a reasoning framework. At a higher level, we can represent languages and reason about them using higher-order syntax. We take advantage of Coq's constructive logic by formulating many definitions as Coq programs. We illustrate the method on two examples: the untyped lambda calculus and quantified propositional logic. For each language, we can define recursion and induction principles that work directly on the higher-order syntax.

Focusing on Binding and Computation

“…For example, the Hybrid approach [Ambler et al, 2002, Capretta and Felty, 2007, Momigliano et al, 2007, which has been implemented in Isabelle and Coq, uses an underlying de Bruijn implementation of binding. This circumvents the problems with using computational functions directly in an inductive definition.…”

Higher-order abstract syntax in type theory

“…We began by replicating their development step by step in Coq, but soon realized that the different underlying meta-theory (the Calculus of Inductive Constructions [10,38], as opposed to higher-order logic) provided us with different tools and led us to diverge from a simple translation of their work. The final result [8] exploits the computational content of the Coq logic: we prove a recursion principle that can be used to program functions on the object language. These functions can be directly computed inside Coq.…”

“…• Definition of a type of signatures for Universal Algebra with binding operations: The general shape of formalizations of various case studies in [2] and [8] was informally explained, but the low level work had to be repeated for each new formal system. Our higher-order signatures generalize Plotkin's binding signatures [29,15].…”

“…• Generation of typed terms: In [2] and [8] terms were generated in an untyped way, any term could be applied to any other term. The set of correct terms was defined by using a well-formedness predicate.…”

“…In our earlier work [8], we expressed induction and recursion principles more directly for each object language, but proving them was not just simple instantiation and instead required some proof effort. In that setting, we illustrated the approach with examples showing that reasoning about object languages was direct and simple.…”