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

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.KeywordsObject LanguageLogical FrameworkProof AssistantInduction PrincipleNegation Normal FormThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.