Higher-order Abstract Syntax in Type Theory

We develop a general tool to formalize and reason about languages expressed using higher-order abstract syntax in a proof-tool based on type theory (Coq). A language is specified by its signature, which consists of sets of sort and operation names and typing rules. These rules prescribe the sorts and bindings of each operation. An algebra of terms is associated to a signature, using de Bruijn notation. Then a higher-order notation is built on top of the de Bruijn level, so that the user can work with meta-variables instead of de Bruijn indices. We also provide recursion and induction principles formulated directly on the higher-order syntax. This generalizes work on the Hybrid approach to higher-order syntax in Isabelle and our earlier work on a constructive extension to Hybrid formalized in Coq. In particular, a large class of theorems that must be repeated for each object language in Hybrid is done once in the present work and can be applied directly to each object language. §