Formalisation in Constructive Type Theory of Barendregt's Variable Convention for Generic Structures with Binders

Abstract

We introduce a universe of regular datatypes with variable binding information, for which we define generic formation and elimination (i.e. induction /recursion) operators. We then define a generic alpha-equivalence relation over the types of the universe based on name-swapping, and derive iteration and induction principles which work modulo alpha-conversion capturing Barendregt's Variable Convention. We instantiate the resulting framework so as to obtain the Lambda Calculus and System F, for which we derive substitution operations and substitution lemmas for alpha-conversion and substitution composition. The whole work is carried out in Constructive Type Theory and machine-checked by the system Agda.