A unified treatment of structural definitions on syntax for capture-avoiding substitution, context application, named substitution, partial differentiation, and so on

Abstract

We introduce a category-theoretic abstraction of a syntax with auxiliary functions, called an admissible monad morphism. Relying on an abstract form of structural recursion, we then design generic tools to construct admissible monad morphisms from basic data. These tools automate ubiquitous standard patterns like (1) defining auxiliary functions in successive, potentially dependent layers, and (2) proving properties of auxiliary functions by induction on syntax. We cover significant examples from the literature, including the standard lambda-calculus with capture-avoiding substitution, a lambda-calculus with binding evaluation contexts, the lambda-mu-calculus with named substitution, and the differential lambda-calculus.