The Mechanisation of Barendregt-Style Equational Proofs (the Residual Perspective)

We show how to mechanise equational proofs about higher-order languages by using the primitive proof principles of first-order abstract syntax over one-sorted variable names. We illustrate the method here by proving (in Isabelle/HOL) a technical property which makes the method widely applicable for the λ-calculus: the residual theory of β is renaming-free up-to an initiality condition akin to the so-called Barendregt Variable Convention. We use our results to give a new diagram-based proof of the development part of the strong finite development property for the λ-calculus. The proof has the same equational implications (e.g., confluence) as the proof of the full property but without the need to prove SN. We account for two other uses of the proof method, as presented elsewhere. One has been mechanised in full in Isabelle/HOL.