University of Edinburgh Division of Informatics

We present the Isabelle/HOL formalisation of some key equa-tional properties of the untyped λ-calculus with one-sorted variable names. Existing machine formalisations of λ-calculus proofs typically rely on alternative representations and/or proof principles to facilitate mechanization and we briefly account for these works. Our own development remains faithful to the standard textbook presentation and the usual pen-and-paper proof practices; we reason purely inductively over the standard first-order syntax of the calculus, using only primitive proof principles of the syntax and the reduction relations under consideration. We prove the confluence property of the λ-calculus at the raw syntactic level and derive confluence of the real λ-calculus (the structural collapse onto equivalence classes of the raw calculus) via a general result about abstract rewrite systems which we also formalise. We then show a technical property of the residual theory of the calculus which suggests the general applicability of the method to other equational properties of the calculus. Finally, we make some proof-technical observations pertaining to the extent to which our Isabelle proofs may be automated.