Machine-checked Proof of the Church-Rosser Theorem for the Lambda Calculus Using the Barendregt Variable Convention in Constructive Type Theory

In this article we continue the work started in [Ernesto Copello, Alvaro Tasistro, Nora Szasz, Ana Bove, and Maribel Fernandez. Alpha-structural induction and recursion for the λ-calculus in constructive type theory. Electronic Notes in Theoretical Computer Science, 323:109–124, 2016], deriving in Constructive Type Theory new induction principles for the λ-calculus, using (the historical) first order syntax with only one sort of names for both bound and free variables, and with α-conversion based upon name swapping. The principles provide a flexible framework for mimicking pen-and-paper proofs within the rigorous formal setting of a proof assistant. We here report on one successful application, namely a complete proof of the Church-Rosser Theorem. The whole development has been machine-checked using the system Agda [Ulf Norell. Towards a Practical Programming Language Based on Dependent Type Theory. PhD thesis, Department of Computer Science and Engineering, Chalmers University of Technology, September 2007].