Least Upper Bounds on the Size of Confluence and Church-Rosser Diagrams in Term Rewriting and λ-Calculus1

We study confluence and the Church-Rosser property in term rewriting and λ-calculus with explicit bounds on term sizes and reduction lengths. Given a system R, we are interested in the lengths of the reductions in the smallest valleys t→∗ s′ ∗← t′ expressed as a function: — for confluence a function vsR(m,n) where the valleys are for peaks t ∗← s→∗ t′ with s of size at most m and the reductions of maximum length n, and — for the Church-Rosser property a function cvsR(m,n) where the valleys are for conversions t↔∗ t′ with t and t′ of size at most m and the conversion of maximum length n.