Consider the TRS R consisting of the rewrite rule 1: app(app(apply,f),x) -> app(f,x) There is one dependency pair: 2: APP(app(apply,f),x) -> APP(f,x) The approximated dependency graph contains one SCC: {2}. - Consider the SCC {2}. There are no usable rules. By taking the polynomial interpretation [apply] = 1 and [app](x,y) = [APP](x,y) = x + y + 1, rule 2 is strictly decreasing. Hence the TRS is terminating.