YES(?,PRIMREC) We are left with following problem, upon which TcT provides the certificate YES(?,PRIMREC). Strict Trs: { first(X1, X2) -> n__first(X1, X2) , first(0(), X) -> nil() , first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z))) , activate(X) -> X , activate(n__first(X1, X2)) -> first(X1, X2) , activate(n__from(X)) -> from(X) , from(X) -> cons(X, n__from(s(X))) , from(X) -> n__from(X) } Obligation: innermost runtime complexity Answer: YES(?,PRIMREC) The input was oriented with the instance of'multiset path order' as induced by the precedence first > s, first > cons, first > n__first, first > from, first > n__from, 0 > first, 0 > s, 0 > cons, 0 > n__first, 0 > activate, 0 > from, 0 > n__from, nil > first, nil > s, nil > cons, nil > n__first, nil > activate, nil > from, nil > n__from, s > cons, n__first > s, n__first > cons, n__first > from, n__first > n__from, activate > s, activate > cons, activate > n__first, activate > from, activate > n__from, from > s, from > cons, from > n__from, n__from > cons, first ~ activate, 0 ~ nil, s ~ n__from . Hurray, we answered YES(?,PRIMREC)