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)