YES(?,PRIMREC)

We are left with following problem, upon which TcT provides the
certificate YES(?,PRIMREC).

Strict Trs:
  { terms(N) -> cons(recip(sqr(N)), n__terms(s(N)))
  , terms(X) -> n__terms(X)
  , sqr(s(X)) -> s(add(sqr(X), dbl(X)))
  , sqr(0()) -> 0()
  , add(s(X), Y) -> s(add(X, Y))
  , add(0(), X) -> X
  , dbl(s(X)) -> s(s(dbl(X)))
  , dbl(0()) -> 0()
  , first(X1, X2) -> n__first(X1, X2)
  , first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
  , first(0(), X) -> nil()
  , activate(X) -> X
  , activate(n__terms(X)) -> terms(X)
  , activate(n__first(X1, X2)) -> first(X1, X2) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,PRIMREC)

The input was oriented with the instance of'multiset path order' as
induced by the precedence

 terms > cons, terms > recip, terms > sqr, terms > n__terms,
 terms > s, terms > add, terms > dbl, terms > n__first,
 recip > cons, recip > sqr, recip > s, recip > add, recip > dbl,
 sqr > cons, sqr > s, sqr > add, sqr > dbl, n__terms > cons,
 n__terms > sqr, n__terms > s, n__terms > add, n__terms > dbl,
 0 > terms, 0 > cons, 0 > recip, 0 > sqr, 0 > n__terms, 0 > s,
 0 > add, 0 > dbl, 0 > first, 0 > nil, 0 > n__first, 0 > activate,
 add > cons, add > s, dbl > cons, dbl > s, first > cons,
 first > recip, first > sqr, first > n__terms, first > s,
 first > add, first > dbl, first > n__first, nil > terms,
 nil > cons, nil > recip, nil > sqr, nil > n__terms, nil > s,
 nil > add, nil > dbl, nil > first, nil > n__first, nil > activate,
 n__first > cons, n__first > sqr, n__first > s, n__first > add,
 n__first > dbl, activate > cons, activate > recip, activate > sqr,
 activate > n__terms, activate > s, activate > add, activate > dbl,
 activate > n__first, terms ~ first, terms ~ activate,
 recip ~ n__terms, recip ~ n__first, n__terms ~ n__first, add ~ dbl,
 first ~ activate .

Hurray, we answered YES(?,PRIMREC)