YES(?,O(n^1))

We are left with following problem, upon which TcT provides the
certificate YES(?,O(n^1)).

Strict Trs:
  { f(X) -> n__f(X)
  , f(0()) -> cons(0(), n__f(s(0())))
  , f(s(0())) -> f(p(s(0())))
  , p(s(X)) -> X
  , activate(X) -> X
  , activate(n__f(X)) -> f(X) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^1))

The problem is match-bounded by 2. The enriched problem is
compatible with the following automaton.
{ f_0(2) -> 1
, f_1(2) -> 1
, f_1(7) -> 1
, 0_0() -> 1
, 0_0() -> 2
, 0_1() -> 3
, 0_1() -> 6
, 0_1() -> 7
, 0_2() -> 8
, 0_2() -> 11
, cons_0(2, 2) -> 1
, cons_0(2, 2) -> 2
, cons_1(3, 4) -> 1
, cons_2(8, 9) -> 1
, n__f_0(2) -> 1
, n__f_0(2) -> 2
, n__f_1(2) -> 1
, n__f_1(5) -> 4
, n__f_2(2) -> 1
, n__f_2(7) -> 1
, n__f_2(10) -> 9
, s_0(2) -> 1
, s_0(2) -> 2
, s_1(6) -> 5
, s_2(11) -> 10
, p_0(2) -> 1
, p_1(5) -> 7
, activate_0(2) -> 1
, 2 -> 1
, 6 -> 7 }

Hurray, we answered YES(?,O(n^1))