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

Strict Trs:
  { f(X) -> n__f(X)
  , f(f(a())) -> f(g(n__f(a())))
  , activate(X) -> X
  , activate(n__f(X)) -> f(X) }
Obligation:
  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_0(3) -> 1
, f_0(4) -> 1
, f_1(2) -> 5
, f_1(3) -> 5
, f_1(4) -> 5
, a_0() -> 2
, a_0() -> 5
, g_0(2) -> 3
, g_0(2) -> 5
, g_0(3) -> 3
, g_0(3) -> 5
, g_0(4) -> 3
, g_0(4) -> 5
, n__f_0(2) -> 4
, n__f_0(2) -> 5
, n__f_0(3) -> 4
, n__f_0(3) -> 5
, n__f_0(4) -> 4
, n__f_0(4) -> 5
, n__f_1(2) -> 1
, n__f_1(3) -> 1
, n__f_1(4) -> 1
, n__f_2(2) -> 5
, n__f_2(3) -> 5
, n__f_2(4) -> 5
, activate_0(2) -> 5
, activate_0(3) -> 5
, activate_0(4) -> 5 }

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