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(n__f(n__a())) -> f(n__g(n__f(n__a())))
  , a() -> n__a()
  , g(X) -> n__g(X)
  , activate(X) -> X
  , activate(n__f(X)) -> f(X)
  , activate(n__a()) -> a()
  , activate(n__g(X)) -> g(activate(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_0(3) -> 1
, f_0(4) -> 1
, f_1(2) -> 7
, f_1(2) -> 11
, f_1(3) -> 7
, f_1(3) -> 11
, f_1(4) -> 7
, f_1(4) -> 11
, f_1(8) -> 1
, f_1(8) -> 7
, f_1(8) -> 11
, n__f_0(2) -> 2
, n__f_0(2) -> 7
, n__f_0(2) -> 11
, n__f_0(3) -> 2
, n__f_0(3) -> 7
, n__f_0(3) -> 11
, n__f_0(4) -> 2
, n__f_0(4) -> 7
, n__f_0(4) -> 11
, n__f_1(2) -> 1
, n__f_1(3) -> 1
, n__f_1(4) -> 1
, n__f_1(10) -> 9
, n__f_2(2) -> 7
, n__f_2(2) -> 11
, n__f_2(3) -> 7
, n__f_2(3) -> 11
, n__f_2(4) -> 7
, n__f_2(4) -> 11
, n__f_2(8) -> 1
, n__f_2(8) -> 7
, n__f_2(8) -> 11
, n__a_0() -> 3
, n__a_0() -> 7
, n__a_0() -> 11
, n__a_1() -> 5
, n__a_1() -> 10
, n__a_2() -> 7
, n__a_2() -> 11
, n__g_0(2) -> 4
, n__g_0(2) -> 7
, n__g_0(2) -> 11
, n__g_0(3) -> 4
, n__g_0(3) -> 7
, n__g_0(3) -> 11
, n__g_0(4) -> 4
, n__g_0(4) -> 7
, n__g_0(4) -> 11
, n__g_1(2) -> 6
, n__g_1(3) -> 6
, n__g_1(4) -> 6
, n__g_1(9) -> 8
, n__g_2(11) -> 7
, n__g_2(11) -> 11
, a_0() -> 5
, a_1() -> 7
, a_1() -> 11
, g_0(2) -> 6
, g_0(3) -> 6
, g_0(4) -> 6
, g_1(11) -> 7
, g_1(11) -> 11
, activate_0(2) -> 7
, activate_0(3) -> 7
, activate_0(4) -> 7
, activate_1(2) -> 11
, activate_1(3) -> 11
, activate_1(4) -> 11
, 2 -> 7
, 2 -> 11
, 3 -> 7
, 3 -> 11
, 4 -> 7
, 4 -> 11 }

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