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(f(X)) -> c(n__f(n__g(n__f(X))))
  , c(X) -> d(activate(X))
  , d(X) -> n__d(X)
  , activate(X) -> X
  , activate(n__f(X)) -> f(activate(X))
  , activate(n__g(X)) -> g(X)
  , activate(n__d(X)) -> d(X)
  , h(X) -> c(n__d(X))
  , g(X) -> n__g(X) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^1))

Arguments of following rules are not normal-forms:

{ f(f(X)) -> c(n__f(n__g(n__f(X)))) }

All above mentioned rules can be savely removed.

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

Strict Trs:
  { f(X) -> n__f(X)
  , c(X) -> d(activate(X))
  , d(X) -> n__d(X)
  , activate(X) -> X
  , activate(n__f(X)) -> f(activate(X))
  , activate(n__g(X)) -> g(X)
  , activate(n__d(X)) -> d(X)
  , h(X) -> c(n__d(X))
  , g(X) -> n__g(X) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^1))

The problem is match-bounded by 3. The enriched problem is
compatible with the following automaton.
{ f_0(3) -> 1
, f_0(4) -> 1
, f_0(8) -> 1
, f_1(10) -> 6
, f_1(10) -> 10
, c_0(3) -> 2
, c_0(4) -> 2
, c_0(8) -> 2
, c_1(5) -> 7
, n__f_0(3) -> 3
, n__f_0(3) -> 6
, n__f_0(3) -> 10
, n__f_0(4) -> 3
, n__f_0(4) -> 6
, n__f_0(4) -> 10
, n__f_0(8) -> 3
, n__f_0(8) -> 6
, n__f_0(8) -> 10
, n__f_1(3) -> 1
, n__f_1(4) -> 1
, n__f_1(8) -> 1
, n__f_2(10) -> 6
, n__f_2(10) -> 10
, n__g_0(3) -> 4
, n__g_0(3) -> 6
, n__g_0(3) -> 10
, n__g_0(4) -> 4
, n__g_0(4) -> 6
, n__g_0(4) -> 10
, n__g_0(8) -> 4
, n__g_0(8) -> 6
, n__g_0(8) -> 10
, n__g_1(3) -> 9
, n__g_1(4) -> 9
, n__g_1(8) -> 9
, n__g_2(3) -> 6
, n__g_2(3) -> 10
, n__g_2(4) -> 6
, n__g_2(4) -> 10
, n__g_2(8) -> 6
, n__g_2(8) -> 10
, d_0(3) -> 5
, d_0(3) -> 11
, d_0(4) -> 5
, d_0(4) -> 11
, d_0(8) -> 5
, d_0(8) -> 11
, d_1(3) -> 6
, d_1(3) -> 10
, d_1(4) -> 6
, d_1(4) -> 10
, d_1(8) -> 6
, d_1(8) -> 10
, d_1(10) -> 2
, d_2(3) -> 11
, d_2(4) -> 11
, d_2(8) -> 11
, d_2(11) -> 7
, activate_0(3) -> 6
, activate_0(4) -> 6
, activate_0(8) -> 6
, activate_1(3) -> 10
, activate_1(4) -> 10
, activate_1(8) -> 10
, activate_2(5) -> 11
, h_0(3) -> 7
, h_0(4) -> 7
, h_0(8) -> 7
, n__d_0(3) -> 6
, n__d_0(3) -> 8
, n__d_0(3) -> 10
, n__d_0(4) -> 6
, n__d_0(4) -> 8
, n__d_0(4) -> 10
, n__d_0(8) -> 6
, n__d_0(8) -> 8
, n__d_0(8) -> 10
, n__d_1(3) -> 5
, n__d_1(3) -> 11
, n__d_1(4) -> 5
, n__d_1(4) -> 11
, n__d_1(8) -> 5
, n__d_1(8) -> 11
, n__d_2(3) -> 6
, n__d_2(3) -> 10
, n__d_2(4) -> 6
, n__d_2(4) -> 10
, n__d_2(8) -> 6
, n__d_2(8) -> 10
, n__d_2(10) -> 2
, n__d_3(3) -> 11
, n__d_3(4) -> 11
, n__d_3(8) -> 11
, n__d_3(11) -> 7
, g_0(3) -> 9
, g_0(4) -> 9
, g_0(8) -> 9
, g_1(3) -> 6
, g_1(3) -> 10
, g_1(4) -> 6
, g_1(4) -> 10
, g_1(8) -> 6
, g_1(8) -> 10
, 3 -> 6
, 3 -> 10
, 4 -> 6
, 4 -> 10
, 5 -> 11
, 8 -> 6
, 8 -> 10 }

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