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

Strict Trs:
  { active(c()) -> mark(f(g(c())))
  , active(f(g(X))) -> mark(g(X))
  , f(ok(X)) -> ok(f(X))
  , g(ok(X)) -> ok(g(X))
  , proper(c()) -> ok(c())
  , proper(f(X)) -> f(proper(X))
  , proper(g(X)) -> g(proper(X))
  , top(mark(X)) -> top(proper(X))
  , top(ok(X)) -> top(active(X)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^1))

The problem is match-bounded by 6. The enriched problem is
compatible with the following automaton.
{ active_0(2) -> 1
, active_1(2) -> 7
, active_2(5) -> 8
, active_3(19) -> 14
, active_4(22) -> 23
, active_5(20) -> 26
, active_6(29) -> 30
, c_0() -> 2
, c_1() -> 5
, c_2() -> 11
, c_3() -> 18
, c_4() -> 28
, mark_0(2) -> 2
, mark_1(3) -> 1
, mark_1(3) -> 7
, mark_2(9) -> 8
, mark_4(21) -> 14
, mark_5(24) -> 23
, f_0(2) -> 1
, f_1(2) -> 6
, f_1(4) -> 3
, f_2(10) -> 9
, f_2(12) -> 8
, f_3(15) -> 14
, f_3(17) -> 19
, f_4(20) -> 22
, g_0(2) -> 1
, g_1(2) -> 6
, g_1(5) -> 4
, g_2(11) -> 10
, g_2(13) -> 12
, g_3(11) -> 17
, g_3(16) -> 15
, g_4(11) -> 21
, g_4(18) -> 20
, g_5(18) -> 24
, g_5(25) -> 23
, g_5(28) -> 29
, g_6(27) -> 26
, proper_0(2) -> 1
, proper_1(2) -> 7
, proper_2(3) -> 8
, proper_2(4) -> 12
, proper_2(5) -> 13
, proper_3(9) -> 14
, proper_3(10) -> 15
, proper_3(11) -> 16
, proper_4(21) -> 23
, proper_5(11) -> 25
, proper_5(24) -> 26
, proper_6(18) -> 27
, ok_0(2) -> 2
, ok_1(5) -> 1
, ok_1(5) -> 7
, ok_1(6) -> 1
, ok_1(6) -> 6
, ok_2(11) -> 13
, ok_3(17) -> 12
, ok_3(18) -> 16
, ok_3(18) -> 25
, ok_3(19) -> 8
, ok_4(20) -> 15
, ok_4(20) -> 23
, ok_4(22) -> 14
, ok_4(28) -> 27
, ok_5(29) -> 26
, top_0(2) -> 1
, top_1(7) -> 1
, top_2(8) -> 1
, top_3(14) -> 1
, top_4(23) -> 1
, top_5(26) -> 1
, top_6(30) -> 1 }

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