YES(?,O(n^1))

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

Strict Trs:
  { active(f(X)) -> f(active(X))
  , active(f(X)) -> mark(if(X, c(), f(true())))
  , active(if(X1, X2, X3)) -> if(X1, active(X2), X3)
  , active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
  , active(if(true(), X, Y)) -> mark(X)
  , active(if(false(), X, Y)) -> mark(Y)
  , f(mark(X)) -> mark(f(X))
  , f(ok(X)) -> ok(f(X))
  , if(X1, mark(X2), X3) -> mark(if(X1, X2, X3))
  , if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
  , if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
  , proper(f(X)) -> f(proper(X))
  , proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
  , proper(c()) -> ok(c())
  , proper(true()) -> ok(true())
  , proper(false()) -> ok(false())
  , 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 2. The enriched problem is
compatible with the following automaton.
{ active_0(3) -> 1
, active_0(5) -> 1
, active_0(6) -> 1
, active_0(7) -> 1
, active_0(9) -> 1
, active_1(3) -> 14
, active_1(5) -> 14
, active_1(6) -> 14
, active_1(7) -> 14
, active_1(9) -> 14
, active_2(13) -> 15
, f_0(3) -> 2
, f_0(5) -> 2
, f_0(6) -> 2
, f_0(7) -> 2
, f_0(9) -> 2
, f_1(3) -> 11
, f_1(5) -> 11
, f_1(6) -> 11
, f_1(7) -> 11
, f_1(9) -> 11
, mark_0(3) -> 3
, mark_0(5) -> 3
, mark_0(6) -> 3
, mark_0(7) -> 3
, mark_0(9) -> 3
, mark_1(11) -> 2
, mark_1(11) -> 11
, mark_1(12) -> 4
, mark_1(12) -> 12
, if_0(3, 3, 3) -> 4
, if_0(3, 3, 5) -> 4
, if_0(3, 3, 6) -> 4
, if_0(3, 3, 7) -> 4
, if_0(3, 3, 9) -> 4
, if_0(3, 5, 3) -> 4
, if_0(3, 5, 5) -> 4
, if_0(3, 5, 6) -> 4
, if_0(3, 5, 7) -> 4
, if_0(3, 5, 9) -> 4
, if_0(3, 6, 3) -> 4
, if_0(3, 6, 5) -> 4
, if_0(3, 6, 6) -> 4
, if_0(3, 6, 7) -> 4
, if_0(3, 6, 9) -> 4
, if_0(3, 7, 3) -> 4
, if_0(3, 7, 5) -> 4
, if_0(3, 7, 6) -> 4
, if_0(3, 7, 7) -> 4
, if_0(3, 7, 9) -> 4
, if_0(3, 9, 3) -> 4
, if_0(3, 9, 5) -> 4
, if_0(3, 9, 6) -> 4
, if_0(3, 9, 7) -> 4
, if_0(3, 9, 9) -> 4
, if_0(5, 3, 3) -> 4
, if_0(5, 3, 5) -> 4
, if_0(5, 3, 6) -> 4
, if_0(5, 3, 7) -> 4
, if_0(5, 3, 9) -> 4
, if_0(5, 5, 3) -> 4
, if_0(5, 5, 5) -> 4
, if_0(5, 5, 6) -> 4
, if_0(5, 5, 7) -> 4
, if_0(5, 5, 9) -> 4
, if_0(5, 6, 3) -> 4
, if_0(5, 6, 5) -> 4
, if_0(5, 6, 6) -> 4
, if_0(5, 6, 7) -> 4
, if_0(5, 6, 9) -> 4
, if_0(5, 7, 3) -> 4
, if_0(5, 7, 5) -> 4
, if_0(5, 7, 6) -> 4
, if_0(5, 7, 7) -> 4
, if_0(5, 7, 9) -> 4
, if_0(5, 9, 3) -> 4
, if_0(5, 9, 5) -> 4
, if_0(5, 9, 6) -> 4
, if_0(5, 9, 7) -> 4
, if_0(5, 9, 9) -> 4
, if_0(6, 3, 3) -> 4
, if_0(6, 3, 5) -> 4
, if_0(6, 3, 6) -> 4
, if_0(6, 3, 7) -> 4
, if_0(6, 3, 9) -> 4
, if_0(6, 5, 3) -> 4
, if_0(6, 5, 5) -> 4
, if_0(6, 5, 6) -> 4
, if_0(6, 5, 7) -> 4
, if_0(6, 5, 9) -> 4
, if_0(6, 6, 3) -> 4
, if_0(6, 6, 5) -> 4
, if_0(6, 6, 6) -> 4
, if_0(6, 6, 7) -> 4
, if_0(6, 6, 9) -> 4
, if_0(6, 7, 3) -> 4
, if_0(6, 7, 5) -> 4
, if_0(6, 7, 6) -> 4
, if_0(6, 7, 7) -> 4
, if_0(6, 7, 9) -> 4
, if_0(6, 9, 3) -> 4
, if_0(6, 9, 5) -> 4
, if_0(6, 9, 6) -> 4
, if_0(6, 9, 7) -> 4
, if_0(6, 9, 9) -> 4
, if_0(7, 3, 3) -> 4
, if_0(7, 3, 5) -> 4
, if_0(7, 3, 6) -> 4
, if_0(7, 3, 7) -> 4
, if_0(7, 3, 9) -> 4
, if_0(7, 5, 3) -> 4
, if_0(7, 5, 5) -> 4
, if_0(7, 5, 6) -> 4
, if_0(7, 5, 7) -> 4
, if_0(7, 5, 9) -> 4
, if_0(7, 6, 3) -> 4
, if_0(7, 6, 5) -> 4
, if_0(7, 6, 6) -> 4
, if_0(7, 6, 7) -> 4
, if_0(7, 6, 9) -> 4
, if_0(7, 7, 3) -> 4
, if_0(7, 7, 5) -> 4
, if_0(7, 7, 6) -> 4
, if_0(7, 7, 7) -> 4
, if_0(7, 7, 9) -> 4
, if_0(7, 9, 3) -> 4
, if_0(7, 9, 5) -> 4
, if_0(7, 9, 6) -> 4
, if_0(7, 9, 7) -> 4
, if_0(7, 9, 9) -> 4
, if_0(9, 3, 3) -> 4
, if_0(9, 3, 5) -> 4
, if_0(9, 3, 6) -> 4
, if_0(9, 3, 7) -> 4
, if_0(9, 3, 9) -> 4
, if_0(9, 5, 3) -> 4
, if_0(9, 5, 5) -> 4
, if_0(9, 5, 6) -> 4
, if_0(9, 5, 7) -> 4
, if_0(9, 5, 9) -> 4
, if_0(9, 6, 3) -> 4
, if_0(9, 6, 5) -> 4
, if_0(9, 6, 6) -> 4
, if_0(9, 6, 7) -> 4
, if_0(9, 6, 9) -> 4
, if_0(9, 7, 3) -> 4
, if_0(9, 7, 5) -> 4
, if_0(9, 7, 6) -> 4
, if_0(9, 7, 7) -> 4
, if_0(9, 7, 9) -> 4
, if_0(9, 9, 3) -> 4
, if_0(9, 9, 5) -> 4
, if_0(9, 9, 6) -> 4
, if_0(9, 9, 7) -> 4
, if_0(9, 9, 9) -> 4
, if_1(3, 3, 3) -> 12
, if_1(3, 3, 5) -> 12
, if_1(3, 3, 6) -> 12
, if_1(3, 3, 7) -> 12
, if_1(3, 3, 9) -> 12
, if_1(3, 5, 3) -> 12
, if_1(3, 5, 5) -> 12
, if_1(3, 5, 6) -> 12
, if_1(3, 5, 7) -> 12
, if_1(3, 5, 9) -> 12
, if_1(3, 6, 3) -> 12
, if_1(3, 6, 5) -> 12
, if_1(3, 6, 6) -> 12
, if_1(3, 6, 7) -> 12
, if_1(3, 6, 9) -> 12
, if_1(3, 7, 3) -> 12
, if_1(3, 7, 5) -> 12
, if_1(3, 7, 6) -> 12
, if_1(3, 7, 7) -> 12
, if_1(3, 7, 9) -> 12
, if_1(3, 9, 3) -> 12
, if_1(3, 9, 5) -> 12
, if_1(3, 9, 6) -> 12
, if_1(3, 9, 7) -> 12
, if_1(3, 9, 9) -> 12
, if_1(5, 3, 3) -> 12
, if_1(5, 3, 5) -> 12
, if_1(5, 3, 6) -> 12
, if_1(5, 3, 7) -> 12
, if_1(5, 3, 9) -> 12
, if_1(5, 5, 3) -> 12
, if_1(5, 5, 5) -> 12
, if_1(5, 5, 6) -> 12
, if_1(5, 5, 7) -> 12
, if_1(5, 5, 9) -> 12
, if_1(5, 6, 3) -> 12
, if_1(5, 6, 5) -> 12
, if_1(5, 6, 6) -> 12
, if_1(5, 6, 7) -> 12
, if_1(5, 6, 9) -> 12
, if_1(5, 7, 3) -> 12
, if_1(5, 7, 5) -> 12
, if_1(5, 7, 6) -> 12
, if_1(5, 7, 7) -> 12
, if_1(5, 7, 9) -> 12
, if_1(5, 9, 3) -> 12
, if_1(5, 9, 5) -> 12
, if_1(5, 9, 6) -> 12
, if_1(5, 9, 7) -> 12
, if_1(5, 9, 9) -> 12
, if_1(6, 3, 3) -> 12
, if_1(6, 3, 5) -> 12
, if_1(6, 3, 6) -> 12
, if_1(6, 3, 7) -> 12
, if_1(6, 3, 9) -> 12
, if_1(6, 5, 3) -> 12
, if_1(6, 5, 5) -> 12
, if_1(6, 5, 6) -> 12
, if_1(6, 5, 7) -> 12
, if_1(6, 5, 9) -> 12
, if_1(6, 6, 3) -> 12
, if_1(6, 6, 5) -> 12
, if_1(6, 6, 6) -> 12
, if_1(6, 6, 7) -> 12
, if_1(6, 6, 9) -> 12
, if_1(6, 7, 3) -> 12
, if_1(6, 7, 5) -> 12
, if_1(6, 7, 6) -> 12
, if_1(6, 7, 7) -> 12
, if_1(6, 7, 9) -> 12
, if_1(6, 9, 3) -> 12
, if_1(6, 9, 5) -> 12
, if_1(6, 9, 6) -> 12
, if_1(6, 9, 7) -> 12
, if_1(6, 9, 9) -> 12
, if_1(7, 3, 3) -> 12
, if_1(7, 3, 5) -> 12
, if_1(7, 3, 6) -> 12
, if_1(7, 3, 7) -> 12
, if_1(7, 3, 9) -> 12
, if_1(7, 5, 3) -> 12
, if_1(7, 5, 5) -> 12
, if_1(7, 5, 6) -> 12
, if_1(7, 5, 7) -> 12
, if_1(7, 5, 9) -> 12
, if_1(7, 6, 3) -> 12
, if_1(7, 6, 5) -> 12
, if_1(7, 6, 6) -> 12
, if_1(7, 6, 7) -> 12
, if_1(7, 6, 9) -> 12
, if_1(7, 7, 3) -> 12
, if_1(7, 7, 5) -> 12
, if_1(7, 7, 6) -> 12
, if_1(7, 7, 7) -> 12
, if_1(7, 7, 9) -> 12
, if_1(7, 9, 3) -> 12
, if_1(7, 9, 5) -> 12
, if_1(7, 9, 6) -> 12
, if_1(7, 9, 7) -> 12
, if_1(7, 9, 9) -> 12
, if_1(9, 3, 3) -> 12
, if_1(9, 3, 5) -> 12
, if_1(9, 3, 6) -> 12
, if_1(9, 3, 7) -> 12
, if_1(9, 3, 9) -> 12
, if_1(9, 5, 3) -> 12
, if_1(9, 5, 5) -> 12
, if_1(9, 5, 6) -> 12
, if_1(9, 5, 7) -> 12
, if_1(9, 5, 9) -> 12
, if_1(9, 6, 3) -> 12
, if_1(9, 6, 5) -> 12
, if_1(9, 6, 6) -> 12
, if_1(9, 6, 7) -> 12
, if_1(9, 6, 9) -> 12
, if_1(9, 7, 3) -> 12
, if_1(9, 7, 5) -> 12
, if_1(9, 7, 6) -> 12
, if_1(9, 7, 7) -> 12
, if_1(9, 7, 9) -> 12
, if_1(9, 9, 3) -> 12
, if_1(9, 9, 5) -> 12
, if_1(9, 9, 6) -> 12
, if_1(9, 9, 7) -> 12
, if_1(9, 9, 9) -> 12
, c_0() -> 5
, c_1() -> 13
, true_0() -> 6
, true_1() -> 13
, false_0() -> 7
, false_1() -> 13
, proper_0(3) -> 8
, proper_0(5) -> 8
, proper_0(6) -> 8
, proper_0(7) -> 8
, proper_0(9) -> 8
, proper_1(3) -> 14
, proper_1(5) -> 14
, proper_1(6) -> 14
, proper_1(7) -> 14
, proper_1(9) -> 14
, ok_0(3) -> 9
, ok_0(5) -> 9
, ok_0(6) -> 9
, ok_0(7) -> 9
, ok_0(9) -> 9
, ok_1(11) -> 2
, ok_1(11) -> 11
, ok_1(12) -> 4
, ok_1(12) -> 12
, ok_1(13) -> 8
, ok_1(13) -> 14
, top_0(3) -> 10
, top_0(5) -> 10
, top_0(6) -> 10
, top_0(7) -> 10
, top_0(9) -> 10
, top_1(14) -> 10
, top_2(15) -> 10 }

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