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

Strict Trs:
  { f(X) -> if(X, c(), n__f(true()))
  , f(X) -> n__f(X)
  , if(true(), X, Y) -> X
  , if(false(), X, Y) -> activate(Y)
  , activate(X) -> X
  , activate(n__f(X)) -> f(X) }
Obligation:
  runtime complexity
Answer:
  YES(?,O(n^1))

The input is overlay and right-linear. Switching to innermost
rewriting.

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

Strict Trs:
  { f(X) -> if(X, c(), n__f(true()))
  , f(X) -> n__f(X)
  , if(true(), X, Y) -> X
  , if(false(), X, Y) -> activate(Y)
  , activate(X) -> X
  , activate(n__f(X)) -> f(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(5) -> 1
, f_0(6) -> 1
, f_1(3) -> 2
, f_1(3) -> 7
, f_1(4) -> 2
, f_1(4) -> 7
, f_1(5) -> 2
, f_1(5) -> 7
, f_1(6) -> 2
, f_1(6) -> 7
, f_2(10) -> 1
, f_2(13) -> 2
, f_2(13) -> 7
, if_0(3, 3, 3) -> 2
, if_0(3, 3, 4) -> 2
, if_0(3, 3, 5) -> 2
, if_0(3, 3, 6) -> 2
, if_0(3, 4, 3) -> 2
, if_0(3, 4, 4) -> 2
, if_0(3, 4, 5) -> 2
, if_0(3, 4, 6) -> 2
, if_0(3, 5, 3) -> 2
, if_0(3, 5, 4) -> 2
, if_0(3, 5, 5) -> 2
, if_0(3, 5, 6) -> 2
, if_0(3, 6, 3) -> 2
, if_0(3, 6, 4) -> 2
, if_0(3, 6, 5) -> 2
, if_0(3, 6, 6) -> 2
, if_0(4, 3, 3) -> 2
, if_0(4, 3, 4) -> 2
, if_0(4, 3, 5) -> 2
, if_0(4, 3, 6) -> 2
, if_0(4, 4, 3) -> 2
, if_0(4, 4, 4) -> 2
, if_0(4, 4, 5) -> 2
, if_0(4, 4, 6) -> 2
, if_0(4, 5, 3) -> 2
, if_0(4, 5, 4) -> 2
, if_0(4, 5, 5) -> 2
, if_0(4, 5, 6) -> 2
, if_0(4, 6, 3) -> 2
, if_0(4, 6, 4) -> 2
, if_0(4, 6, 5) -> 2
, if_0(4, 6, 6) -> 2
, if_0(5, 3, 3) -> 2
, if_0(5, 3, 4) -> 2
, if_0(5, 3, 5) -> 2
, if_0(5, 3, 6) -> 2
, if_0(5, 4, 3) -> 2
, if_0(5, 4, 4) -> 2
, if_0(5, 4, 5) -> 2
, if_0(5, 4, 6) -> 2
, if_0(5, 5, 3) -> 2
, if_0(5, 5, 4) -> 2
, if_0(5, 5, 5) -> 2
, if_0(5, 5, 6) -> 2
, if_0(5, 6, 3) -> 2
, if_0(5, 6, 4) -> 2
, if_0(5, 6, 5) -> 2
, if_0(5, 6, 6) -> 2
, if_0(6, 3, 3) -> 2
, if_0(6, 3, 4) -> 2
, if_0(6, 3, 5) -> 2
, if_0(6, 3, 6) -> 2
, if_0(6, 4, 3) -> 2
, if_0(6, 4, 4) -> 2
, if_0(6, 4, 5) -> 2
, if_0(6, 4, 6) -> 2
, if_0(6, 5, 3) -> 2
, if_0(6, 5, 4) -> 2
, if_0(6, 5, 5) -> 2
, if_0(6, 5, 6) -> 2
, if_0(6, 6, 3) -> 2
, if_0(6, 6, 4) -> 2
, if_0(6, 6, 5) -> 2
, if_0(6, 6, 6) -> 2
, if_1(3, 8, 9) -> 1
, if_1(4, 8, 9) -> 1
, if_1(5, 8, 9) -> 1
, if_1(6, 8, 9) -> 1
, if_2(3, 11, 12) -> 2
, if_2(3, 11, 12) -> 7
, if_2(4, 11, 12) -> 2
, if_2(4, 11, 12) -> 7
, if_2(5, 11, 12) -> 2
, if_2(5, 11, 12) -> 7
, if_2(6, 11, 12) -> 2
, if_2(6, 11, 12) -> 7
, if_3(10, 14, 15) -> 1
, if_3(13, 14, 15) -> 2
, if_3(13, 14, 15) -> 7
, c_0() -> 2
, c_0() -> 3
, c_0() -> 7
, c_1() -> 1
, c_1() -> 8
, c_2() -> 2
, c_2() -> 7
, c_2() -> 11
, c_3() -> 1
, c_3() -> 2
, c_3() -> 7
, c_3() -> 14
, n__f_0(3) -> 2
, n__f_0(3) -> 4
, n__f_0(3) -> 7
, n__f_0(4) -> 2
, n__f_0(4) -> 4
, n__f_0(4) -> 7
, n__f_0(5) -> 2
, n__f_0(5) -> 4
, n__f_0(5) -> 7
, n__f_0(6) -> 2
, n__f_0(6) -> 4
, n__f_0(6) -> 7
, n__f_1(3) -> 1
, n__f_1(4) -> 1
, n__f_1(5) -> 1
, n__f_1(6) -> 1
, n__f_1(10) -> 1
, n__f_1(10) -> 9
, n__f_2(3) -> 2
, n__f_2(3) -> 7
, n__f_2(4) -> 2
, n__f_2(4) -> 7
, n__f_2(5) -> 2
, n__f_2(5) -> 7
, n__f_2(6) -> 2
, n__f_2(6) -> 7
, n__f_2(13) -> 2
, n__f_2(13) -> 7
, n__f_2(13) -> 12
, n__f_3(10) -> 1
, n__f_3(13) -> 2
, n__f_3(13) -> 7
, n__f_3(16) -> 15
, true_0() -> 2
, true_0() -> 5
, true_0() -> 7
, true_1() -> 10
, true_2() -> 13
, true_3() -> 16
, false_0() -> 2
, false_0() -> 6
, false_0() -> 7
, activate_0(3) -> 7
, activate_0(4) -> 7
, activate_0(5) -> 7
, activate_0(6) -> 7
, activate_1(3) -> 2
, activate_1(4) -> 2
, activate_1(5) -> 2
, activate_1(6) -> 2
, activate_1(9) -> 1
, activate_1(12) -> 2
, activate_1(12) -> 7 }

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