YES
O(n)
TRS:
 {   f(f(X)) -> f(g(f(g(f(X))))),
  f(g(f(X))) -> f(g(X))}
 DUP: We consider a non-duplicating system.
  Trs:
   {   f(f(X)) -> f(g(f(g(f(X))))),
    f(g(f(X))) -> f(g(X))}
  BOUND:
   Automaton:
    {
     g_3(89) -> 90,
     g_3(67) -> 68,
     g_3(44) -> 45,
     g_3(39) -> 40,
     g_2(85) -> 86,
     g_2(81) -> 82,
     g_2(65) -> 66,
     g_2(61) -> 62,
     g_2(57) -> 58,
     g_2(37) -> 38,
     g_2(22) -> 23,
     g_2(19) -> 20,
     g_1(52) -> 53,
     g_1(33) -> 34,
     g_1(15) -> 16,
      g_1(3) -> 4,
      g_0(2) -> 2,
     f_3(40) -> 41,
     f_2(75) -> 76,
     f_2(26) -> 27,
     f_2(20) -> 21,
     f_1(71) -> 72,
     f_1(46) -> 47,
      f_1(9) -> 10,
      f_1(4) -> 5,
      f_0(2) -> 2,
          90 -> 40,
          86 -> 20,
          82 -> 20,
          76 -> 65,
          75 -> 89 | 85,
          72 -> 52,
          71 -> 81,
          68 -> 40,
          66 -> 20,
          62 -> 20,
          58 -> 20,
          53 -> 4,
          47 -> 33,
          46 -> 65,
          45 -> 40,
          41 -> 21,
          40 -> 75 | 71 | 67 | 61 | 52,
          38 -> 20,
          34 -> 4,
          27 -> 22,
          26 -> 57 | 44,
          23 -> 20,
          21 -> 5,
          20 -> 46 | 39 | 37 | 33,
          16 -> 4,
          10 -> 3,
           9 -> 22,
           5 -> 47 | 27 | 21 | 10 | 2,
           4 -> 26 | 19 | 15,
           2 -> 9 | 3
    }
   Strict:
    {}
   Qed