YES(?,O(n^1))

Problem:
 f(f(X)) -> f(g(f(g(f(X)))))
 f(g(f(X))) -> f(g(X))

Proof:
 Bounds Processor:
  bound: 3
  enrichment: match
  automaton:
   final states: {2,1}
   transitions:
    f3(70) -> 71*
    f3(88) -> 89*
    f3(78) -> 79*
    f1(4) -> 5*
    f1(6) -> 7*
    f1(18) -> 19*
    f1(8) -> 9*
    g3(102) -> 103*
    g3(92) -> 93*
    g3(87) -> 88*
    g3(77) -> 78*
    g3(69) -> 70*
    g3(100) -> 101*
    g1(20) -> 21*
    g1(5) -> 6*
    g1(7) -> 8*
    g1(53) -> 54*
    g1(28) -> 29*
    f2(30) -> 31*
    f2(47) -> 48*
    f2(32) -> 33*
    f2(34) -> 35*
    f2(58) -> 59*
    f0(2) -> 1*
    f0(1) -> 1*
    g2(60) -> 61*
    g2(67) -> 68*
    g2(57) -> 58*
    g2(96) -> 97*
    g2(51) -> 52*
    g2(46) -> 47*
    g2(31) -> 32*
    g2(33) -> 34*
    g2(85) -> 86*
    g2(80) -> 81*
    g0(2) -> 2*
    g0(1) -> 2*
    1 -> 28,18
    2 -> 20,4
    4 -> 46*
    6 -> 60*
    8 -> 57,53,30
    9 -> 19,5,1
    18 -> 51*
    19 -> 5*
    21 -> 8*
    29 -> 8*
    30 -> 87*
    32 -> 77*
    34 -> 85*
    35 -> 19,5
    47 -> 102,80
    48 -> 67,7
    52 -> 47*
    54 -> 8*
    58 -> 69*
    59 -> 31,48,9,1
    61 -> 58*
    68 -> 34*
    70 -> 92*
    71 -> 35,33,31,9,48
    78 -> 96*
    79 -> 35*
    81 -> 58*
    86 -> 47*
    88 -> 100*
    89 -> 33*
    93 -> 70*
    97 -> 47*
    101 -> 70*
    103 -> 70*
  problem:
   
  Qed