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: 4
  enrichment: match
  automaton:
   final states: {2,1}
   transitions:
    g3(122) -> 123*
    g3(97) -> 98*
    g3(149) -> 150*
    g3(124) -> 125*
    g3(114) -> 115*
    g3(176) -> 177*
    g3(146) -> 147*
    g3(126) -> 127*
    g3(76) -> 77*
    g3(71) -> 72*
    g3(238) -> 239*
    g3(170) -> 171*
    g3(110) -> 111*
    f4(191) -> 192*
    g4(199) -> 200*
    g4(236) -> 237*
    g4(211) -> 212*
    g4(201) -> 202*
    g4(193) -> 194*
    g4(205) -> 206*
    g4(190) -> 191*
    f0(2) -> 1*
    f0(1) -> 1*
    g0(2) -> 2*
    g0(1) -> 2*
    f1(179) -> 180*
    f1(4) -> 5*
    f1(6) -> 7*
    f1(18) -> 19*
    f1(8) -> 9*
    g1(20) -> 21*
    g1(5) -> 6*
    g1(132) -> 133*
    g1(7) -> 8*
    g1(89) -> 90*
    g1(246) -> 247*
    g1(186) -> 187*
    g1(136) -> 137*
    g1(178) -> 179*
    g1(138) -> 139*
    g1(53) -> 54*
    g1(28) -> 29*
    g1(130) -> 131*
    f2(30) -> 31*
    f2(102) -> 103*
    f2(47) -> 48*
    f2(32) -> 33*
    f2(34) -> 35*
    f2(108) -> 109*
    f2(58) -> 59*
    g2(60) -> 61*
    g2(87) -> 88*
    g2(67) -> 68*
    g2(57) -> 58*
    g2(244) -> 245*
    g2(79) -> 80*
    g2(51) -> 52*
    g2(46) -> 47*
    g2(31) -> 32*
    g2(223) -> 224*
    g2(118) -> 119*
    g2(33) -> 34*
    g2(120) -> 121*
    g2(95) -> 96*
    f3(147) -> 148*
    f3(77) -> 78*
    f3(72) -> 73*
    f3(151) -> 152*
    f3(158) -> 159*
    f3(98) -> 99*
    f3(160) -> 161*
    1 -> 28,18
    2 -> 20,4
    4 -> 136,102,46
    5 -> 103,109,1,28,18,51
    6 -> 60*
    8 -> 57,53,30
    9 -> 103,19,5,1
    18 -> 138,108,51
    19 -> 5*
    21 -> 8*
    29 -> 8*
    30 -> 97*
    32 -> 76*
    34 -> 158,132,114,95
    35 -> 103,109,31,9,19,5
    47 -> 186,151,124,79
    48 -> 103,31,9,19,67,7
    52 -> 47*
    54 -> 8*
    58 -> 160,89,87,71
    59 -> 73,152,35,149,31,48,9,1
    61 -> 58*
    68 -> 34*
    72 -> 190,178,126,120
    73 -> 152,159,109,149,35,33,48
    77 -> 193,130,118,110
    78 -> 159,146,35
    80 -> 58*
    88 -> 58*
    90 -> 4*
    96 -> 47*
    98 -> 122*
    99 -> 33*
    102 -> 170*
    103 -> 31*
    108 -> 176*
    109 -> 31*
    111 -> 72*
    115 -> 72*
    119 -> 47*
    121 -> 34*
    123 -> 72*
    125 -> 72*
    127 -> 72*
    131 -> 4*
    133 -> 4*
    137 -> 4*
    139 -> 4*
    147 -> 211*
    148 -> 71*
    150 -> 147*
    151 -> 199*
    152 -> 146*
    158 -> 201*
    159 -> 146*
    160 -> 205*
    161 -> 146*
    171 -> 72*
    177 -> 72*
    179 -> 223*
    180 -> 103*
    187 -> 179*
    191 -> 246,244,238,236
    192 -> 148,73,35,33,48,149,71
    194 -> 191*
    200 -> 191*
    202 -> 191*
    206 -> 191*
    212 -> 191*
    224 -> 32*
    237 -> 191*
    239 -> 72*
    245 -> 47*
    247 -> 4*
  problem:
   
  Qed