YES(?,O(n^1))

Problem:
 a(a(a(b(b(b(x1)))))) -> b(b(b(b(a(a(a(a(x1))))))))

Proof:
 Bounds Processor:
  bound: 5
  enrichment: match
  automaton:
   final states: {2,1}
   transitions:
    a3(167) -> 168*
    a3(157) -> 158*
    a3(169) -> 170*
    a3(159) -> 160*
    a3(149) -> 150*
    a3(119) -> 120*
    a3(166) -> 167*
    a3(151) -> 152*
    a3(121) -> 122*
    a3(168) -> 169*
    a3(158) -> 159*
    a3(148) -> 149*
    a3(118) -> 119*
    a3(160) -> 161*
    a3(150) -> 151*
    a3(135) -> 136*
    a3(120) -> 121*
    b4(192) -> 193*
    b4(182) -> 183*
    b4(179) -> 180*
    b4(191) -> 192*
    b4(181) -> 182*
    b4(193) -> 194*
    b4(190) -> 191*
    b4(180) -> 181*
    a4(187) -> 188*
    a4(177) -> 178*
    a4(189) -> 190*
    a4(186) -> 187*
    a4(176) -> 177*
    a4(188) -> 189*
    a4(178) -> 179*
    a4(175) -> 176*
    a0(2) -> 1*
    a0(1) -> 1*
    b5(202) -> 203*
    b5(199) -> 200*
    b5(201) -> 202*
    b5(200) -> 201*
    b0(2) -> 2*
    b0(1) -> 2*
    a5(197) -> 198*
    a5(196) -> 197*
    a5(198) -> 199*
    a5(195) -> 196*
    b1(40) -> 41*
    b1(10) -> 11*
    b1(37) -> 38*
    b1(39) -> 40*
    b1(9) -> 10*
    b1(11) -> 12*
    b1(38) -> 39*
    b1(8) -> 9*
    a1(35) -> 36*
    a1(5) -> 6*
    a1(7) -> 8*
    a1(79) -> 80*
    a1(34) -> 35*
    a1(4) -> 5*
    a1(81) -> 82*
    a1(46) -> 47*
    a1(36) -> 37*
    a1(31) -> 32*
    a1(6) -> 7*
    a1(48) -> 49*
    a1(33) -> 34*
    a1(85) -> 86*
    b2(55) -> 56*
    b2(142) -> 143*
    b2(97) -> 98*
    b2(57) -> 58*
    b2(144) -> 145*
    b2(114) -> 115*
    b2(54) -> 55*
    b2(141) -> 142*
    b2(116) -> 117*
    b2(96) -> 97*
    b2(56) -> 57*
    b2(143) -> 144*
    b2(113) -> 114*
    b2(98) -> 99*
    b2(115) -> 116*
    b2(95) -> 96*
    a2(50) -> 51*
    a2(137) -> 138*
    a2(127) -> 128*
    a2(112) -> 113*
    a2(92) -> 93*
    a2(87) -> 88*
    a2(52) -> 53*
    a2(139) -> 140*
    a2(129) -> 130*
    a2(109) -> 110*
    a2(94) -> 95*
    a2(146) -> 147*
    a2(111) -> 112*
    a2(91) -> 92*
    a2(51) -> 52*
    a2(138) -> 139*
    a2(93) -> 94*
    a2(53) -> 54*
    a2(140) -> 141*
    a2(110) -> 111*
    b3(172) -> 173*
    b3(162) -> 163*
    b3(152) -> 153*
    b3(122) -> 123*
    b3(164) -> 165*
    b3(154) -> 155*
    b3(124) -> 125*
    b3(171) -> 172*
    b3(161) -> 162*
    b3(173) -> 174*
    b3(163) -> 164*
    b3(153) -> 154*
    b3(123) -> 124*
    b3(170) -> 171*
    b3(155) -> 156*
    b3(125) -> 126*
    1 -> 31*
    2 -> 4*
    8 -> 91*
    9 -> 50,48
    10 -> 46*
    11 -> 33*
    12 -> 6,7,32,1
    32 -> 5*
    37 -> 127*
    38 -> 87,85
    39 -> 81*
    40 -> 79*
    41 -> 6,1,31,7,32
    47 -> 34*
    49 -> 34*
    54 -> 135*
    55 -> 118*
    56 -> 109*
    57 -> 129*
    58 -> 93,92,8,7
    80 -> 5*
    82 -> 5*
    86 -> 5*
    88 -> 51*
    96 -> 146*
    99 -> 36*
    114 -> 166,137
    117 -> 36,53
    124 -> 157*
    126 -> 95,93,94
    128 -> 51*
    130 -> 92*
    136 -> 119*
    142 -> 148*
    145 -> 52*
    147 -> 138*
    153 -> 175*
    156 -> 136*
    165 -> 140*
    174 -> 120*
    179 -> 195*
    182 -> 186*
    183 -> 122*
    194 -> 160*
    203 -> 189*
  problem:
   
  Qed