YES

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

Proof:
 DP Processor:
  DPs:
   f#(f(X)) -> f#(g(f(X)))
   f#(f(X)) -> f#(g(f(g(f(X)))))
   f#(g(f(X))) -> f#(g(X))
  TRS:
   f(f(X)) -> f(g(f(g(f(X)))))
   f(g(f(X))) -> f(g(X))
  EDG Processor:
   DPs:
    f#(f(X)) -> f#(g(f(X)))
    f#(f(X)) -> f#(g(f(g(f(X)))))
    f#(g(f(X))) -> f#(g(X))
   TRS:
    f(f(X)) -> f(g(f(g(f(X)))))
    f(g(f(X))) -> f(g(X))
   graph:
    f#(g(f(X))) -> f#(g(X)) -> f#(g(f(X))) -> f#(g(X))
    f#(f(X)) -> f#(g(f(g(f(X))))) -> f#(g(f(X))) -> f#(g(X))
    f#(f(X)) -> f#(g(f(X))) -> f#(g(f(X))) -> f#(g(X))
   SCC Processor:
    #sccs: 1
    #rules: 1
    #arcs: 3/9
    DPs:
     f#(g(f(X))) -> f#(g(X))
    TRS:
     f(f(X)) -> f(g(f(g(f(X)))))
     f(g(f(X))) -> f(g(X))
    Bounds Processor:
     bound: 3
     enrichment: match
     automaton:
      final states: {3}
      transitions:
       f3(117) -> 118*
       f3(112) -> 113*
       f3(129) -> 130*
       f1(30) -> 31*
       f1(20) -> 21*
       f1(32) -> 33*
       f1(22) -> 23*
       f1(24) -> 25*
       g3(162) -> 163*
       g3(114) -> 115*
       g3(166) -> 167*
       g3(156) -> 157*
       g3(131) -> 132*
       g3(116) -> 117*
       g3(111) -> 112*
       g3(158) -> 159*
       g3(128) -> 129*
       g3(160) -> 161*
       g1(92) -> 93*
       g1(94) -> 95*
       g1(39) -> 40*
       g1(41) -> 42*
       g1(21) -> 22*
       g1(6) -> 7*
       g1(148) -> 149*
       g1(23) -> 24*
       g1(18) -> 19*
       f{#,1}(7) -> 8*
       f{#,1}(174) -> 175*
       f{#,1}(168) -> 169*
       f{#,1}(150) -> 151*
       f2(75) -> 76*
       f2(60) -> 61*
       f2(82) -> 83*
       f2(67) -> 68*
       f2(62) -> 63*
       f2(58) -> 59*
       f2(80) -> 81*
       f{#,0}(2) -> 3*
       f{#,0}(1) -> 3*
       g2(45) -> 46*
       g2(77) -> 78*
       g2(154) -> 155*
       g2(124) -> 125*
       g2(109) -> 110*
       g2(79) -> 80*
       g2(74) -> 75*
       g2(59) -> 60*
       g2(136) -> 137*
       g2(101) -> 102*
       g2(61) -> 62*
       g2(103) -> 104*
       g2(48) -> 49*
       g0(2) -> 1*
       g0(1) -> 1*
       f{#,2}(182) -> 183*
       f{#,2}(152) -> 153*
       f{#,2}(176) -> 177*
       f{#,2}(46) -> 47*
       f0(2) -> 2*
       f0(1) -> 2*
       1 -> 30,18
       2 -> 20,6
       7 -> 32*
       8 -> 3*
       19 -> 7*
       20 -> 77*
       22 -> 79*
       24 -> 67,45,41
       25 -> 21,2
       30 -> 74*
       31 -> 21*
       32 -> 58,48,39
       33 -> 31,21,2
       40 -> 7*
       42 -> 7*
       46 -> 82*
       47 -> 8,3
       49 -> 46*
       58 -> 114*
       60 -> 116*
       62 -> 158,124
       63 -> 68,59,25,2,20,6,77,21
       67 -> 111*
       68 -> 59*
       75 -> 162,136
       76 -> 33,31,109,23
       78 -> 75*
       80 -> 128,101,94
       81 -> 68,59,25,2
       82 -> 131,103,92
       83 -> 76,59,23,33,2
       93 -> 7*
       95 -> 7*
       102 -> 46*
       104 -> 46*
       110 -> 62*
       112 -> 166*
       113 -> 61*
       115 -> 112*
       117 -> 156,154,148
       118 -> 76,61,33,23,31,109,63
       125 -> 152,150,75
       129 -> 160*
       130 -> 61,63,76,23
       132 -> 129*
       137 -> 176,174,62
       149 -> 7*
       151 -> 3*
       153 -> 8,3
       155 -> 46*
       157 -> 129*
       159 -> 129*
       161 -> 182,168,117
       163 -> 117*
       167 -> 117*
       169 -> 3*
       175 -> 3*
       177 -> 8,3
       183 -> 8,3
     problem:
      DPs:
       
      TRS:
       f(f(X)) -> f(g(f(g(f(X)))))
       f(g(f(X))) -> f(g(X))
     Qed