YES

Problem:
 b(b(x1)) -> c(d(x1))
 c(c(x1)) -> d(d(d(x1)))
 c(x1) -> g(x1)
 d(d(x1)) -> c(f(x1))
 d(d(d(x1))) -> g(c(x1))
 f(x1) -> a(g(x1))
 g(x1) -> d(a(b(x1)))
 g(g(x1)) -> b(c(x1))

Proof:
 Matrix Interpretation Processor: dim=2
  
  interpretation:
             [1 2]     [1]
   [f](x0) = [0 0]x0 + [0],
   
             [1 2]     [0]
   [d](x0) = [0 0]x0 + [1],
   
             [1 2]     [1]
   [g](x0) = [0 0]x0 + [1],
   
             [1 0]  
   [a](x0) = [0 0]x0,
   
             [1 2]     [1]
   [b](x0) = [0 0]x0 + [1],
   
             [1 2]     [1]
   [c](x0) = [0 0]x0 + [1]
  orientation:
              [1 2]     [4]    [1 2]     [3]           
   b(b(x1)) = [0 0]x1 + [1] >= [0 0]x1 + [1] = c(d(x1))
   
              [1 2]     [4]    [1 2]     [4]              
   c(c(x1)) = [0 0]x1 + [1] >= [0 0]x1 + [1] = d(d(d(x1)))
   
           [1 2]     [1]    [1 2]     [1]        
   c(x1) = [0 0]x1 + [1] >= [0 0]x1 + [1] = g(x1)
   
              [1 2]     [2]    [1 2]     [2]           
   d(d(x1)) = [0 0]x1 + [1] >= [0 0]x1 + [1] = c(f(x1))
   
                 [1 2]     [4]    [1 2]     [4]           
   d(d(d(x1))) = [0 0]x1 + [1] >= [0 0]x1 + [1] = g(c(x1))
   
           [1 2]     [1]    [1 2]     [1]           
   f(x1) = [0 0]x1 + [0] >= [0 0]x1 + [0] = a(g(x1))
   
           [1 2]     [1]    [1 2]     [1]              
   g(x1) = [0 0]x1 + [1] >= [0 0]x1 + [1] = d(a(b(x1)))
   
              [1 2]     [4]    [1 2]     [4]           
   g(g(x1)) = [0 0]x1 + [1] >= [0 0]x1 + [1] = b(c(x1))
  problem:
   c(c(x1)) -> d(d(d(x1)))
   c(x1) -> g(x1)
   d(d(x1)) -> c(f(x1))
   d(d(d(x1))) -> g(c(x1))
   f(x1) -> a(g(x1))
   g(x1) -> d(a(b(x1)))
   g(g(x1)) -> b(c(x1))
  Matrix Interpretation Processor: dim=3
   
   interpretation:
              [1 0 1]     [0]
    [f](x0) = [0 0 0]x0 + [1]
              [0 1 0]     [0],
    
              [1 0 1]     [0]
    [d](x0) = [0 0 0]x0 + [1]
              [0 1 0]     [1],
    
              [1 0 1]     [0]
    [g](x0) = [0 0 0]x0 + [1]
              [0 0 0]     [1],
    
              [1 0 0]  
    [a](x0) = [0 0 0]x0
              [0 0 0]  ,
    
              [1 0 0]  
    [b](x0) = [0 0 0]x0
              [0 0 0]  ,
    
              [1 0 1]     [1]
    [c](x0) = [0 0 0]x0 + [1]
              [0 1 0]     [1]
   orientation:
               [1 1 1]     [3]    [1 1 1]     [3]              
    c(c(x1)) = [0 0 0]x1 + [1] >= [0 0 0]x1 + [1] = d(d(d(x1)))
               [0 0 0]     [2]    [0 0 0]     [2]              
    
            [1 0 1]     [1]    [1 0 1]     [0]        
    c(x1) = [0 0 0]x1 + [1] >= [0 0 0]x1 + [1] = g(x1)
            [0 1 0]     [1]    [0 0 0]     [1]        
    
               [1 1 1]     [1]    [1 1 1]     [1]           
    d(d(x1)) = [0 0 0]x1 + [1] >= [0 0 0]x1 + [1] = c(f(x1))
               [0 0 0]     [2]    [0 0 0]     [2]           
    
                  [1 1 1]     [3]    [1 1 1]     [2]           
    d(d(d(x1))) = [0 0 0]x1 + [1] >= [0 0 0]x1 + [1] = g(c(x1))
                  [0 0 0]     [2]    [0 0 0]     [1]           
    
            [1 0 1]     [0]    [1 0 1]             
    f(x1) = [0 0 0]x1 + [1] >= [0 0 0]x1 = a(g(x1))
            [0 1 0]     [0]    [0 0 0]             
    
            [1 0 1]     [0]    [1 0 0]     [0]              
    g(x1) = [0 0 0]x1 + [1] >= [0 0 0]x1 + [1] = d(a(b(x1)))
            [0 0 0]     [1]    [0 0 0]     [1]              
    
               [1 0 1]     [1]    [1 0 1]     [1]           
    g(g(x1)) = [0 0 0]x1 + [1] >= [0 0 0]x1 + [0] = b(c(x1))
               [0 0 0]     [1]    [0 0 0]     [0]           
   problem:
    c(c(x1)) -> d(d(d(x1)))
    d(d(x1)) -> c(f(x1))
    f(x1) -> a(g(x1))
    g(x1) -> d(a(b(x1)))
    g(g(x1)) -> b(c(x1))
   Bounds Processor:
    bound: 5
    enrichment: match
    automaton:
     final states: {7}
     transitions:
      b2(78) -> 79*
      c0(7) -> 7*
      b5(216) -> 217*
      b5(208) -> 209*
      b5(196) -> 197*
      f3(143) -> 144*
      f3(96) -> 97*
      f3(105) -> 106*
      g4(158) -> 159*
      g4(187) -> 188*
      g4(147) -> 148*
      g2(75) -> 76*
      g2(115) -> 116*
      d1(86) -> 87*
      d1(14) -> 15*
      d1(13) -> 14*
      d1(12) -> 13*
      d1(36) -> 37*
      a5(197) -> 198*
      a5(209) -> 210*
      a5(217) -> 218*
      d3(135) -> 136*
      d3(167) -> 168*
      g1(23) -> 24*
      f0(7) -> 7*
      d5(218) -> 219*
      d5(198) -> 199*
      d5(210) -> 211*
      d4(163) -> 164*
      d4(181) -> 182*
      d4(177) -> 178*
      d4(222) -> 223*
      d4(206) -> 207*
      d0(7) -> 7*
      c1(17) -> 18*
      c1(33) -> 34*
      b1(34) -> 35*
      b1(26) -> 27*
      a4(162) -> 163*
      a4(176) -> 177*
      a4(148) -> 149*
      a4(159) -> 160*
      a4(180) -> 181*
      a4(221) -> 222*
      a4(205) -> 206*
      a0(7) -> 7*
      a3(166) -> 167*
      a3(113) -> 114*
      a3(127) -> 128*
      a3(134) -> 135*
      a3(124) -> 125*
      f2(141) -> 142*
      f2(103) -> 104*
      f2(60) -> 61*
      f2(63) -> 64*
      f2(51) -> 52*
      b0(7) -> 7*
      b3(133) -> 134*
      b3(165) -> 166*
      c3(97) -> 98*
      c3(106) -> 107*
      g0(7) -> 7*
      f1(16) -> 17*
      f1(54) -> 55*
      b4(220) -> 221*
      b4(175) -> 176*
      b4(161) -> 162*
      b4(204) -> 205*
      b4(179) -> 180*
      g3(126) -> 127*
      g3(150) -> 151*
      g3(123) -> 124*
      g3(191) -> 192*
      g3(112) -> 113*
      a2(116) -> 117*
      a2(76) -> 77*
      a2(79) -> 80*
      d2(44) -> 45*
      d2(94) -> 95*
      d2(43) -> 44*
      d2(80) -> 81*
      d2(45) -> 46*
      c2(52) -> 53*
      c2(61) -> 62*
      c2(64) -> 65*
      a1(24) -> 25*
      a1(27) -> 28*
      192 -> 124*
      191 -> 220*
      94 -> 143*
      151 -> 124*
      43 -> 105*
      144 -> 106*
      142 -> 52*
      168 -> 116*
      164 -> 124*
      160 -> 97*
      17 -> 43,36
      115 -> 165*
      158 -> 208*
      35 -> 124,76,24,7
      46 -> 34*
      126 -> 179*
      182 -> 127*
      7 -> 33,26,23,16,12
      104 -> 52*
      77 -> 17*
      112 -> 175*
      12 -> 51*
      63 -> 112*
      14 -> 63,54
      114 -> 64*
      16 -> 75*
      188 -> 148*
      28 -> 14*
      117 -> 55*
      53 -> 14*
      143 -> 187*
      105 -> 147*
      75 -> 133*
      54 -> 115*
      207 -> 151*
      150 -> 204*
      178 -> 113*
      141 -> 191*
      36 -> 103*
      86 -> 141*
      125 -> 142,104,52
      44 -> 96*
      60 -> 126*
      13 -> 60*
      123 -> 161*
      62 -> 15,7
      107 -> 45*
      55 -> 17*
      199 -> 148*
      65 -> 13*
      95 -> 44*
      37 -> 13*
      147 -> 196*
      219 -> 188*
      187 -> 216*
      18 -> 13,7
      211 -> 159*
      149 -> 144,106
      136 -> 76*
      96 -> 158*
      15 -> 34,7
      51 -> 123*
      98 -> 46*
      103 -> 150*
      87 -> 13*
      81 -> 24*
      128 -> 61*
      61 -> 94,86
      25 -> 7*
      23 -> 78*
      223 -> 192*
    problem:
     
    Qed