YES

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

Proof:
 String Reversal Processor:
  a(a(x1)) -> c(b(x1))
  b(a(x1)) -> a(c(x1))
  c(c(x1)) -> b(a(x1))
  DP Processor:
   DPs:
    a#(a(x1)) -> b#(x1)
    a#(a(x1)) -> c#(b(x1))
    b#(a(x1)) -> c#(x1)
    b#(a(x1)) -> a#(c(x1))
    c#(c(x1)) -> a#(x1)
    c#(c(x1)) -> b#(a(x1))
   TRS:
    a(a(x1)) -> c(b(x1))
    b(a(x1)) -> a(c(x1))
    c(c(x1)) -> b(a(x1))
   TDG Processor:
    DPs:
     a#(a(x1)) -> b#(x1)
     a#(a(x1)) -> c#(b(x1))
     b#(a(x1)) -> c#(x1)
     b#(a(x1)) -> a#(c(x1))
     c#(c(x1)) -> a#(x1)
     c#(c(x1)) -> b#(a(x1))
    TRS:
     a(a(x1)) -> c(b(x1))
     b(a(x1)) -> a(c(x1))
     c(c(x1)) -> b(a(x1))
    graph:
     c#(c(x1)) -> b#(a(x1)) -> b#(a(x1)) -> a#(c(x1))
     c#(c(x1)) -> b#(a(x1)) -> b#(a(x1)) -> c#(x1)
     c#(c(x1)) -> a#(x1) -> a#(a(x1)) -> c#(b(x1))
     c#(c(x1)) -> a#(x1) -> a#(a(x1)) -> b#(x1)
     b#(a(x1)) -> c#(x1) -> c#(c(x1)) -> b#(a(x1))
     b#(a(x1)) -> c#(x1) -> c#(c(x1)) -> a#(x1)
     b#(a(x1)) -> a#(c(x1)) -> a#(a(x1)) -> c#(b(x1))
     b#(a(x1)) -> a#(c(x1)) -> a#(a(x1)) -> b#(x1)
     a#(a(x1)) -> c#(b(x1)) -> c#(c(x1)) -> b#(a(x1))
     a#(a(x1)) -> c#(b(x1)) -> c#(c(x1)) -> a#(x1)
     a#(a(x1)) -> b#(x1) -> b#(a(x1)) -> a#(c(x1))
     a#(a(x1)) -> b#(x1) -> b#(a(x1)) -> c#(x1)
    Arctic Interpretation Processor:
     dimension: 2
     usable rules:
      a(a(x1)) -> c(b(x1))
      b(a(x1)) -> a(c(x1))
      c(c(x1)) -> b(a(x1))
     interpretation:
      [c#](x0) = [0  -&]x0,
      
      [b#](x0) = [0 0]x0 + [0],
      
      [a#](x0) = [0 0]x0 + [0],
      
                [3  0 ]     [-&]
      [b](x0) = [-& 3 ]x0 + [0 ],
      
                [3  0 ]     [0]
      [c](x0) = [3  -&]x0 + [0],
      
                [3 0]     [0]
      [a](x0) = [3 0]x0 + [0]
     orientation:
      a#(a(x1)) = [3 0]x1 + [0] >= [0 0]x1 + [0] = b#(x1)
      
      a#(a(x1)) = [3 0]x1 + [0] >= [3 0]x1 = c#(b(x1))
      
      b#(a(x1)) = [3 0]x1 + [0] >= [0  -&]x1 = c#(x1)
      
      b#(a(x1)) = [3 0]x1 + [0] >= [3 0]x1 + [0] = a#(c(x1))
      
      c#(c(x1)) = [3 0]x1 + [0] >= [0 0]x1 + [0] = a#(x1)
      
      c#(c(x1)) = [3 0]x1 + [0] >= [3 0]x1 + [0] = b#(a(x1))
      
                 [6 3]     [3]    [6 3]     [0]           
      a(a(x1)) = [6 3]x1 + [3] >= [6 3]x1 + [0] = c(b(x1))
      
                 [6 3]     [3]    [6 3]     [3]           
      b(a(x1)) = [6 3]x1 + [3] >= [6 3]x1 + [3] = a(c(x1))
      
                 [6 3]     [3]    [6 3]     [3]           
      c(c(x1)) = [6 3]x1 + [3] >= [6 3]x1 + [3] = b(a(x1))
     problem:
      DPs:
       a#(a(x1)) -> b#(x1)
       a#(a(x1)) -> c#(b(x1))
       b#(a(x1)) -> a#(c(x1))
       c#(c(x1)) -> a#(x1)
       c#(c(x1)) -> b#(a(x1))
      TRS:
       a(a(x1)) -> c(b(x1))
       b(a(x1)) -> a(c(x1))
       c(c(x1)) -> b(a(x1))
     Restore Modifier:
      DPs:
       a#(a(x1)) -> b#(x1)
       a#(a(x1)) -> c#(b(x1))
       b#(a(x1)) -> a#(c(x1))
       c#(c(x1)) -> a#(x1)
       c#(c(x1)) -> b#(a(x1))
      TRS:
       a(a(x1)) -> c(b(x1))
       b(a(x1)) -> a(c(x1))
       c(c(x1)) -> b(a(x1))
      EDG Processor:
       DPs:
        a#(a(x1)) -> b#(x1)
        a#(a(x1)) -> c#(b(x1))
        b#(a(x1)) -> a#(c(x1))
        c#(c(x1)) -> a#(x1)
        c#(c(x1)) -> b#(a(x1))
       TRS:
        a(a(x1)) -> c(b(x1))
        b(a(x1)) -> a(c(x1))
        c(c(x1)) -> b(a(x1))
       graph:
        c#(c(x1)) -> b#(a(x1)) -> b#(a(x1)) -> a#(c(x1))
        c#(c(x1)) -> a#(x1) -> a#(a(x1)) -> b#(x1)
        c#(c(x1)) -> a#(x1) -> a#(a(x1)) -> c#(b(x1))
        b#(a(x1)) -> a#(c(x1)) -> a#(a(x1)) -> b#(x1)
        b#(a(x1)) -> a#(c(x1)) -> a#(a(x1)) -> c#(b(x1))
        a#(a(x1)) -> c#(b(x1)) -> c#(c(x1)) -> a#(x1)
        a#(a(x1)) -> c#(b(x1)) -> c#(c(x1)) -> b#(a(x1))
        a#(a(x1)) -> b#(x1) -> b#(a(x1)) -> a#(c(x1))
       Arctic Interpretation Processor:
        dimension: 2
        usable rules:
         a(a(x1)) -> c(b(x1))
         b(a(x1)) -> a(c(x1))
         c(c(x1)) -> b(a(x1))
        interpretation:
         [c#](x0) = [0 0]x0 + [0],
         
         [b#](x0) = [0 0]x0 + [0],
         
         [a#](x0) = [0 0]x0,
         
                   [0 0]     [0 ]
         [b](x0) = [2 1]x0 + [-&],
         
                   [0 0]     [0]
         [c](x0) = [2 1]x0 + [0],
         
                   [0 0]     [0]
         [a](x0) = [2 1]x0 + [0]
        orientation:
         a#(a(x1)) = [2 1]x1 + [0] >= [0 0]x1 + [0] = b#(x1)
         
         a#(a(x1)) = [2 1]x1 + [0] >= [2 1]x1 + [0] = c#(b(x1))
         
         b#(a(x1)) = [2 1]x1 + [0] >= [2 1]x1 + [0] = a#(c(x1))
         
         c#(c(x1)) = [2 1]x1 + [0] >= [0 0]x1 = a#(x1)
         
         c#(c(x1)) = [2 1]x1 + [0] >= [2 1]x1 + [0] = b#(a(x1))
         
                    [2 1]     [0]    [2 1]     [0]           
         a(a(x1)) = [3 2]x1 + [2] >= [3 2]x1 + [2] = c(b(x1))
         
                    [2 1]     [0]    [2 1]     [0]           
         b(a(x1)) = [3 2]x1 + [2] >= [3 2]x1 + [2] = a(c(x1))
         
                    [2 1]     [0]    [2 1]     [0]           
         c(c(x1)) = [3 2]x1 + [2] >= [3 2]x1 + [2] = b(a(x1))
        problem:
         DPs:
          a#(a(x1)) -> b#(x1)
          a#(a(x1)) -> c#(b(x1))
          b#(a(x1)) -> a#(c(x1))
          c#(c(x1)) -> b#(a(x1))
         TRS:
          a(a(x1)) -> c(b(x1))
          b(a(x1)) -> a(c(x1))
          c(c(x1)) -> b(a(x1))
        Restore Modifier:
         DPs:
          a#(a(x1)) -> b#(x1)
          a#(a(x1)) -> c#(b(x1))
          b#(a(x1)) -> a#(c(x1))
          c#(c(x1)) -> b#(a(x1))
         TRS:
          a(a(x1)) -> c(b(x1))
          b(a(x1)) -> a(c(x1))
          c(c(x1)) -> b(a(x1))
         EDG Processor:
          DPs:
           a#(a(x1)) -> b#(x1)
           a#(a(x1)) -> c#(b(x1))
           b#(a(x1)) -> a#(c(x1))
           c#(c(x1)) -> b#(a(x1))
          TRS:
           a(a(x1)) -> c(b(x1))
           b(a(x1)) -> a(c(x1))
           c(c(x1)) -> b(a(x1))
          graph:
           c#(c(x1)) -> b#(a(x1)) -> b#(a(x1)) -> a#(c(x1))
           b#(a(x1)) -> a#(c(x1)) -> a#(a(x1)) -> c#(b(x1))
           b#(a(x1)) -> a#(c(x1)) -> a#(a(x1)) -> b#(x1)
           a#(a(x1)) -> c#(b(x1)) -> c#(c(x1)) -> b#(a(x1))
           a#(a(x1)) -> b#(x1) -> b#(a(x1)) -> a#(c(x1))
          Arctic Interpretation Processor:
           dimension: 3
           usable rules:
            a(a(x1)) -> c(b(x1))
            b(a(x1)) -> a(c(x1))
            c(c(x1)) -> b(a(x1))
           interpretation:
            [c#](x0) = [1 1 0]x0 + [0],
            
            [b#](x0) = [1 1 0]x0 + [0],
            
            [a#](x0) = [1 1 0]x0 + [0],
            
                      [0  1  -&]     [-&]
            [b](x0) = [1  0  0 ]x0 + [-&]
                      [0  0  1 ]     [0 ],
            
                      [0  1  0 ]     [0]
            [c](x0) = [1  -& 0 ]x0 + [0]
                      [0  -& 1 ]     [1],
            
                      [-& 1  0 ]     [0]
            [a](x0) = [1  0  0 ]x0 + [0]
                      [-& 0  1 ]     [1]
           orientation:
            a#(a(x1)) = [2 2 1]x1 + [1] >= [1 1 0]x1 + [0] = b#(x1)
            
            a#(a(x1)) = [2 2 1]x1 + [1] >= [2 2 1]x1 + [0] = c#(b(x1))
            
            b#(a(x1)) = [2 2 1]x1 + [1] >= [2 2 1]x1 + [1] = a#(c(x1))
            
            c#(c(x1)) = [2 2 1]x1 + [1] >= [2 2 1]x1 + [1] = b#(a(x1))
            
                       [2 1 1]     [1]    [2 1 1]     [0]           
            a(a(x1)) = [1 2 1]x1 + [1] >= [1 2 1]x1 + [0] = c(b(x1))
                       [1 1 2]     [2]    [1 1 2]     [1]           
            
                       [2 1 1]     [1]    [2  -& 1 ]     [1]           
            b(a(x1)) = [1 2 1]x1 + [1] >= [1  2  1 ]x1 + [1] = a(c(x1))
                       [1 1 2]     [2]    [1  -& 2 ]     [2]           
            
                       [2 1 1]     [1]    [2 1 1]     [1]           
            c(c(x1)) = [1 2 1]x1 + [1] >= [1 2 1]x1 + [1] = b(a(x1))
                       [1 1 2]     [2]    [1 1 2]     [2]           
           problem:
            DPs:
             a#(a(x1)) -> c#(b(x1))
             b#(a(x1)) -> a#(c(x1))
             c#(c(x1)) -> b#(a(x1))
            TRS:
             a(a(x1)) -> c(b(x1))
             b(a(x1)) -> a(c(x1))
             c(c(x1)) -> b(a(x1))
           Restore Modifier:
            DPs:
             a#(a(x1)) -> c#(b(x1))
             b#(a(x1)) -> a#(c(x1))
             c#(c(x1)) -> b#(a(x1))
            TRS:
             a(a(x1)) -> c(b(x1))
             b(a(x1)) -> a(c(x1))
             c(c(x1)) -> b(a(x1))
            EDG Processor:
             DPs:
              a#(a(x1)) -> c#(b(x1))
              b#(a(x1)) -> a#(c(x1))
              c#(c(x1)) -> b#(a(x1))
             TRS:
              a(a(x1)) -> c(b(x1))
              b(a(x1)) -> a(c(x1))
              c(c(x1)) -> b(a(x1))
             graph:
              c#(c(x1)) -> b#(a(x1)) -> b#(a(x1)) -> a#(c(x1))
              b#(a(x1)) -> a#(c(x1)) -> a#(a(x1)) -> c#(b(x1))
              a#(a(x1)) -> c#(b(x1)) -> c#(c(x1)) -> b#(a(x1))
             Arctic Interpretation Processor:
              dimension: 3
              usable rules:
               a(a(x1)) -> c(b(x1))
               b(a(x1)) -> a(c(x1))
               c(c(x1)) -> b(a(x1))
              interpretation:
               [c#](x0) = [1 0 0]x0 + [1],
               
               [b#](x0) = [0 1 0]x0 + [1],
               
               [a#](x0) = [0  -& 0 ]x0 + [0],
               
                         [0  0  -&]     [-&]
               [b](x0) = [0  1  0 ]x0 + [0 ]
                         [0  1  0 ]     [0 ],
               
                         [1 0 0]     [0]
               [c](x0) = [1 0 0]x0 + [0]
                         [1 0 0]     [0],
               
                         [0 0 0]     [0]
               [a](x0) = [1 0 0]x0 + [0]
                         [1 1 0]     [1]
              orientation:
               a#(a(x1)) = [1 1 0]x1 + [1] >= [1 1 0]x1 + [1] = c#(b(x1))
               
               b#(a(x1)) = [2 1 1]x1 + [1] >= [1 0 0]x1 + [0] = a#(c(x1))
               
               c#(c(x1)) = [2 1 1]x1 + [1] >= [2 1 1]x1 + [1] = b#(a(x1))
               
                          [1 1 0]     [1]    [1 1 0]     [0]           
               a(a(x1)) = [1 1 1]x1 + [1] >= [1 1 0]x1 + [0] = c(b(x1))
                          [2 1 1]     [1]    [1 1 0]     [0]           
               
                          [1 0 0]     [0]    [1 0 0]     [0]           
               b(a(x1)) = [2 1 1]x1 + [1] >= [2 1 1]x1 + [1] = a(c(x1))
                          [2 1 1]     [1]    [2 1 1]     [1]           
               
                          [2 1 1]     [1]    [1 0 0]     [0]           
               c(c(x1)) = [2 1 1]x1 + [1] >= [2 1 1]x1 + [1] = b(a(x1))
                          [2 1 1]     [1]    [2 1 1]     [1]           
              problem:
               DPs:
                a#(a(x1)) -> c#(b(x1))
                c#(c(x1)) -> b#(a(x1))
               TRS:
                a(a(x1)) -> c(b(x1))
                b(a(x1)) -> a(c(x1))
                c(c(x1)) -> b(a(x1))
              Restore Modifier:
               DPs:
                a#(a(x1)) -> c#(b(x1))
                c#(c(x1)) -> b#(a(x1))
               TRS:
                a(a(x1)) -> c(b(x1))
                b(a(x1)) -> a(c(x1))
                c(c(x1)) -> b(a(x1))
               EDG Processor:
                DPs:
                 a#(a(x1)) -> c#(b(x1))
                 c#(c(x1)) -> b#(a(x1))
                TRS:
                 a(a(x1)) -> c(b(x1))
                 b(a(x1)) -> a(c(x1))
                 c(c(x1)) -> b(a(x1))
                graph:
                 a#(a(x1)) -> c#(b(x1)) -> c#(c(x1)) -> b#(a(x1))
                SCC Processor:
                 #sccs: 0
                 #rules: 0
                 #arcs: 1/4