YES

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

Proof:
 String Reversal Processor:
  a(a(x1)) -> b(x1)
  b(a(x1)) -> a(c(x1))
  c(b(x1)) -> a(b(c(x1)))
  DP Processor:
   DPs:
    a#(a(x1)) -> b#(x1)
    b#(a(x1)) -> c#(x1)
    b#(a(x1)) -> a#(c(x1))
    c#(b(x1)) -> c#(x1)
    c#(b(x1)) -> b#(c(x1))
    c#(b(x1)) -> a#(b(c(x1)))
   TRS:
    a(a(x1)) -> b(x1)
    b(a(x1)) -> a(c(x1))
    c(b(x1)) -> a(b(c(x1)))
   TDG Processor:
    DPs:
     a#(a(x1)) -> b#(x1)
     b#(a(x1)) -> c#(x1)
     b#(a(x1)) -> a#(c(x1))
     c#(b(x1)) -> c#(x1)
     c#(b(x1)) -> b#(c(x1))
     c#(b(x1)) -> a#(b(c(x1)))
    TRS:
     a(a(x1)) -> b(x1)
     b(a(x1)) -> a(c(x1))
     c(b(x1)) -> a(b(c(x1)))
    graph:
     c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> a#(b(c(x1)))
     c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> b#(c(x1))
     c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> c#(x1)
     c#(b(x1)) -> b#(c(x1)) -> b#(a(x1)) -> a#(c(x1))
     c#(b(x1)) -> b#(c(x1)) -> b#(a(x1)) -> c#(x1)
     c#(b(x1)) -> a#(b(c(x1))) -> a#(a(x1)) -> b#(x1)
     b#(a(x1)) -> c#(x1) -> c#(b(x1)) -> a#(b(c(x1)))
     b#(a(x1)) -> c#(x1) -> c#(b(x1)) -> b#(c(x1))
     b#(a(x1)) -> c#(x1) -> c#(b(x1)) -> c#(x1)
     b#(a(x1)) -> a#(c(x1)) -> a#(a(x1)) -> b#(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
     interpretation:
      [c#](x0) = [0 1]x0 + [-3],
      
      [b#](x0) = [0  -4]x0 + [-3],
      
      [a#](x0) = [0 0]x0 + [-4],
      
                [0  -&]     [-&]
      [c](x0) = [0  0 ]x0 + [-3],
      
                [1  1 ]     [-2]
      [b](x0) = [-& 0 ]x0 + [-&],
      
                [0  1 ]     [-&]
      [a](x0) = [0  -&]x0 + [-3]
     orientation:
      a#(a(x1)) = [0 1]x1 + [-3] >= [0  -4]x1 + [-3] = b#(x1)
      
      b#(a(x1)) = [0 1]x1 + [-3] >= [0 1]x1 + [-3] = c#(x1)
      
      b#(a(x1)) = [0 1]x1 + [-3] >= [0 0]x1 + [-3] = a#(c(x1))
      
      c#(b(x1)) = [1 1]x1 + [-2] >= [0 1]x1 + [-3] = c#(x1)
      
      c#(b(x1)) = [1 1]x1 + [-2] >= [0  -4]x1 + [-3] = b#(c(x1))
      
      c#(b(x1)) = [1 1]x1 + [-2] >= [1 1]x1 + [-2] = a#(b(c(x1)))
      
                 [1 1]     [-2]    [1  1 ]     [-2]        
      a(a(x1)) = [0 1]x1 + [-3] >= [-& 0 ]x1 + [-&] = b(x1)
      
                 [1  2 ]     [-2]    [1  1 ]     [-2]           
      b(a(x1)) = [0  -&]x1 + [-3] >= [0  -&]x1 + [-3] = a(c(x1))
      
                 [1 1]     [-2]    [1 1]     [-2]              
      c(b(x1)) = [1 1]x1 + [-2] >= [1 1]x1 + [-2] = a(b(c(x1)))
     problem:
      DPs:
       a#(a(x1)) -> b#(x1)
       b#(a(x1)) -> c#(x1)
       b#(a(x1)) -> a#(c(x1))
       c#(b(x1)) -> c#(x1)
       c#(b(x1)) -> a#(b(c(x1)))
      TRS:
       a(a(x1)) -> b(x1)
       b(a(x1)) -> a(c(x1))
       c(b(x1)) -> a(b(c(x1)))
     EDG Processor:
      DPs:
       a#(a(x1)) -> b#(x1)
       b#(a(x1)) -> c#(x1)
       b#(a(x1)) -> a#(c(x1))
       c#(b(x1)) -> c#(x1)
       c#(b(x1)) -> a#(b(c(x1)))
      TRS:
       a(a(x1)) -> b(x1)
       b(a(x1)) -> a(c(x1))
       c(b(x1)) -> a(b(c(x1)))
      graph:
       c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> c#(x1)
       c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> a#(b(c(x1)))
       c#(b(x1)) -> a#(b(c(x1))) -> a#(a(x1)) -> b#(x1)
       b#(a(x1)) -> c#(x1) -> c#(b(x1)) -> c#(x1)
       b#(a(x1)) -> c#(x1) -> c#(b(x1)) -> a#(b(c(x1)))
       b#(a(x1)) -> a#(c(x1)) -> a#(a(x1)) -> b#(x1)
       a#(a(x1)) -> b#(x1) -> b#(a(x1)) -> c#(x1)
       a#(a(x1)) -> b#(x1) -> b#(a(x1)) -> a#(c(x1))
      Arctic Interpretation Processor:
       dimension: 2
       interpretation:
        [c#](x0) = [0 0]x0,
        
        [b#](x0) = [0  -&]x0 + [1],
        
        [a#](x0) = [0 0]x0 + [0],
        
                  [0  0 ]     [0]
        [c](x0) = [-& 0 ]x0 + [1],
        
                  [-& 0 ]     [0]
        [b](x0) = [0  2 ]x0 + [3],
        
                  [0 0]     [-&]
        [a](x0) = [2 0]x0 + [3 ]
       orientation:
        a#(a(x1)) = [2 0]x1 + [3] >= [0  -&]x1 + [1] = b#(x1)
        
        b#(a(x1)) = [0 0]x1 + [1] >= [0 0]x1 = c#(x1)
        
        b#(a(x1)) = [0 0]x1 + [1] >= [0 0]x1 + [1] = a#(c(x1))
        
        c#(b(x1)) = [0 2]x1 + [3] >= [0 0]x1 = c#(x1)
        
        c#(b(x1)) = [0 2]x1 + [3] >= [0 2]x1 + [3] = a#(b(c(x1)))
        
                   [2 0]     [3]    [-& 0 ]     [0]        
        a(a(x1)) = [2 2]x1 + [3] >= [0  2 ]x1 + [3] = b(x1)
        
                   [2 0]     [3]    [0 0]     [1]           
        b(a(x1)) = [4 2]x1 + [5] >= [2 2]x1 + [3] = a(c(x1))
        
                   [0 2]     [3]    [0 2]     [3]              
        c(b(x1)) = [0 2]x1 + [3] >= [0 2]x1 + [3] = a(b(c(x1)))
       problem:
        DPs:
         b#(a(x1)) -> c#(x1)
         b#(a(x1)) -> a#(c(x1))
         c#(b(x1)) -> c#(x1)
         c#(b(x1)) -> a#(b(c(x1)))
        TRS:
         a(a(x1)) -> b(x1)
         b(a(x1)) -> a(c(x1))
         c(b(x1)) -> a(b(c(x1)))
       EDG Processor:
        DPs:
         b#(a(x1)) -> c#(x1)
         b#(a(x1)) -> a#(c(x1))
         c#(b(x1)) -> c#(x1)
         c#(b(x1)) -> a#(b(c(x1)))
        TRS:
         a(a(x1)) -> b(x1)
         b(a(x1)) -> a(c(x1))
         c(b(x1)) -> a(b(c(x1)))
        graph:
         c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> a#(b(c(x1)))
         c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> c#(x1)
         b#(a(x1)) -> c#(x1) -> c#(b(x1)) -> a#(b(c(x1)))
         b#(a(x1)) -> c#(x1) -> c#(b(x1)) -> c#(x1)
        SCC Processor:
         #sccs: 1
         #rules: 1
         #arcs: 4/16
         DPs:
          c#(b(x1)) -> c#(x1)
         TRS:
          a(a(x1)) -> b(x1)
          b(a(x1)) -> a(c(x1))
          c(b(x1)) -> a(b(c(x1)))
         Arctic Interpretation Processor:
          dimension: 2
          interpretation:
           [c#](x0) = [1  -&]x0,
           
                     [0  -&]     [-&]
           [c](x0) = [0  -&]x0 + [0 ],
           
                     [1  -&]     [2]
           [b](x0) = [0  -&]x0 + [0],
           
                     [0 1]     [2]
           [a](x0) = [0 0]x0 + [2]
          orientation:
           c#(b(x1)) = [2  -&]x1 + [3] >= [1  -&]x1 = c#(x1)
           
                      [1 1]     [3]    [1  -&]     [2]        
           a(a(x1)) = [0 1]x1 + [2] >= [0  -&]x1 + [0] = b(x1)
           
                      [1 2]     [3]    [1  -&]     [2]           
           b(a(x1)) = [0 1]x1 + [2] >= [0  -&]x1 + [2] = a(c(x1))
           
                      [1  -&]     [2]    [1  -&]     [2]              
           c(b(x1)) = [1  -&]x1 + [2] >= [1  -&]x1 + [2] = a(b(c(x1)))
          problem:
           DPs:
            
           TRS:
            a(a(x1)) -> b(x1)
            b(a(x1)) -> a(c(x1))
            c(b(x1)) -> a(b(c(x1)))
          Qed