YES

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

Proof:
 String Reversal Processor:
  a(x1) -> b(x1)
  a(x1) -> c(x1)
  b(a(x1)) -> c(a(b(x1)))
  c(c(x1)) -> a(x1)
  DP Processor:
   DPs:
    a#(x1) -> b#(x1)
    a#(x1) -> c#(x1)
    b#(a(x1)) -> b#(x1)
    b#(a(x1)) -> a#(b(x1))
    b#(a(x1)) -> c#(a(b(x1)))
    c#(c(x1)) -> a#(x1)
   TRS:
    a(x1) -> b(x1)
    a(x1) -> c(x1)
    b(a(x1)) -> c(a(b(x1)))
    c(c(x1)) -> a(x1)
   TDG Processor:
    DPs:
     a#(x1) -> b#(x1)
     a#(x1) -> c#(x1)
     b#(a(x1)) -> b#(x1)
     b#(a(x1)) -> a#(b(x1))
     b#(a(x1)) -> c#(a(b(x1)))
     c#(c(x1)) -> a#(x1)
    TRS:
     a(x1) -> b(x1)
     a(x1) -> c(x1)
     b(a(x1)) -> c(a(b(x1)))
     c(c(x1)) -> a(x1)
    graph:
     c#(c(x1)) -> a#(x1) -> a#(x1) -> c#(x1)
     c#(c(x1)) -> a#(x1) -> a#(x1) -> b#(x1)
     b#(a(x1)) -> c#(a(b(x1))) -> c#(c(x1)) -> a#(x1)
     b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> c#(a(b(x1)))
     b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> a#(b(x1))
     b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> b#(x1)
     b#(a(x1)) -> a#(b(x1)) -> a#(x1) -> c#(x1)
     b#(a(x1)) -> a#(b(x1)) -> a#(x1) -> b#(x1)
     a#(x1) -> c#(x1) -> c#(c(x1)) -> a#(x1)
     a#(x1) -> b#(x1) -> b#(a(x1)) -> c#(a(b(x1)))
     a#(x1) -> b#(x1) -> b#(a(x1)) -> a#(b(x1))
     a#(x1) -> b#(x1) -> b#(a(x1)) -> b#(x1)
    Arctic Interpretation Processor:
     dimension: 2
     usable rules:
      a(x1) -> b(x1)
      a(x1) -> c(x1)
      b(a(x1)) -> c(a(b(x1)))
      c(c(x1)) -> a(x1)
     interpretation:
      [b#](x0) = [0  -&]x0 + [0],
      
                [0  -&]     [-&]
      [b](x0) = [2  0 ]x0 + [0 ],
      
      [a#](x0) = [0 0]x0 + [0],
      
      [c#](x0) = [0 0]x0 + [0],
      
                [2 0]     [1]
      [a](x0) = [2 0]x0 + [0],
      
                [-& 0 ]     [1]
      [c](x0) = [2  0 ]x0 + [0]
     orientation:
      a#(x1) = [0 0]x1 + [0] >= [0  -&]x1 + [0] = b#(x1)
      
      a#(x1) = [0 0]x1 + [0] >= [0 0]x1 + [0] = c#(x1)
      
      b#(a(x1)) = [2 0]x1 + [1] >= [0  -&]x1 + [0] = b#(x1)
      
      b#(a(x1)) = [2 0]x1 + [1] >= [2 0]x1 + [0] = a#(b(x1))
      
      b#(a(x1)) = [2 0]x1 + [1] >= [2 0]x1 + [1] = c#(a(b(x1)))
      
      c#(c(x1)) = [2 0]x1 + [1] >= [0 0]x1 + [0] = a#(x1)
      
              [2 0]     [1]    [0  -&]     [-&]        
      a(x1) = [2 0]x1 + [0] >= [2  0 ]x1 + [0 ] = b(x1)
      
              [2 0]     [1]    [-& 0 ]     [1]        
      a(x1) = [2 0]x1 + [0] >= [2  0 ]x1 + [0] = c(x1)
      
                 [2 0]     [1]    [2 0]     [1]              
      b(a(x1)) = [4 2]x1 + [3] >= [4 2]x1 + [3] = c(a(b(x1)))
      
                 [2 0]     [1]    [2 0]     [1]        
      c(c(x1)) = [2 2]x1 + [3] >= [2 0]x1 + [0] = a(x1)
     problem:
      DPs:
       a#(x1) -> b#(x1)
       a#(x1) -> c#(x1)
       b#(a(x1)) -> a#(b(x1))
       b#(a(x1)) -> c#(a(b(x1)))
       c#(c(x1)) -> a#(x1)
      TRS:
       a(x1) -> b(x1)
       a(x1) -> c(x1)
       b(a(x1)) -> c(a(b(x1)))
       c(c(x1)) -> a(x1)
     Restore Modifier:
      DPs:
       a#(x1) -> b#(x1)
       a#(x1) -> c#(x1)
       b#(a(x1)) -> a#(b(x1))
       b#(a(x1)) -> c#(a(b(x1)))
       c#(c(x1)) -> a#(x1)
      TRS:
       a(x1) -> b(x1)
       a(x1) -> c(x1)
       b(a(x1)) -> c(a(b(x1)))
       c(c(x1)) -> a(x1)
      EDG Processor:
       DPs:
        a#(x1) -> b#(x1)
        a#(x1) -> c#(x1)
        b#(a(x1)) -> a#(b(x1))
        b#(a(x1)) -> c#(a(b(x1)))
        c#(c(x1)) -> a#(x1)
       TRS:
        a(x1) -> b(x1)
        a(x1) -> c(x1)
        b(a(x1)) -> c(a(b(x1)))
        c(c(x1)) -> a(x1)
       graph:
        c#(c(x1)) -> a#(x1) -> a#(x1) -> b#(x1)
        c#(c(x1)) -> a#(x1) -> a#(x1) -> c#(x1)
        b#(a(x1)) -> c#(a(b(x1))) -> c#(c(x1)) -> a#(x1)
        b#(a(x1)) -> a#(b(x1)) -> a#(x1) -> b#(x1)
        b#(a(x1)) -> a#(b(x1)) -> a#(x1) -> c#(x1)
        a#(x1) -> c#(x1) -> c#(c(x1)) -> a#(x1)
        a#(x1) -> b#(x1) -> b#(a(x1)) -> a#(b(x1))
        a#(x1) -> b#(x1) -> b#(a(x1)) -> c#(a(b(x1)))
       Arctic Interpretation Processor:
        dimension: 2
        usable rules:
         a(x1) -> b(x1)
         a(x1) -> c(x1)
         b(a(x1)) -> c(a(b(x1)))
         c(c(x1)) -> a(x1)
        interpretation:
         [b#](x0) = [0 0]x0,
         
                   [0  -&]  
         [b](x0) = [0  0 ]x0,
         
         [a#](x0) = [0 1]x0,
         
         [c#](x0) = [0 1]x0,
         
                   [2 2]     [0 ]
         [a](x0) = [0 0]x0 + [-&],
         
                   [-& 2 ]     [0 ]
         [c](x0) = [0  0 ]x0 + [-&]
        orientation:
         a#(x1) = [0 1]x1 >= [0 0]x1 = b#(x1)
         
         a#(x1) = [0 1]x1 >= [0 1]x1 = c#(x1)
         
         b#(a(x1)) = [2 2]x1 + [0] >= [1 1]x1 = a#(b(x1))
         
         b#(a(x1)) = [2 2]x1 + [0] >= [2 2]x1 + [0] = c#(a(b(x1)))
         
         c#(c(x1)) = [1 2]x1 + [0] >= [0 1]x1 = a#(x1)
         
                 [2 2]     [0 ]    [0  -&]          
         a(x1) = [0 0]x1 + [-&] >= [0  0 ]x1 = b(x1)
         
                 [2 2]     [0 ]    [-& 2 ]     [0 ]        
         a(x1) = [0 0]x1 + [-&] >= [0  0 ]x1 + [-&] = c(x1)
         
                    [2 2]     [0]    [2 2]     [0]              
         b(a(x1)) = [2 2]x1 + [0] >= [2 2]x1 + [0] = c(a(b(x1)))
         
                    [2 2]     [0]    [2 2]     [0 ]        
         c(c(x1)) = [0 2]x1 + [0] >= [0 0]x1 + [-&] = a(x1)
        problem:
         DPs:
          a#(x1) -> b#(x1)
          a#(x1) -> c#(x1)
          b#(a(x1)) -> c#(a(b(x1)))
         TRS:
          a(x1) -> b(x1)
          a(x1) -> c(x1)
          b(a(x1)) -> c(a(b(x1)))
          c(c(x1)) -> a(x1)
        Restore Modifier:
         DPs:
          a#(x1) -> b#(x1)
          a#(x1) -> c#(x1)
          b#(a(x1)) -> c#(a(b(x1)))
         TRS:
          a(x1) -> b(x1)
          a(x1) -> c(x1)
          b(a(x1)) -> c(a(b(x1)))
          c(c(x1)) -> a(x1)
         EDG Processor:
          DPs:
           a#(x1) -> b#(x1)
           a#(x1) -> c#(x1)
           b#(a(x1)) -> c#(a(b(x1)))
          TRS:
           a(x1) -> b(x1)
           a(x1) -> c(x1)
           b(a(x1)) -> c(a(b(x1)))
           c(c(x1)) -> a(x1)
          graph:
           a#(x1) -> b#(x1) -> b#(a(x1)) -> c#(a(b(x1)))
          SCC Processor:
           #sccs: 0
           #rules: 0
           #arcs: 1/9