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
     interpretation:
      [c#](x0) = [1 2]x0,
      
      [b#](x0) = [0 2]x0,
      
      [a#](x0) = [1 2]x0,
      
                [0  1 ]     [-&]
      [c](x0) = [1  -&]x0 + [2 ],
      
                [0  1 ]  
      [b](x0) = [-& 0 ]x0,
      
                [0 1]     [-&]
      [a](x0) = [1 2]x0 + [2 ]
     orientation:
      a#(x1) = [1 2]x1 >= [0 2]x1 = b#(x1)
      
      a#(x1) = [1 2]x1 >= [1 2]x1 = c#(x1)
      
      b#(a(x1)) = [3 4]x1 + [4] >= [0 2]x1 = b#(x1)
      
      b#(a(x1)) = [3 4]x1 + [4] >= [1 2]x1 = a#(b(x1))
      
      b#(a(x1)) = [3 4]x1 + [4] >= [3 4]x1 + [4] = c#(a(b(x1)))
      
      c#(c(x1)) = [3 2]x1 + [4] >= [1 2]x1 = a#(x1)
      
              [0 1]     [-&]    [0  1 ]          
      a(x1) = [1 2]x1 + [2 ] >= [-& 0 ]x1 = b(x1)
      
              [0 1]     [-&]    [0  1 ]     [-&]        
      a(x1) = [1 2]x1 + [2 ] >= [1  -&]x1 + [2 ] = c(x1)
      
                 [2 3]     [3]    [2 3]     [3]              
      b(a(x1)) = [1 2]x1 + [2] >= [1 2]x1 + [2] = c(a(b(x1)))
      
                 [2 1]     [3]    [0 1]     [-&]        
      c(c(x1)) = [1 2]x1 + [2] >= [1 2]x1 + [2 ] = a(x1)
     problem:
      DPs:
       a#(x1) -> b#(x1)
       a#(x1) -> c#(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)) -> 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)
       a#(x1) -> c#(x1) -> c#(c(x1)) -> a#(x1)
       a#(x1) -> b#(x1) -> b#(a(x1)) -> c#(a(b(x1)))
      Arctic Interpretation Processor:
       dimension: 2
       interpretation:
        [c#](x0) = [0 0]x0,
        
        [b#](x0) = [0  -&]x0,
        
        [a#](x0) = [2 0]x0,
        
                  [0  0 ]     [0 ]
        [c](x0) = [2  -&]x0 + [-&],
        
                  [0  -&]  
        [b](x0) = [2  -&]x0,
        
                  [2 0]     [0 ]
        [a](x0) = [2 0]x0 + [-&]
       orientation:
        a#(x1) = [2 0]x1 >= [0  -&]x1 = b#(x1)
        
        a#(x1) = [2 0]x1 >= [0 0]x1 = c#(x1)
        
        b#(a(x1)) = [2 0]x1 + [0] >= [2  -&]x1 + [0] = c#(a(b(x1)))
        
        c#(c(x1)) = [2 0]x1 + [0] >= [2 0]x1 = a#(x1)
        
                [2 0]     [0 ]    [0  -&]          
        a(x1) = [2 0]x1 + [-&] >= [2  -&]x1 = b(x1)
        
                [2 0]     [0 ]    [0  0 ]     [0 ]        
        a(x1) = [2 0]x1 + [-&] >= [2  -&]x1 + [-&] = c(x1)
        
                   [2 0]     [0]    [2  -&]     [0]              
        b(a(x1)) = [4 2]x1 + [2] >= [4  -&]x1 + [2] = c(a(b(x1)))
        
                   [2 0]     [0]    [2 0]     [0 ]        
        c(c(x1)) = [2 2]x1 + [2] >= [2 0]x1 + [-&] = a(x1)
       problem:
        DPs:
         a#(x1) -> c#(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) -> c#(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)
         b#(a(x1)) -> c#(a(b(x1))) -> c#(c(x1)) -> a#(x1)
         a#(x1) -> c#(x1) -> c#(c(x1)) -> a#(x1)
        CDG Processor:
         DPs:
          a#(x1) -> c#(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)
          b#(a(x1)) -> c#(a(b(x1))) -> c#(c(x1)) -> a#(x1)
         SCC Processor:
          #sccs: 0
          #rules: 0
          #arcs: 2/9