YES

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

Proof:
 String Reversal Processor:
  a(b(a(x1))) -> c(b(x1))
  b(b(b(x1))) -> b(c(x1))
  c(x1) -> b(a(x1))
  d(c(x1)) -> a(b(c(d(x1))))
  Matrix Interpretation Processor: dim=3
   
   interpretation:
              [1 0 1]     [0]
    [d](x0) = [1 1 1]x0 + [1]
              [1 1 1]     [0],
    
              [1 0 0]     [0]
    [c](x0) = [1 1 1]x0 + [1]
              [1 0 1]     [1],
    
              [1 0 0]  
    [b](x0) = [1 0 0]x0
              [0 0 0]  ,
    
              [1 0 0]     [0]
    [a](x0) = [1 1 0]x0 + [1]
              [1 1 0]     [1]
   orientation:
                  [1 0 0]     [0]    [1 0 0]     [0]           
    a(b(a(x1))) = [2 0 0]x1 + [1] >= [2 0 0]x1 + [1] = c(b(x1))
                  [2 0 0]     [1]    [1 0 0]     [1]           
    
                  [1 0 0]      [1 0 0]             
    b(b(b(x1))) = [1 0 0]x1 >= [1 0 0]x1 = b(c(x1))
                  [0 0 0]      [0 0 0]             
    
            [1 0 0]     [0]    [1 0 0]             
    c(x1) = [1 1 1]x1 + [1] >= [1 0 0]x1 = b(a(x1))
            [1 0 1]     [1]    [0 0 0]             
    
               [2 0 1]     [1]    [1 0 1]     [0]                 
    d(c(x1)) = [3 1 2]x1 + [3] >= [2 0 2]x1 + [1] = a(b(c(d(x1))))
               [3 1 2]     [2]    [2 0 2]     [1]                 
   problem:
    a(b(a(x1))) -> c(b(x1))
    b(b(b(x1))) -> b(c(x1))
    c(x1) -> b(a(x1))
   String Reversal Processor:
    a(b(a(x1))) -> b(c(x1))
    b(b(b(x1))) -> c(b(x1))
    c(x1) -> a(b(x1))
    DP Processor:
     DPs:
      a#(b(a(x1))) -> c#(x1)
      a#(b(a(x1))) -> b#(c(x1))
      b#(b(b(x1))) -> c#(b(x1))
      c#(x1) -> b#(x1)
      c#(x1) -> a#(b(x1))
     TRS:
      a(b(a(x1))) -> b(c(x1))
      b(b(b(x1))) -> c(b(x1))
      c(x1) -> a(b(x1))
     TDG Processor:
      DPs:
       a#(b(a(x1))) -> c#(x1)
       a#(b(a(x1))) -> b#(c(x1))
       b#(b(b(x1))) -> c#(b(x1))
       c#(x1) -> b#(x1)
       c#(x1) -> a#(b(x1))
      TRS:
       a(b(a(x1))) -> b(c(x1))
       b(b(b(x1))) -> c(b(x1))
       c(x1) -> a(b(x1))
      graph:
       b#(b(b(x1))) -> c#(b(x1)) -> c#(x1) -> a#(b(x1))
       b#(b(b(x1))) -> c#(b(x1)) -> c#(x1) -> b#(x1)
       c#(x1) -> b#(x1) -> b#(b(b(x1))) -> c#(b(x1))
       c#(x1) -> a#(b(x1)) -> a#(b(a(x1))) -> b#(c(x1))
       c#(x1) -> a#(b(x1)) -> a#(b(a(x1))) -> c#(x1)
       a#(b(a(x1))) -> b#(c(x1)) -> b#(b(b(x1))) -> c#(b(x1))
       a#(b(a(x1))) -> c#(x1) -> c#(x1) -> a#(b(x1))
       a#(b(a(x1))) -> c#(x1) -> c#(x1) -> b#(x1)
      Arctic Interpretation Processor:
       dimension: 1
       interpretation:
        [b#](x0) = -2x0 + 0,
        
        [c#](x0) = x0 + 1,
        
        [a#](x0) = -2x0 + 0,
        
        [c](x0) = 4x0 + 3,
        
        [b](x0) = 2x0 + 1,
        
        [a](x0) = 2x0 + 1
       orientation:
        a#(b(a(x1))) = 2x1 + 1 >= x1 + 1 = c#(x1)
        
        a#(b(a(x1))) = 2x1 + 1 >= 2x1 + 1 = b#(c(x1))
        
        b#(b(b(x1))) = 2x1 + 1 >= 2x1 + 1 = c#(b(x1))
        
        c#(x1) = x1 + 1 >= -2x1 + 0 = b#(x1)
        
        c#(x1) = x1 + 1 >= x1 + 0 = a#(b(x1))
        
        a(b(a(x1))) = 6x1 + 5 >= 6x1 + 5 = b(c(x1))
        
        b(b(b(x1))) = 6x1 + 5 >= 6x1 + 5 = c(b(x1))
        
        c(x1) = 4x1 + 3 >= 4x1 + 3 = a(b(x1))
       problem:
        DPs:
         a#(b(a(x1))) -> c#(x1)
         a#(b(a(x1))) -> b#(c(x1))
         b#(b(b(x1))) -> c#(b(x1))
         c#(x1) -> a#(b(x1))
        TRS:
         a(b(a(x1))) -> b(c(x1))
         b(b(b(x1))) -> c(b(x1))
         c(x1) -> a(b(x1))
       EDG Processor:
        DPs:
         a#(b(a(x1))) -> c#(x1)
         a#(b(a(x1))) -> b#(c(x1))
         b#(b(b(x1))) -> c#(b(x1))
         c#(x1) -> a#(b(x1))
        TRS:
         a(b(a(x1))) -> b(c(x1))
         b(b(b(x1))) -> c(b(x1))
         c(x1) -> a(b(x1))
        graph:
         b#(b(b(x1))) -> c#(b(x1)) -> c#(x1) -> a#(b(x1))
         c#(x1) -> a#(b(x1)) -> a#(b(a(x1))) -> c#(x1)
         c#(x1) -> a#(b(x1)) -> a#(b(a(x1))) -> b#(c(x1))
         a#(b(a(x1))) -> b#(c(x1)) -> b#(b(b(x1))) -> c#(b(x1))
         a#(b(a(x1))) -> c#(x1) -> c#(x1) -> a#(b(x1))
        Arctic Interpretation Processor:
         dimension: 2
         interpretation:
          [b#](x0) = [-& 0 ]x0 + [2],
          
          [c#](x0) = [-& 1 ]x0 + [0],
          
          [a#](x0) = [-& 0 ]x0 + [0],
          
                    [0  1 ]     [1]
          [c](x0) = [-& 2 ]x0 + [1],
          
                    [0  0 ]     [0]
          [b](x0) = [-& 1 ]x0 + [0],
          
                    [0  0 ]     [1]
          [a](x0) = [-& 1 ]x0 + [1]
         orientation:
          a#(b(a(x1))) = [-& 2 ]x1 + [2] >= [-& 1 ]x1 + [0] = c#(x1)
          
          a#(b(a(x1))) = [-& 2 ]x1 + [2] >= [-& 2 ]x1 + [2] = b#(c(x1))
          
          b#(b(b(x1))) = [-& 2 ]x1 + [2] >= [-& 2 ]x1 + [1] = c#(b(x1))
          
          c#(x1) = [-& 1 ]x1 + [0] >= [-& 1 ]x1 + [0] = a#(b(x1))
          
                        [0  2 ]     [2]    [0  2 ]     [1]           
          a(b(a(x1))) = [-& 3 ]x1 + [3] >= [-& 3 ]x1 + [2] = b(c(x1))
          
                        [0  2 ]     [1]    [0  2 ]     [1]           
          b(b(b(x1))) = [-& 3 ]x1 + [2] >= [-& 3 ]x1 + [2] = c(b(x1))
          
                  [0  1 ]     [1]    [0  1 ]     [1]           
          c(x1) = [-& 2 ]x1 + [1] >= [-& 2 ]x1 + [1] = a(b(x1))
         problem:
          DPs:
           a#(b(a(x1))) -> b#(c(x1))
           b#(b(b(x1))) -> c#(b(x1))
           c#(x1) -> a#(b(x1))
          TRS:
           a(b(a(x1))) -> b(c(x1))
           b(b(b(x1))) -> c(b(x1))
           c(x1) -> a(b(x1))
         EDG Processor:
          DPs:
           a#(b(a(x1))) -> b#(c(x1))
           b#(b(b(x1))) -> c#(b(x1))
           c#(x1) -> a#(b(x1))
          TRS:
           a(b(a(x1))) -> b(c(x1))
           b(b(b(x1))) -> c(b(x1))
           c(x1) -> a(b(x1))
          graph:
           b#(b(b(x1))) -> c#(b(x1)) -> c#(x1) -> a#(b(x1))
           c#(x1) -> a#(b(x1)) -> a#(b(a(x1))) -> b#(c(x1))
           a#(b(a(x1))) -> b#(c(x1)) -> b#(b(b(x1))) -> c#(b(x1))
          Arctic Interpretation Processor:
           dimension: 3
           interpretation:
            [b#](x0) = [1 1 0]x0 + [0],
            
            [c#](x0) = [1 1 1]x0 + [1],
            
            [a#](x0) = [0 1 1]x0 + [0],
            
                      [0 0 1]     [0]
            [c](x0) = [0 0 1]x0 + [0]
                      [1 0 0]     [1],
            
                      [1  0  0 ]     [1 ]
            [b](x0) = [-& -& 0 ]x0 + [-&]
                      [0  -& -&]     [0 ],
            
                      [-& 1  0 ]     [0]
            [a](x0) = [-& 1  0 ]x0 + [0]
                      [0  0  1 ]     [0]
           orientation:
            a#(b(a(x1))) = [1 2 2]x1 + [1] >= [1 1 2]x1 + [1] = b#(c(x1))
            
            b#(b(b(x1))) = [3 2 2]x1 + [3] >= [2 1 1]x1 + [2] = c#(b(x1))
            
            c#(x1) = [1 1 1]x1 + [1] >= [1 0 1]x1 + [1] = a#(b(x1))
            
                          [1 1 2]     [1]    [1 1 2]     [1]           
            a(b(a(x1))) = [1 1 2]x1 + [1] >= [1 0 0]x1 + [1] = b(c(x1))
                          [0 2 1]     [1]    [0 0 1]     [0]           
            
                          [3 2 2]     [3]    [1 0 0]     [1]           
            b(b(b(x1))) = [1 0 0]x1 + [1] >= [1 0 0]x1 + [1] = c(b(x1))
                          [2 1 1]     [2]    [2 1 1]     [2]           
            
                    [0 0 1]     [0]    [0  -& 1 ]     [0]           
            c(x1) = [0 0 1]x1 + [0] >= [0  -& 1 ]x1 + [0] = a(b(x1))
                    [1 0 0]     [1]    [1  0  0 ]     [1]           
           problem:
            DPs:
             a#(b(a(x1))) -> b#(c(x1))
             c#(x1) -> a#(b(x1))
            TRS:
             a(b(a(x1))) -> b(c(x1))
             b(b(b(x1))) -> c(b(x1))
             c(x1) -> a(b(x1))
           EDG Processor:
            DPs:
             a#(b(a(x1))) -> b#(c(x1))
             c#(x1) -> a#(b(x1))
            TRS:
             a(b(a(x1))) -> b(c(x1))
             b(b(b(x1))) -> c(b(x1))
             c(x1) -> a(b(x1))
            graph:
             c#(x1) -> a#(b(x1)) -> a#(b(a(x1))) -> b#(c(x1))
            SCC Processor:
             #sccs: 0
             #rules: 0
             #arcs: 1/4