YES

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

Proof:
 String Reversal Processor:
  a(x1) -> x1
  b(b(a(a(x1)))) -> c(a(c(b(x1))))
  c(x1) -> a(b(x1))
  DP Processor:
   DPs:
    b#(b(a(a(x1)))) -> b#(x1)
    b#(b(a(a(x1)))) -> c#(b(x1))
    b#(b(a(a(x1)))) -> a#(c(b(x1)))
    b#(b(a(a(x1)))) -> c#(a(c(b(x1))))
    c#(x1) -> b#(x1)
    c#(x1) -> a#(b(x1))
   TRS:
    a(x1) -> x1
    b(b(a(a(x1)))) -> c(a(c(b(x1))))
    c(x1) -> a(b(x1))
   TDG Processor:
    DPs:
     b#(b(a(a(x1)))) -> b#(x1)
     b#(b(a(a(x1)))) -> c#(b(x1))
     b#(b(a(a(x1)))) -> a#(c(b(x1)))
     b#(b(a(a(x1)))) -> c#(a(c(b(x1))))
     c#(x1) -> b#(x1)
     c#(x1) -> a#(b(x1))
    TRS:
     a(x1) -> x1
     b(b(a(a(x1)))) -> c(a(c(b(x1))))
     c(x1) -> a(b(x1))
    graph:
     c#(x1) -> b#(x1) -> b#(b(a(a(x1)))) -> c#(a(c(b(x1))))
     c#(x1) -> b#(x1) -> b#(b(a(a(x1)))) -> a#(c(b(x1)))
     c#(x1) -> b#(x1) -> b#(b(a(a(x1)))) -> c#(b(x1))
     c#(x1) -> b#(x1) -> b#(b(a(a(x1)))) -> b#(x1)
     b#(b(a(a(x1)))) -> c#(b(x1)) -> c#(x1) -> a#(b(x1))
     b#(b(a(a(x1)))) -> c#(b(x1)) -> c#(x1) -> b#(x1)
     b#(b(a(a(x1)))) -> c#(a(c(b(x1)))) -> c#(x1) -> a#(b(x1))
     b#(b(a(a(x1)))) -> c#(a(c(b(x1)))) -> c#(x1) -> b#(x1)
     b#(b(a(a(x1)))) -> b#(x1) -> b#(b(a(a(x1)))) -> c#(a(c(b(x1))))
     b#(b(a(a(x1)))) -> b#(x1) -> b#(b(a(a(x1)))) -> a#(c(b(x1)))
     b#(b(a(a(x1)))) -> b#(x1) -> b#(b(a(a(x1)))) -> c#(b(x1))
     b#(b(a(a(x1)))) -> b#(x1) -> b#(b(a(a(x1)))) -> b#(x1)
    SCC Processor:
     #sccs: 1
     #rules: 4
     #arcs: 12/36
     DPs:
      c#(x1) -> b#(x1)
      b#(b(a(a(x1)))) -> b#(x1)
      b#(b(a(a(x1)))) -> c#(b(x1))
      b#(b(a(a(x1)))) -> c#(a(c(b(x1))))
     TRS:
      a(x1) -> x1
      b(b(a(a(x1)))) -> c(a(c(b(x1))))
      c(x1) -> a(b(x1))
     Arctic Interpretation Processor:
      dimension: 4
      usable rules:
       a(x1) -> x1
       b(b(a(a(x1)))) -> c(a(c(b(x1))))
       c(x1) -> a(b(x1))
      interpretation:
       [c#](x0) = [0 0 0 0]x0 + [0],
       
       [b#](x0) = [0  -& -& 0 ]x0 + [0],
       
                 [0 0 0 0]     [0]
                 [0 0 0 0]     [0]
       [c](x0) = [0 0 0 0]x0 + [0]
                 [0 0 1 0]     [1],
       
                 [-& -& 0  -&]     [0 ]
                 [0  0  0  0 ]     [-&]
       [b](x0) = [0  0  0  0 ]x0 + [0 ]
                 [0  0  0  0 ]     [0 ],
       
                 [0  0  0  0 ]     [0]
                 [0  0  0  0 ]     [0]
       [a](x0) = [0  0  0  -&]x0 + [0]
                 [1  0  0  0 ]     [1]
      orientation:
       c#(x1) = [0 0 0 0]x1 + [0] >= [0  -& -& 0 ]x1 + [0] = b#(x1)
       
       b#(b(a(a(x1)))) = [1 1 1 1]x1 + [1] >= [0  -& -& 0 ]x1 + [0] = b#(x1)
       
       b#(b(a(a(x1)))) = [1 1 1 1]x1 + [1] >= [0 0 0 0]x1 + [0] = c#(b(x1))
       
       b#(b(a(a(x1)))) = [1 1 1 1]x1 + [1] >= [1 1 1 1]x1 + [1] = c#(a(c(b(x1))))
       
               [0  0  0  0 ]     [0]           
               [0  0  0  0 ]     [0]           
       a(x1) = [0  0  0  -&]x1 + [0] >= x1 = x1
               [1  0  0  0 ]     [1]           
       
                        [1 1 1 1]     [1]    [1 1 1 1]     [1]                 
                        [1 1 1 1]     [1]    [1 1 1 1]     [1]                 
       b(b(a(a(x1)))) = [1 1 1 1]x1 + [1] >= [1 1 1 1]x1 + [1] = c(a(c(b(x1))))
                        [1 1 1 1]     [1]    [1 1 1 1]     [1]                 
       
               [0 0 0 0]     [0]    [0 0 0 0]     [0]           
               [0 0 0 0]     [0]    [0 0 0 0]     [0]           
       c(x1) = [0 0 0 0]x1 + [0] >= [0 0 0 0]x1 + [0] = a(b(x1))
               [0 0 1 0]     [1]    [0 0 1 0]     [1]           
      problem:
       DPs:
        c#(x1) -> b#(x1)
        b#(b(a(a(x1)))) -> c#(a(c(b(x1))))
       TRS:
        a(x1) -> x1
        b(b(a(a(x1)))) -> c(a(c(b(x1))))
        c(x1) -> a(b(x1))
      Restore Modifier:
       DPs:
        c#(x1) -> b#(x1)
        b#(b(a(a(x1)))) -> c#(a(c(b(x1))))
       TRS:
        a(x1) -> x1
        b(b(a(a(x1)))) -> c(a(c(b(x1))))
        c(x1) -> a(b(x1))
       EDG Processor:
        DPs:
         c#(x1) -> b#(x1)
         b#(b(a(a(x1)))) -> c#(a(c(b(x1))))
        TRS:
         a(x1) -> x1
         b(b(a(a(x1)))) -> c(a(c(b(x1))))
         c(x1) -> a(b(x1))
        graph:
         c#(x1) -> b#(x1) -> b#(b(a(a(x1)))) -> c#(a(c(b(x1))))
         b#(b(a(a(x1)))) -> c#(a(c(b(x1)))) -> c#(x1) -> b#(x1)
        Arctic Interpretation Processor:
         dimension: 3
         usable rules:
          a(x1) -> x1
          b(b(a(a(x1)))) -> c(a(c(b(x1))))
          c(x1) -> a(b(x1))
         interpretation:
          [c#](x0) = [-& 0  0 ]x0 + [0],
          
          [b#](x0) = [-& 0  0 ]x0 + [0],
          
                    [0  1  0 ]     [0]
          [c](x0) = [-& 0  0 ]x0 + [0]
                    [0  1  0 ]     [0],
          
                    [-& 0  0 ]     [0 ]
          [b](x0) = [-& 0  -&]x0 + [-&]
                    [0  1  0 ]     [0 ],
          
                    [0  1  0 ]     [0]
          [a](x0) = [0  0  -&]x0 + [0]
                    [-& 0  0 ]     [0]
         orientation:
          c#(x1) = [-& 0  0 ]x1 + [0] >= [-& 0  0 ]x1 + [0] = b#(x1)
          
          b#(b(a(a(x1)))) = [1 2 1]x1 + [1] >= [0 1 0]x1 + [0] = c#(a(c(b(x1))))
          
                  [0  1  0 ]     [0]           
          a(x1) = [0  0  -&]x1 + [0] >= x1 = x1
                  [-& 0  0 ]     [0]           
          
                           [1 2 1]     [1]    [1 2 1]     [1]                 
          b(b(a(a(x1)))) = [0 1 0]x1 + [0] >= [0 1 0]x1 + [0] = c(a(c(b(x1))))
                           [1 2 1]     [1]    [1 2 1]     [1]                 
          
                  [0  1  0 ]     [0]    [0  1  0 ]     [0]           
          c(x1) = [-& 0  0 ]x1 + [0] >= [-& 0  0 ]x1 + [0] = a(b(x1))
                  [0  1  0 ]     [0]    [0  1  0 ]     [0]           
         problem:
          DPs:
           c#(x1) -> b#(x1)
          TRS:
           a(x1) -> x1
           b(b(a(a(x1)))) -> c(a(c(b(x1))))
           c(x1) -> a(b(x1))
         Restore Modifier:
          DPs:
           c#(x1) -> b#(x1)
          TRS:
           a(x1) -> x1
           b(b(a(a(x1)))) -> c(a(c(b(x1))))
           c(x1) -> a(b(x1))
          EDG Processor:
           DPs:
            c#(x1) -> b#(x1)
           TRS:
            a(x1) -> x1
            b(b(a(a(x1)))) -> c(a(c(b(x1))))
            c(x1) -> a(b(x1))
           graph:
            
           Qed