YES

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

Proof:
 String Reversal Processor:
  a(x1) -> c(b(c(b(b(x1)))))
  b(b(c(x1))) -> a(x1)
  c(c(x1)) -> x1
  DP Processor:
   DPs:
    a#(x1) -> b#(x1)
    a#(x1) -> b#(b(x1))
    a#(x1) -> c#(b(b(x1)))
    a#(x1) -> b#(c(b(b(x1))))
    a#(x1) -> c#(b(c(b(b(x1)))))
    b#(b(c(x1))) -> a#(x1)
   TRS:
    a(x1) -> c(b(c(b(b(x1)))))
    b(b(c(x1))) -> a(x1)
    c(c(x1)) -> x1
   TDG Processor:
    DPs:
     a#(x1) -> b#(x1)
     a#(x1) -> b#(b(x1))
     a#(x1) -> c#(b(b(x1)))
     a#(x1) -> b#(c(b(b(x1))))
     a#(x1) -> c#(b(c(b(b(x1)))))
     b#(b(c(x1))) -> a#(x1)
    TRS:
     a(x1) -> c(b(c(b(b(x1)))))
     b(b(c(x1))) -> a(x1)
     c(c(x1)) -> x1
    graph:
     b#(b(c(x1))) -> a#(x1) -> a#(x1) -> c#(b(c(b(b(x1)))))
     b#(b(c(x1))) -> a#(x1) -> a#(x1) -> b#(c(b(b(x1))))
     b#(b(c(x1))) -> a#(x1) -> a#(x1) -> c#(b(b(x1)))
     b#(b(c(x1))) -> a#(x1) -> a#(x1) -> b#(b(x1))
     b#(b(c(x1))) -> a#(x1) -> a#(x1) -> b#(x1)
     a#(x1) -> b#(b(x1)) -> b#(b(c(x1))) -> a#(x1)
     a#(x1) -> b#(c(b(b(x1)))) -> b#(b(c(x1))) -> a#(x1)
     a#(x1) -> b#(x1) -> b#(b(c(x1))) -> a#(x1)
    SCC Processor:
     #sccs: 1
     #rules: 4
     #arcs: 8/36
     DPs:
      b#(b(c(x1))) -> a#(x1)
      a#(x1) -> b#(x1)
      a#(x1) -> b#(b(x1))
      a#(x1) -> b#(c(b(b(x1))))
     TRS:
      a(x1) -> c(b(c(b(b(x1)))))
      b(b(c(x1))) -> a(x1)
      c(c(x1)) -> x1
     Arctic Interpretation Processor:
      dimension: 2
      usable rules:
       a(x1) -> c(b(c(b(b(x1)))))
       b(b(c(x1))) -> a(x1)
       c(c(x1)) -> x1
      interpretation:
       [b#](x0) = [0  -&]x0 + [1],
       
                 [-& 0 ]     [1]
       [c](x0) = [0  1 ]x0 + [2],
       
       [a#](x0) = [0 1]x0 + [2],
       
                 [-& 0 ]     [1]
       [a](x0) = [0  1 ]x0 + [2],
       
                 [-& 0 ]     [0]
       [b](x0) = [0  -&]x0 + [0]
      orientation:
       b#(b(c(x1))) = [0 1]x1 + [2] >= [0 1]x1 + [2] = a#(x1)
       
       a#(x1) = [0 1]x1 + [2] >= [0  -&]x1 + [1] = b#(x1)
       
       a#(x1) = [0 1]x1 + [2] >= [-& 0 ]x1 + [1] = b#(b(x1))
       
       a#(x1) = [0 1]x1 + [2] >= [-& 0 ]x1 + [1] = b#(c(b(b(x1))))
       
               [-& 0 ]     [1]    [-& 0 ]     [1]                    
       a(x1) = [0  1 ]x1 + [2] >= [0  1 ]x1 + [2] = c(b(c(b(b(x1)))))
       
                     [-& 0 ]     [1]    [-& 0 ]     [1]        
       b(b(c(x1))) = [0  1 ]x1 + [2] >= [0  1 ]x1 + [2] = a(x1)
       
                  [0 1]     [2]           
       c(c(x1)) = [1 2]x1 + [3] >= x1 = x1
      problem:
       DPs:
        b#(b(c(x1))) -> a#(x1)
        a#(x1) -> b#(x1)
       TRS:
        a(x1) -> c(b(c(b(b(x1)))))
        b(b(c(x1))) -> a(x1)
        c(c(x1)) -> x1
      Restore Modifier:
       DPs:
        b#(b(c(x1))) -> a#(x1)
        a#(x1) -> b#(x1)
       TRS:
        a(x1) -> c(b(c(b(b(x1)))))
        b(b(c(x1))) -> a(x1)
        c(c(x1)) -> x1
       EDG Processor:
        DPs:
         b#(b(c(x1))) -> a#(x1)
         a#(x1) -> b#(x1)
        TRS:
         a(x1) -> c(b(c(b(b(x1)))))
         b(b(c(x1))) -> a(x1)
         c(c(x1)) -> x1
        graph:
         b#(b(c(x1))) -> a#(x1) -> a#(x1) -> b#(x1)
         a#(x1) -> b#(x1) -> b#(b(c(x1))) -> a#(x1)
        Usable Rule Processor:
         DPs:
          b#(b(c(x1))) -> a#(x1)
          a#(x1) -> b#(x1)
         TRS:
          
         Arctic Interpretation Processor:
          dimension: 1
          usable rules:
           
          interpretation:
           [b#](x0) = x0,
           
           [c](x0) = 8x0 + 8,
           
           [a#](x0) = x0,
           
           [b](x0) = x0 + 9
          orientation:
           b#(b(c(x1))) = 8x1 + 9 >= x1 = a#(x1)
           
           a#(x1) = x1 >= x1 = b#(x1)
          problem:
           DPs:
            a#(x1) -> b#(x1)
           TRS:
            
          Restore Modifier:
           DPs:
            a#(x1) -> b#(x1)
           TRS:
            a(x1) -> c(b(c(b(b(x1)))))
            b(b(c(x1))) -> a(x1)
            c(c(x1)) -> x1
           EDG Processor:
            DPs:
             a#(x1) -> b#(x1)
            TRS:
             a(x1) -> c(b(c(b(b(x1)))))
             b(b(c(x1))) -> a(x1)
             c(c(x1)) -> x1
            graph:
             
            SCC Processor:
             #sccs: 0
             #rules: 0
             #arcs: 0/1