YES

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

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