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
     interpretation:
      [b#](x0) = [1 0]x0 + [1],
      
      [a#](x0) = [0 1]x0 + [0],
      
                [3 0]     [0]
      [b](x0) = [2 0]x0 + [3],
      
                [-& 0 ]     [0 ]
      [c](x0) = [0  -&]x0 + [-&],
      
                [-& 0 ]     [0]
      [a](x0) = [0  2 ]x0 + [3]
     orientation:
      a#(a(x1)) = [1 3]x1 + [4] >= [0 2]x1 + [3] = b#(a(x1))
      
      a#(a(x1)) = [1 3]x1 + [4] >= [1 3]x1 + [4] = a#(b(a(x1)))
      
      b#(c(x1)) = [0 1]x1 + [1] >= [0 1]x1 + [0] = a#(x1)
      
                 [0 2]     [3]    [0 2]     [3]                    
      a(a(x1)) = [2 4]x1 + [5] >= [2 4]x1 + [5] = c(c(a(b(a(x1)))))
      
                 [0  -&]     [0]           
      a(c(x1)) = [2  0 ]x1 + [3] >= x1 = x1
      
                 [0 3]     [3]    [-& 0 ]     [0]        
      b(c(x1)) = [0 2]x1 + [3] >= [0  2 ]x1 + [3] = 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)
     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: 2
        interpretation:
         [a#](x0) = [-1 0 ]x0 + [0],
         
                   [-1 1 ]     [0]
         [b](x0) = [-2 -1]x0 + [0],
         
                   [-& 1 ]     [0]
         [c](x0) = [0  -&]x0 + [1],
         
                   [1  0 ]     [2 ]
         [a](x0) = [-1 -&]x0 + [-&]
        orientation:
         a#(a(x1)) = [0  -1]x1 + [1] >= [-1 -2]x1 + [0] = a#(b(a(x1)))
         
                    [2  1 ]     [3]    [2  1 ]     [3]                    
         a(a(x1)) = [0  -1]x1 + [1] >= [0  -1]x1 + [1] = c(c(a(b(a(x1)))))
         
                    [0  2 ]     [2 ]           
         a(c(x1)) = [-& 0 ]x1 + [-1] >= x1 = x1
         
                    [1  0 ]     [2]    [1  0 ]     [2 ]        
         b(c(x1)) = [-1 -1]x1 + [0] >= [-1 -&]x1 + [-&] = a(x1)
        problem:
         DPs:
          
         TRS:
          a(a(x1)) -> c(c(a(b(a(x1)))))
          a(c(x1)) -> x1
          b(c(x1)) -> a(x1)
        Qed