YES

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

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