YES

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

Proof:
 DP Processor:
  DPs:
   a#(b(b(x1))) -> b#(c(x1))
   a#(b(b(x1))) -> a#(b(c(x1)))
   a#(b(b(x1))) -> b#(a(b(c(x1))))
   a#(b(b(x1))) -> b#(b(a(b(c(x1)))))
   b#(c(x1)) -> a#(x1)
  TRS:
   a(x1) -> x1
   a(b(b(x1))) -> b(b(a(b(c(x1)))))
   b(c(x1)) -> a(x1)
  TDG Processor:
   DPs:
    a#(b(b(x1))) -> b#(c(x1))
    a#(b(b(x1))) -> a#(b(c(x1)))
    a#(b(b(x1))) -> b#(a(b(c(x1))))
    a#(b(b(x1))) -> b#(b(a(b(c(x1)))))
    b#(c(x1)) -> a#(x1)
   TRS:
    a(x1) -> x1
    a(b(b(x1))) -> b(b(a(b(c(x1)))))
    b(c(x1)) -> a(x1)
   graph:
    b#(c(x1)) -> a#(x1) -> a#(b(b(x1))) -> b#(b(a(b(c(x1)))))
    b#(c(x1)) -> a#(x1) -> a#(b(b(x1))) -> b#(a(b(c(x1))))
    b#(c(x1)) -> a#(x1) -> a#(b(b(x1))) -> a#(b(c(x1)))
    b#(c(x1)) -> a#(x1) -> a#(b(b(x1))) -> b#(c(x1))
    a#(b(b(x1))) -> b#(c(x1)) -> b#(c(x1)) -> a#(x1)
    a#(b(b(x1))) -> b#(b(a(b(c(x1))))) -> b#(c(x1)) -> a#(x1)
    a#(b(b(x1))) -> b#(a(b(c(x1)))) -> b#(c(x1)) -> a#(x1)
    a#(b(b(x1))) -> a#(b(c(x1))) -> a#(b(b(x1))) -> b#(b(a(b(c(x1)))))
    a#(b(b(x1))) -> a#(b(c(x1))) -> a#(b(b(x1))) -> b#(a(b(c(x1))))
    a#(b(b(x1))) -> a#(b(c(x1))) -> a#(b(b(x1))) -> a#(b(c(x1)))
    a#(b(b(x1))) -> a#(b(c(x1))) -> a#(b(b(x1))) -> b#(c(x1))
   Arctic Interpretation Processor:
    dimension: 2
    interpretation:
     [b#](x0) = [0  -&]x0 + [2],
     
     [a#](x0) = [3 0]x0,
     
               [3  0 ]     [3]
     [c](x0) = [0  -&]x0 + [0],
     
               [-& 0 ]     [0 ]
     [b](x0) = [0  -&]x0 + [-&],
     
               [0  -&]     [-&]
     [a](x0) = [3  0 ]x0 + [0 ]
    orientation:
     a#(b(b(x1))) = [3 0]x1 + [3] >= [3 0]x1 + [3] = b#(c(x1))
     
     a#(b(b(x1))) = [3 0]x1 + [3] >= [3 0]x1 + [3] = a#(b(c(x1)))
     
     a#(b(b(x1))) = [3 0]x1 + [3] >= [0  -&]x1 + [2] = b#(a(b(c(x1))))
     
     a#(b(b(x1))) = [3 0]x1 + [3] >= [3 0]x1 + [3] = b#(b(a(b(c(x1)))))
     
     b#(c(x1)) = [3 0]x1 + [3] >= [3 0]x1 = a#(x1)
     
             [0  -&]     [-&]           
     a(x1) = [3  0 ]x1 + [0 ] >= x1 = x1
     
                   [0  -&]     [0]    [0  -&]     [0]                    
     a(b(b(x1))) = [3  0 ]x1 + [3] >= [3  0 ]x1 + [3] = b(b(a(b(c(x1)))))
     
                [0  -&]     [0]    [0  -&]     [-&]        
     b(c(x1)) = [3  0 ]x1 + [3] >= [3  0 ]x1 + [0 ] = a(x1)
    problem:
     DPs:
      a#(b(b(x1))) -> b#(c(x1))
      a#(b(b(x1))) -> a#(b(c(x1)))
      a#(b(b(x1))) -> b#(b(a(b(c(x1)))))
      b#(c(x1)) -> a#(x1)
     TRS:
      a(x1) -> x1
      a(b(b(x1))) -> b(b(a(b(c(x1)))))
      b(c(x1)) -> a(x1)
    EDG Processor:
     DPs:
      a#(b(b(x1))) -> b#(c(x1))
      a#(b(b(x1))) -> a#(b(c(x1)))
      a#(b(b(x1))) -> b#(b(a(b(c(x1)))))
      b#(c(x1)) -> a#(x1)
     TRS:
      a(x1) -> x1
      a(b(b(x1))) -> b(b(a(b(c(x1)))))
      b(c(x1)) -> a(x1)
     graph:
      b#(c(x1)) -> a#(x1) -> a#(b(b(x1))) -> b#(c(x1))
      b#(c(x1)) -> a#(x1) -> a#(b(b(x1))) -> a#(b(c(x1)))
      b#(c(x1)) -> a#(x1) -> a#(b(b(x1))) -> b#(b(a(b(c(x1)))))
      a#(b(b(x1))) -> b#(c(x1)) -> b#(c(x1)) -> a#(x1)
      a#(b(b(x1))) -> b#(b(a(b(c(x1))))) -> b#(c(x1)) -> a#(x1)
      a#(b(b(x1))) -> a#(b(c(x1))) -> a#(b(b(x1))) -> b#(c(x1))
      a#(b(b(x1))) -> a#(b(c(x1))) -> a#(b(b(x1))) -> a#(b(c(x1)))
      a#(b(b(x1))) -> a#(b(c(x1))) -> a#(b(b(x1))) -> b#(b(a(b(c(x1)))))
     Arctic Interpretation Processor:
      dimension: 2
      interpretation:
       [b#](x0) = [0 0]x0 + [1],
       
       [a#](x0) = [0 0]x0 + [0],
       
                 [-& -&]     [0]
       [c](x0) = [0  0 ]x0 + [0],
       
                 [0 0]     [2]
       [b](x0) = [1 0]x0 + [2],
       
                 [0 0]  
       [a](x0) = [0 0]x0
      orientation:
       a#(b(b(x1))) = [1 1]x1 + [3] >= [0 0]x1 + [1] = b#(c(x1))
       
       a#(b(b(x1))) = [1 1]x1 + [3] >= [0 0]x1 + [2] = a#(b(c(x1)))
       
       a#(b(b(x1))) = [1 1]x1 + [3] >= [1 1]x1 + [3] = b#(b(a(b(c(x1)))))
       
       b#(c(x1)) = [0 0]x1 + [1] >= [0 0]x1 + [0] = a#(x1)
       
               [0 0]             
       a(x1) = [0 0]x1 >= x1 = x1
       
                     [1 1]     [3]    [1 1]     [3]                    
       a(b(b(x1))) = [1 1]x1 + [3] >= [1 1]x1 + [3] = b(b(a(b(c(x1)))))
       
                  [0 0]     [2]    [0 0]          
       b(c(x1)) = [0 0]x1 + [2] >= [0 0]x1 = a(x1)
      problem:
       DPs:
        a#(b(b(x1))) -> b#(b(a(b(c(x1)))))
        b#(c(x1)) -> a#(x1)
       TRS:
        a(x1) -> x1
        a(b(b(x1))) -> b(b(a(b(c(x1)))))
        b(c(x1)) -> a(x1)
      EDG Processor:
       DPs:
        a#(b(b(x1))) -> b#(b(a(b(c(x1)))))
        b#(c(x1)) -> a#(x1)
       TRS:
        a(x1) -> x1
        a(b(b(x1))) -> b(b(a(b(c(x1)))))
        b(c(x1)) -> a(x1)
       graph:
        b#(c(x1)) -> a#(x1) -> a#(b(b(x1))) -> b#(b(a(b(c(x1)))))
        a#(b(b(x1))) -> b#(b(a(b(c(x1))))) -> b#(c(x1)) -> a#(x1)
       Matrix Interpretation Processor: dim=1
        
        interpretation:
         [b#](x0) = 1/2x0,
         
         [a#](x0) = x0 + 3,
         
         [c](x0) = 2x0 + 6,
         
         [b](x0) = 1/2x0,
         
         [a](x0) = x0 + 3
        orientation:
         a#(b(b(x1))) = 1/4x1 + 3 >= 1/4x1 + 3/2 = b#(b(a(b(c(x1)))))
         
         b#(c(x1)) = x1 + 3 >= x1 + 3 = a#(x1)
         
         a(x1) = x1 + 3 >= x1 = x1
         
         a(b(b(x1))) = 1/4x1 + 3 >= 1/4x1 + 3/2 = b(b(a(b(c(x1)))))
         
         b(c(x1)) = x1 + 3 >= x1 + 3 = a(x1)
        problem:
         DPs:
          b#(c(x1)) -> a#(x1)
         TRS:
          a(x1) -> x1
          a(b(b(x1))) -> b(b(a(b(c(x1)))))
          b(c(x1)) -> a(x1)
        EDG Processor:
         DPs:
          b#(c(x1)) -> a#(x1)
         TRS:
          a(x1) -> x1
          a(b(b(x1))) -> b(b(a(b(c(x1)))))
          b(c(x1)) -> a(x1)
         graph:
          
         Qed