YES

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

Proof:
 DP Processor:
  DPs:
   a#(b(b(x1))) -> a#(x1)
   a#(b(b(x1))) -> b#(c(a(x1)))
   a#(b(b(x1))) -> b#(b(c(a(x1))))
   a#(b(b(x1))) -> b#(b(b(c(a(x1)))))
   b#(c(x1)) -> a#(x1)
  TRS:
   a(x1) -> x1
   a(b(b(x1))) -> b(b(b(c(a(x1)))))
   b(c(x1)) -> a(x1)
  TDG Processor:
   DPs:
    a#(b(b(x1))) -> a#(x1)
    a#(b(b(x1))) -> b#(c(a(x1)))
    a#(b(b(x1))) -> b#(b(c(a(x1))))
    a#(b(b(x1))) -> b#(b(b(c(a(x1)))))
    b#(c(x1)) -> a#(x1)
   TRS:
    a(x1) -> x1
    a(b(b(x1))) -> b(b(b(c(a(x1)))))
    b(c(x1)) -> a(x1)
   graph:
    b#(c(x1)) -> a#(x1) -> a#(b(b(x1))) -> b#(b(b(c(a(x1)))))
    b#(c(x1)) -> a#(x1) -> a#(b(b(x1))) -> b#(b(c(a(x1))))
    b#(c(x1)) -> a#(x1) -> a#(b(b(x1))) -> b#(c(a(x1)))
    b#(c(x1)) -> a#(x1) -> a#(b(b(x1))) -> a#(x1)
    a#(b(b(x1))) -> b#(c(a(x1))) -> b#(c(x1)) -> a#(x1)
    a#(b(b(x1))) -> b#(b(c(a(x1)))) -> b#(c(x1)) -> a#(x1)
    a#(b(b(x1))) -> b#(b(b(c(a(x1))))) -> b#(c(x1)) -> a#(x1)
    a#(b(b(x1))) -> a#(x1) -> a#(b(b(x1))) -> b#(b(b(c(a(x1)))))
    a#(b(b(x1))) -> a#(x1) -> a#(b(b(x1))) -> b#(b(c(a(x1))))
    a#(b(b(x1))) -> a#(x1) -> a#(b(b(x1))) -> b#(c(a(x1)))
    a#(b(b(x1))) -> a#(x1) -> a#(b(b(x1))) -> a#(x1)
   Arctic Interpretation Processor:
    dimension: 2
    interpretation:
     [b#](x0) = [3 0]x0 + [0],
     
     [a#](x0) = [3  -&]x0 + [0],
     
               [0  0 ]  
     [c](x0) = [-& -&]x0,
     
               [0 1]     [0]
     [b](x0) = [0 0]x0 + [3],
     
               [0 0]  
     [a](x0) = [0 0]x0
    orientation:
     a#(b(b(x1))) = [4 4]x1 + [7] >= [3  -&]x1 + [0] = a#(x1)
     
     a#(b(b(x1))) = [4 4]x1 + [7] >= [3 3]x1 + [0] = b#(c(a(x1)))
     
     a#(b(b(x1))) = [4 4]x1 + [7] >= [3 3]x1 + [3] = b#(b(c(a(x1))))
     
     a#(b(b(x1))) = [4 4]x1 + [7] >= [4 4]x1 + [7] = b#(b(b(c(a(x1)))))
     
     b#(c(x1)) = [3 3]x1 + [0] >= [3  -&]x1 + [0] = a#(x1)
     
             [0 0]             
     a(x1) = [0 0]x1 >= x1 = x1
     
                   [1 1]     [4]    [1 1]     [4]                    
     a(b(b(x1))) = [1 1]x1 + [4] >= [1 1]x1 + [4] = b(b(b(c(a(x1)))))
     
                [0 0]     [0]    [0 0]          
     b(c(x1)) = [0 0]x1 + [3] >= [0 0]x1 = a(x1)
    problem:
     DPs:
      a#(b(b(x1))) -> b#(b(b(c(a(x1)))))
      b#(c(x1)) -> a#(x1)
     TRS:
      a(x1) -> x1
      a(b(b(x1))) -> b(b(b(c(a(x1)))))
      b(c(x1)) -> a(x1)
    EDG Processor:
     DPs:
      a#(b(b(x1))) -> b#(b(b(c(a(x1)))))
      b#(c(x1)) -> a#(x1)
     TRS:
      a(x1) -> x1
      a(b(b(x1))) -> b(b(b(c(a(x1)))))
      b(c(x1)) -> a(x1)
     graph:
      b#(c(x1)) -> a#(x1) -> a#(b(b(x1))) -> b#(b(b(c(a(x1)))))
      a#(b(b(x1))) -> b#(b(b(c(a(x1))))) -> b#(c(x1)) -> a#(x1)
     Matrix Interpretation Processor: dim=1
      
      interpretation:
       [b#](x0) = x0 + 1,
       
       [a#](x0) = 2x0 + 3,
       
       [c](x0) = 2x0 + 6,
       
       [b](x0) = 1/2x0,
       
       [a](x0) = x0 + 1
      orientation:
       a#(b(b(x1))) = 1/2x1 + 3 >= 1/2x1 + 3 = b#(b(b(c(a(x1)))))
       
       b#(c(x1)) = 2x1 + 7 >= 2x1 + 3 = a#(x1)
       
       a(x1) = x1 + 1 >= x1 = x1
       
       a(b(b(x1))) = 1/4x1 + 1 >= 1/4x1 + 1 = b(b(b(c(a(x1)))))
       
       b(c(x1)) = x1 + 3 >= x1 + 1 = a(x1)
      problem:
       DPs:
        a#(b(b(x1))) -> b#(b(b(c(a(x1)))))
       TRS:
        a(x1) -> x1
        a(b(b(x1))) -> b(b(b(c(a(x1)))))
        b(c(x1)) -> a(x1)
      EDG Processor:
       DPs:
        a#(b(b(x1))) -> b#(b(b(c(a(x1)))))
       TRS:
        a(x1) -> x1
        a(b(b(x1))) -> b(b(b(c(a(x1)))))
        b(c(x1)) -> a(x1)
       graph:
        
       Qed