YES

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

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