YES

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

Proof:
 DP Processor:
  DPs:
   a#(a(x1)) -> b#(x1)
   b#(c(x1)) -> a#(x1)
   c#(b(x1)) -> a#(x1)
   c#(b(x1)) -> c#(a(x1))
   c#(b(x1)) -> c#(c(a(x1)))
   c#(b(x1)) -> b#(c(c(a(x1))))
  TRS:
   a(a(x1)) -> b(x1)
   b(c(x1)) -> a(x1)
   c(b(x1)) -> b(c(c(a(x1))))
  TDG Processor:
   DPs:
    a#(a(x1)) -> b#(x1)
    b#(c(x1)) -> a#(x1)
    c#(b(x1)) -> a#(x1)
    c#(b(x1)) -> c#(a(x1))
    c#(b(x1)) -> c#(c(a(x1)))
    c#(b(x1)) -> b#(c(c(a(x1))))
   TRS:
    a(a(x1)) -> b(x1)
    b(c(x1)) -> a(x1)
    c(b(x1)) -> b(c(c(a(x1))))
   graph:
    c#(b(x1)) -> c#(c(a(x1))) -> c#(b(x1)) -> b#(c(c(a(x1))))
    c#(b(x1)) -> c#(c(a(x1))) -> c#(b(x1)) -> c#(c(a(x1)))
    c#(b(x1)) -> c#(c(a(x1))) -> c#(b(x1)) -> c#(a(x1))
    c#(b(x1)) -> c#(c(a(x1))) -> c#(b(x1)) -> a#(x1)
    c#(b(x1)) -> c#(a(x1)) -> c#(b(x1)) -> b#(c(c(a(x1))))
    c#(b(x1)) -> c#(a(x1)) -> c#(b(x1)) -> c#(c(a(x1)))
    c#(b(x1)) -> c#(a(x1)) -> c#(b(x1)) -> c#(a(x1))
    c#(b(x1)) -> c#(a(x1)) -> c#(b(x1)) -> a#(x1)
    c#(b(x1)) -> b#(c(c(a(x1)))) -> b#(c(x1)) -> a#(x1)
    c#(b(x1)) -> a#(x1) -> a#(a(x1)) -> b#(x1)
    b#(c(x1)) -> a#(x1) -> a#(a(x1)) -> b#(x1)
    a#(a(x1)) -> b#(x1) -> b#(c(x1)) -> a#(x1)
   CDG Processor:
    DPs:
     a#(a(x1)) -> b#(x1)
     b#(c(x1)) -> a#(x1)
     c#(b(x1)) -> a#(x1)
     c#(b(x1)) -> c#(a(x1))
     c#(b(x1)) -> c#(c(a(x1)))
     c#(b(x1)) -> b#(c(c(a(x1))))
    TRS:
     a(a(x1)) -> b(x1)
     b(c(x1)) -> a(x1)
     c(b(x1)) -> b(c(c(a(x1))))
    graph:
     c#(b(x1)) -> c#(c(a(x1))) -> c#(b(x1)) -> a#(x1)
     c#(b(x1)) -> c#(c(a(x1))) -> c#(b(x1)) -> c#(a(x1))
     c#(b(x1)) -> c#(c(a(x1))) -> c#(b(x1)) -> c#(c(a(x1)))
     c#(b(x1)) -> c#(c(a(x1))) -> c#(b(x1)) -> b#(c(c(a(x1))))
     c#(b(x1)) -> c#(a(x1)) -> c#(b(x1)) -> a#(x1)
     c#(b(x1)) -> c#(a(x1)) -> c#(b(x1)) -> c#(a(x1))
     c#(b(x1)) -> c#(a(x1)) -> c#(b(x1)) -> c#(c(a(x1)))
     c#(b(x1)) -> c#(a(x1)) -> c#(b(x1)) -> b#(c(c(a(x1))))
     c#(b(x1)) -> b#(c(c(a(x1)))) -> b#(c(x1)) -> a#(x1)
    SCC Processor:
     #sccs: 1
     #rules: 2
     #arcs: 9/36
     DPs:
      c#(b(x1)) -> c#(c(a(x1)))
      c#(b(x1)) -> c#(a(x1))
     TRS:
      a(a(x1)) -> b(x1)
      b(c(x1)) -> a(x1)
      c(b(x1)) -> b(c(c(a(x1))))
     Arctic Interpretation Processor:
      dimension: 3
      interpretation:
       [c#](x0) = [-& 0  0 ]x0 + [0],
       
                 [-& 0  0 ]     [0]
       [c](x0) = [-& -& 0 ]x0 + [0]
                 [0  -& 0 ]     [0],
       
                 [0 0 0]     [0]
       [b](x0) = [1 0 0]x0 + [0]
                 [0 1 0]     [1],
       
                 [-& 0  -&]     [0]
       [a](x0) = [0  1  0 ]x0 + [1]
                 [-& 0  -&]     [0]
      orientation:
       c#(b(x1)) = [1 1 0]x1 + [1] >= [-& 0  -&]x1 + [0] = c#(c(a(x1)))
       
       c#(b(x1)) = [1 1 0]x1 + [1] >= [0 1 0]x1 + [1] = c#(a(x1))
       
                  [0 1 0]     [1]    [0 0 0]     [0]        
       a(a(x1)) = [1 2 1]x1 + [2] >= [1 0 0]x1 + [0] = b(x1)
                  [0 1 0]     [1]    [0 1 0]     [1]        
       
                  [0 0 0]     [0]    [-& 0  -&]     [0]        
       b(c(x1)) = [0 1 1]x1 + [1] >= [0  1  0 ]x1 + [1] = a(x1)
                  [0 0 1]     [1]    [-& 0  -&]     [0]        
       
                  [1 1 0]     [1]    [0 1 0]     [1]                 
       c(b(x1)) = [0 1 0]x1 + [1] >= [0 1 0]x1 + [1] = b(c(c(a(x1))))
                  [0 1 0]     [1]    [0 1 0]     [1]                 
      problem:
       DPs:
        c#(b(x1)) -> c#(a(x1))
       TRS:
        a(a(x1)) -> b(x1)
        b(c(x1)) -> a(x1)
        c(b(x1)) -> b(c(c(a(x1))))
      EDG Processor:
       DPs:
        c#(b(x1)) -> c#(a(x1))
       TRS:
        a(a(x1)) -> b(x1)
        b(c(x1)) -> a(x1)
        c(b(x1)) -> b(c(c(a(x1))))
       graph:
        c#(b(x1)) -> c#(a(x1)) -> c#(b(x1)) -> c#(a(x1))
       Arctic Interpretation Processor:
        dimension: 3
        interpretation:
         [c#](x0) = [-& 0  0 ]x0 + [0],
         
                   [-& 0  0 ]     [0]
         [c](x0) = [0  1  1 ]x0 + [1]
                   [0  -& -&]     [0],
         
                   [0  -& 1 ]     [-&]
         [b](x0) = [1  0  0 ]x0 + [1 ]
                   [-& -& -&]     [0 ],
         
                   [1  0  0 ]     [1 ]
         [a](x0) = [0  -& -&]x0 + [-&]
                   [-& -& -&]     [0 ]
        orientation:
         c#(b(x1)) = [1 0 0]x1 + [1] >= [0  -& -&]x1 + [0] = c#(a(x1))
         
                    [2  1  1 ]     [2]    [0  -& 1 ]     [-&]        
         a(a(x1)) = [1  0  0 ]x1 + [1] >= [1  0  0 ]x1 + [1 ] = b(x1)
                    [-& -& -&]     [0]    [-& -& -&]     [0 ]        
         
                    [1  0  0 ]     [1]    [1  0  0 ]     [1 ]        
         b(c(x1)) = [0  1  1 ]x1 + [1] >= [0  -& -&]x1 + [-&] = a(x1)
                    [-& -& -&]     [0]    [-& -& -&]     [0 ]        
         
                    [1  0  0 ]     [1]    [1  0  0 ]     [1]                 
         c(b(x1)) = [2  1  1 ]x1 + [2] >= [2  1  1 ]x1 + [2] = b(c(c(a(x1))))
                    [0  -& 1 ]     [0]    [-& -& -&]     [0]                 
        problem:
         DPs:
          
         TRS:
          a(a(x1)) -> b(x1)
          b(c(x1)) -> a(x1)
          c(b(x1)) -> b(c(c(a(x1))))
        Qed