YES

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

Proof:
 DP Processor:
  DPs:
   a#(b(x1)) -> a#(x1)
   a#(b(x1)) -> b#(a(x1))
   a#(b(x1)) -> c#(b(a(x1)))
   a#(b(x1)) -> b#(c(b(a(x1))))
   b#(x1) -> a#(x1)
  TRS:
   a(x1) -> x1
   a(b(x1)) -> b(c(b(a(x1))))
   b(x1) -> a(x1)
   c(c(c(x1))) -> x1
  TDG Processor:
   DPs:
    a#(b(x1)) -> a#(x1)
    a#(b(x1)) -> b#(a(x1))
    a#(b(x1)) -> c#(b(a(x1)))
    a#(b(x1)) -> b#(c(b(a(x1))))
    b#(x1) -> a#(x1)
   TRS:
    a(x1) -> x1
    a(b(x1)) -> b(c(b(a(x1))))
    b(x1) -> a(x1)
    c(c(c(x1))) -> x1
   graph:
    b#(x1) -> a#(x1) -> a#(b(x1)) -> b#(c(b(a(x1))))
    b#(x1) -> a#(x1) -> a#(b(x1)) -> c#(b(a(x1)))
    b#(x1) -> a#(x1) -> a#(b(x1)) -> b#(a(x1))
    b#(x1) -> a#(x1) -> a#(b(x1)) -> a#(x1)
    a#(b(x1)) -> b#(c(b(a(x1)))) -> b#(x1) -> a#(x1)
    a#(b(x1)) -> b#(a(x1)) -> b#(x1) -> a#(x1)
    a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> b#(c(b(a(x1))))
    a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> c#(b(a(x1)))
    a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> b#(a(x1))
    a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> a#(x1)
   SCC Processor:
    #sccs: 1
    #rules: 4
    #arcs: 10/25
    DPs:
     b#(x1) -> a#(x1)
     a#(b(x1)) -> a#(x1)
     a#(b(x1)) -> b#(a(x1))
     a#(b(x1)) -> b#(c(b(a(x1))))
    TRS:
     a(x1) -> x1
     a(b(x1)) -> b(c(b(a(x1))))
     b(x1) -> a(x1)
     c(c(c(x1))) -> x1
    Arctic Interpretation Processor:
     dimension: 2
     interpretation:
      [b#](x0) = [-4 0 ]x0 + [1],
      
      [a#](x0) = [-4 -1]x0 + [0],
      
                [1  -1]     [2]
      [c](x0) = [0  -&]x0 + [0],
      
                [0 0]     [-&]
      [b](x0) = [1 2]x0 + [2 ],
      
                [0 0]     [-&]
      [a](x0) = [0 1]x0 + [-3]
     orientation:
      b#(x1) = [-4 0 ]x1 + [1] >= [-4 -1]x1 + [0] = a#(x1)
      
      a#(b(x1)) = [0 1]x1 + [1] >= [-4 -1]x1 + [0] = a#(x1)
      
      a#(b(x1)) = [0 1]x1 + [1] >= [0 1]x1 + [1] = b#(a(x1))
      
      a#(b(x1)) = [0 1]x1 + [1] >= [0 1]x1 + [1] = b#(c(b(a(x1))))
      
              [0 0]     [-&]           
      a(x1) = [0 1]x1 + [-3] >= x1 = x1
      
                 [1 2]     [2]    [1 2]     [2]                 
      a(b(x1)) = [2 3]x1 + [3] >= [2 3]x1 + [3] = b(c(b(a(x1))))
      
              [0 0]     [-&]    [0 0]     [-&]        
      b(x1) = [1 2]x1 + [2 ] >= [0 1]x1 + [-3] = a(x1)
      
                    [3 1]     [4]           
      c(c(c(x1))) = [2 0]x1 + [3] >= x1 = x1
     problem:
      DPs:
       b#(x1) -> a#(x1)
       a#(b(x1)) -> b#(a(x1))
       a#(b(x1)) -> b#(c(b(a(x1))))
      TRS:
       a(x1) -> x1
       a(b(x1)) -> b(c(b(a(x1))))
       b(x1) -> a(x1)
       c(c(c(x1))) -> x1
     EDG Processor:
      DPs:
       b#(x1) -> a#(x1)
       a#(b(x1)) -> b#(a(x1))
       a#(b(x1)) -> b#(c(b(a(x1))))
      TRS:
       a(x1) -> x1
       a(b(x1)) -> b(c(b(a(x1))))
       b(x1) -> a(x1)
       c(c(c(x1))) -> x1
      graph:
       b#(x1) -> a#(x1) -> a#(b(x1)) -> b#(a(x1))
       b#(x1) -> a#(x1) -> a#(b(x1)) -> b#(c(b(a(x1))))
       a#(b(x1)) -> b#(c(b(a(x1)))) -> b#(x1) -> a#(x1)
       a#(b(x1)) -> b#(a(x1)) -> b#(x1) -> a#(x1)
      Arctic Interpretation Processor:
       dimension: 2
       interpretation:
        [b#](x0) = [0  -&]x0 + [1],
        
        [a#](x0) = [0  -&]x0,
        
                  [-& 0 ]     [1]
        [c](x0) = [-1 1 ]x0 + [0],
        
                  [2 1]     [3]
        [b](x0) = [0 0]x0 + [0],
        
                  [1  -4]     [1 ]
        [a](x0) = [0  0 ]x0 + [-4]
       orientation:
        b#(x1) = [0  -&]x1 + [1] >= [0  -&]x1 = a#(x1)
        
        a#(b(x1)) = [2 1]x1 + [3] >= [1  -4]x1 + [1] = b#(a(x1))
        
        a#(b(x1)) = [2 1]x1 + [3] >= [1 0]x1 + [1] = b#(c(b(a(x1))))
        
                [1  -4]     [1 ]           
        a(x1) = [0  0 ]x1 + [-4] >= x1 = x1
        
                   [3 2]     [4]    [3 2]     [3]                 
        a(b(x1)) = [2 1]x1 + [3] >= [2 1]x1 + [2] = b(c(b(a(x1))))
        
                [2 1]     [3]    [1  -4]     [1 ]        
        b(x1) = [0 0]x1 + [0] >= [0  0 ]x1 + [-4] = a(x1)
        
                      [0 2]     [1]           
        c(c(c(x1))) = [1 3]x1 + [2] >= x1 = x1
       problem:
        DPs:
         b#(x1) -> a#(x1)
        TRS:
         a(x1) -> x1
         a(b(x1)) -> b(c(b(a(x1))))
         b(x1) -> a(x1)
         c(c(c(x1))) -> x1
       EDG Processor:
        DPs:
         b#(x1) -> a#(x1)
        TRS:
         a(x1) -> x1
         a(b(x1)) -> b(c(b(a(x1))))
         b(x1) -> a(x1)
         c(c(c(x1))) -> x1
        graph:
         
        Qed