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
     usable rules:
      a(x1) -> x1
      a(b(x1)) -> b(c(b(a(x1))))
      b(x1) -> a(x1)
      c(c(c(x1))) -> x1
     interpretation:
      [b#](x0) = [3 2]x0 + [0],
      
      [a#](x0) = [3 1]x0,
      
                [-& -1]     [0]
      [c](x0) = [0  1 ]x0 + [1],
      
                [2 0]     [3]
      [b](x0) = [1 0]x0 + [3],
      
                [1  -&]     [-&]
      [a](x0) = [1  0 ]x0 + [3 ]
     orientation:
      b#(x1) = [3 2]x1 + [0] >= [3 1]x1 = a#(x1)
      
      a#(b(x1)) = [5 3]x1 + [6] >= [3 1]x1 = a#(x1)
      
      a#(b(x1)) = [5 3]x1 + [6] >= [4 2]x1 + [5] = b#(a(x1))
      
      a#(b(x1)) = [5 3]x1 + [6] >= [5 3]x1 + [6] = b#(c(b(a(x1))))
      
              [1  -&]     [-&]           
      a(x1) = [1  0 ]x1 + [3 ] >= x1 = x1
      
                 [3 1]     [4]    [3 1]     [4]                 
      a(b(x1)) = [3 1]x1 + [4] >= [3 1]x1 + [4] = b(c(b(a(x1))))
      
              [2 0]     [3]    [1  -&]     [-&]        
      b(x1) = [1 0]x1 + [3] >= [1  0 ]x1 + [3 ] = a(x1)
      
                    [0 1]     [1]           
      c(c(c(x1))) = [2 3]x1 + [3] >= x1 = x1
     problem:
      DPs:
       b#(x1) -> 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
     Restore Modifier:
      DPs:
       b#(x1) -> 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#(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#(c(b(a(x1))))
        a#(b(x1)) -> b#(c(b(a(x1)))) -> b#(x1) -> a#(x1)
       Arctic Interpretation Processor:
        dimension: 2
        usable rules:
         a(x1) -> x1
         a(b(x1)) -> b(c(b(a(x1))))
         b(x1) -> a(x1)
         c(c(c(x1))) -> x1
        interpretation:
         [b#](x0) = [2 1]x0 + [0],
         
         [a#](x0) = [2  -2]x0 + [0],
         
                   [-& 0 ]     [0]
         [c](x0) = [-1 1 ]x0 + [0],
         
                   [2 1]     [2]
         [b](x0) = [0 0]x0 + [1],
         
                   [1 0]     [-&]
         [a](x0) = [0 0]x0 + [1 ]
        orientation:
         b#(x1) = [2 1]x1 + [0] >= [2  -2]x1 + [0] = a#(x1)
         
         a#(b(x1)) = [4 3]x1 + [4] >= [3 2]x1 + [3] = b#(c(b(a(x1))))
         
                 [1 0]     [-&]           
         a(x1) = [0 0]x1 + [1 ] >= x1 = x1
         
                    [3 2]     [3]    [3 2]     [3]                 
         a(b(x1)) = [2 1]x1 + [2] >= [2 1]x1 + [2] = b(c(b(a(x1))))
         
                 [2 1]     [2]    [1 0]     [-&]        
         b(x1) = [0 0]x1 + [1] >= [0 0]x1 + [1 ] = 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
        Restore Modifier:
         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