YES

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

Proof:
 DP Processor:
  DPs:
   a#(b(x1)) -> a#(x1)
   a#(b(x1)) -> b#(a(x1))
   a#(b(x1)) -> b#(b(a(x1)))
   b#(x1) -> c#(x1)
   b#(x1) -> a#(c(x1))
   b#(x1) -> c#(a(c(x1)))
  TRS:
   a(x1) -> x1
   a(b(x1)) -> b(b(a(x1)))
   b(x1) -> c(a(c(x1)))
   c(c(x1)) -> x1
  TDG Processor:
   DPs:
    a#(b(x1)) -> a#(x1)
    a#(b(x1)) -> b#(a(x1))
    a#(b(x1)) -> b#(b(a(x1)))
    b#(x1) -> c#(x1)
    b#(x1) -> a#(c(x1))
    b#(x1) -> c#(a(c(x1)))
   TRS:
    a(x1) -> x1
    a(b(x1)) -> b(b(a(x1)))
    b(x1) -> c(a(c(x1)))
    c(c(x1)) -> x1
   graph:
    b#(x1) -> a#(c(x1)) -> a#(b(x1)) -> b#(b(a(x1)))
    b#(x1) -> a#(c(x1)) -> a#(b(x1)) -> b#(a(x1))
    b#(x1) -> a#(c(x1)) -> a#(b(x1)) -> a#(x1)
    a#(b(x1)) -> b#(b(a(x1))) -> b#(x1) -> c#(a(c(x1)))
    a#(b(x1)) -> b#(b(a(x1))) -> b#(x1) -> a#(c(x1))
    a#(b(x1)) -> b#(b(a(x1))) -> b#(x1) -> c#(x1)
    a#(b(x1)) -> b#(a(x1)) -> b#(x1) -> c#(a(c(x1)))
    a#(b(x1)) -> b#(a(x1)) -> b#(x1) -> a#(c(x1))
    a#(b(x1)) -> b#(a(x1)) -> b#(x1) -> c#(x1)
    a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> b#(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: 12/36
    DPs:
     b#(x1) -> a#(c(x1))
     a#(b(x1)) -> a#(x1)
     a#(b(x1)) -> b#(a(x1))
     a#(b(x1)) -> b#(b(a(x1)))
    TRS:
     a(x1) -> x1
     a(b(x1)) -> b(b(a(x1)))
     b(x1) -> c(a(c(x1)))
     c(c(x1)) -> x1
    Arctic Interpretation Processor:
     dimension: 2
     usable rules:
      a(x1) -> x1
      a(b(x1)) -> b(b(a(x1)))
      b(x1) -> c(a(c(x1)))
      c(c(x1)) -> x1
     interpretation:
      [b#](x0) = [-& 0 ]x0 + [0],
      
                [0  2 ]     [2]
      [b](x0) = [-& 0 ]x0 + [0],
      
      [a#](x0) = [0  -&]x0 + [-4],
      
                [0  -&]     [-&]
      [a](x0) = [-2 0 ]x0 + [0 ],
      
                [-& 0 ]     [0 ]
      [c](x0) = [0  -&]x0 + [-&]
     orientation:
      b#(x1) = [-& 0 ]x1 + [0] >= [-& 0 ]x1 + [0] = a#(c(x1))
      
      a#(b(x1)) = [0 2]x1 + [2] >= [0  -&]x1 + [-4] = a#(x1)
      
      a#(b(x1)) = [0 2]x1 + [2] >= [-2 0 ]x1 + [0] = b#(a(x1))
      
      a#(b(x1)) = [0 2]x1 + [2] >= [-2 0 ]x1 + [0] = b#(b(a(x1)))
      
              [0  -&]     [-&]           
      a(x1) = [-2 0 ]x1 + [0 ] >= x1 = x1
      
                 [0  2 ]     [2]    [0  2 ]     [2]              
      a(b(x1)) = [-2 0 ]x1 + [0] >= [-2 0 ]x1 + [0] = b(b(a(x1)))
      
              [0  2 ]     [2]    [0  -2]     [0]              
      b(x1) = [-& 0 ]x1 + [0] >= [-& 0 ]x1 + [0] = c(a(c(x1)))
      
                      [0]           
      c(c(x1)) = x1 + [0] >= x1 = x1
     problem:
      DPs:
       b#(x1) -> a#(c(x1))
       a#(b(x1)) -> a#(x1)
      TRS:
       a(x1) -> x1
       a(b(x1)) -> b(b(a(x1)))
       b(x1) -> c(a(c(x1)))
       c(c(x1)) -> x1
     Restore Modifier:
      DPs:
       b#(x1) -> a#(c(x1))
       a#(b(x1)) -> a#(x1)
      TRS:
       a(x1) -> x1
       a(b(x1)) -> b(b(a(x1)))
       b(x1) -> c(a(c(x1)))
       c(c(x1)) -> x1
      EDG Processor:
       DPs:
        b#(x1) -> a#(c(x1))
        a#(b(x1)) -> a#(x1)
       TRS:
        a(x1) -> x1
        a(b(x1)) -> b(b(a(x1)))
        b(x1) -> c(a(c(x1)))
        c(c(x1)) -> x1
       graph:
        b#(x1) -> a#(c(x1)) -> a#(b(x1)) -> a#(x1)
        a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> a#(x1)
       SCC Processor:
        #sccs: 1
        #rules: 1
        #arcs: 2/4
        DPs:
         a#(b(x1)) -> a#(x1)
        TRS:
         a(x1) -> x1
         a(b(x1)) -> b(b(a(x1)))
         b(x1) -> c(a(c(x1)))
         c(c(x1)) -> x1
        Usable Rule Processor:
         DPs:
          a#(b(x1)) -> a#(x1)
         TRS:
          
         Arctic Interpretation Processor:
          dimension: 4
          usable rules:
           
          interpretation:
                     [1 0 0 0]     [0]
                     [0 0 0 0]     [0]
           [b](x0) = [1 0 0 0]x0 + [0]
                     [0 0 0 0]     [0],
           
           [a#](x0) = [0  -& -& -&]x0
          orientation:
           a#(b(x1)) = [1 0 0 0]x1 + [0] >= [0  -& -& -&]x1 = a#(x1)
          problem:
           DPs:
            
           TRS:
            
          Qed