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
     interpretation:
      [b#](x0) = [2 0]x0 + [0],
      
      [a#](x0) = [0 2]x0 + [0],
      
                [-& 0 ]     [0 ]
      [c](x0) = [0  -&]x0 + [-&],
      
                [0  -&]     [0]
      [b](x0) = [1  0 ]x0 + [2],
      
                  
      [a](x0) = x0
     orientation:
      b#(x1) = [2 0]x1 + [0] >= [2 0]x1 + [0] = a#(c(x1))
      
      a#(b(x1)) = [3 2]x1 + [4] >= [0 2]x1 + [0] = a#(x1)
      
      a#(b(x1)) = [3 2]x1 + [4] >= [2 0]x1 + [0] = b#(a(x1))
      
      a#(b(x1)) = [3 2]x1 + [4] >= [2 0]x1 + [2] = b#(b(a(x1)))
      
                           
      a(x1) = x1 >= x1 = x1
      
                 [0  -&]     [0]    [0  -&]     [0]              
      a(b(x1)) = [1  0 ]x1 + [2] >= [1  0 ]x1 + [2] = b(b(a(x1)))
      
              [0  -&]     [0]         [0]              
      b(x1) = [1  0 ]x1 + [2] >= 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
     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)
      CDG 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:
        
       Qed