YES

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

Proof:
 DP Processor:
  DPs:
   b#(b(x1)) -> c#(x1)
   b#(b(x1)) -> c#(c(x1))
   b#(b(x1)) -> c#(c(c(x1)))
   b#(b(x1)) -> c#(c(c(c(x1))))
   c#(c(x1)) -> b#(x1)
   c#(c(x1)) -> c#(b(x1))
   c#(c(x1)) -> a#(c(b(x1)))
  TRS:
   a(a(x1)) -> x1
   b(b(x1)) -> c(c(c(c(x1))))
   c(c(x1)) -> a(c(b(x1)))
  TDG Processor:
   DPs:
    b#(b(x1)) -> c#(x1)
    b#(b(x1)) -> c#(c(x1))
    b#(b(x1)) -> c#(c(c(x1)))
    b#(b(x1)) -> c#(c(c(c(x1))))
    c#(c(x1)) -> b#(x1)
    c#(c(x1)) -> c#(b(x1))
    c#(c(x1)) -> a#(c(b(x1)))
   TRS:
    a(a(x1)) -> x1
    b(b(x1)) -> c(c(c(c(x1))))
    c(c(x1)) -> a(c(b(x1)))
   graph:
    c#(c(x1)) -> c#(b(x1)) -> c#(c(x1)) -> a#(c(b(x1)))
    c#(c(x1)) -> c#(b(x1)) -> c#(c(x1)) -> c#(b(x1))
    c#(c(x1)) -> c#(b(x1)) -> c#(c(x1)) -> b#(x1)
    c#(c(x1)) -> b#(x1) -> b#(b(x1)) -> c#(c(c(c(x1))))
    c#(c(x1)) -> b#(x1) -> b#(b(x1)) -> c#(c(c(x1)))
    c#(c(x1)) -> b#(x1) -> b#(b(x1)) -> c#(c(x1))
    c#(c(x1)) -> b#(x1) -> b#(b(x1)) -> c#(x1)
    b#(b(x1)) -> c#(c(c(c(x1)))) -> c#(c(x1)) -> a#(c(b(x1)))
    b#(b(x1)) -> c#(c(c(c(x1)))) -> c#(c(x1)) -> c#(b(x1))
    b#(b(x1)) -> c#(c(c(c(x1)))) -> c#(c(x1)) -> b#(x1)
    b#(b(x1)) -> c#(c(c(x1))) -> c#(c(x1)) -> a#(c(b(x1)))
    b#(b(x1)) -> c#(c(c(x1))) -> c#(c(x1)) -> c#(b(x1))
    b#(b(x1)) -> c#(c(c(x1))) -> c#(c(x1)) -> b#(x1)
    b#(b(x1)) -> c#(c(x1)) -> c#(c(x1)) -> a#(c(b(x1)))
    b#(b(x1)) -> c#(c(x1)) -> c#(c(x1)) -> c#(b(x1))
    b#(b(x1)) -> c#(c(x1)) -> c#(c(x1)) -> b#(x1)
    b#(b(x1)) -> c#(x1) -> c#(c(x1)) -> a#(c(b(x1)))
    b#(b(x1)) -> c#(x1) -> c#(c(x1)) -> c#(b(x1))
    b#(b(x1)) -> c#(x1) -> c#(c(x1)) -> b#(x1)
   SCC Processor:
    #sccs: 1
    #rules: 6
    #arcs: 19/49
    DPs:
     c#(c(x1)) -> c#(b(x1))
     c#(c(x1)) -> b#(x1)
     b#(b(x1)) -> c#(x1)
     b#(b(x1)) -> c#(c(x1))
     b#(b(x1)) -> c#(c(c(x1)))
     b#(b(x1)) -> c#(c(c(c(x1))))
    TRS:
     a(a(x1)) -> x1
     b(b(x1)) -> c(c(c(c(x1))))
     c(c(x1)) -> a(c(b(x1)))
    Arctic Interpretation Processor:
     dimension: 2
     interpretation:
      [c#](x0) = [1 0]x0 + [0],
      
      [b#](x0) = [0 0]x0,
      
                [0  0 ]     [2]
      [c](x0) = [0  -&]x0 + [0],
      
                [-1 -1]     [1]
      [b](x0) = [1  1 ]x0 + [3],
      
                [-& 1 ]     [0 ]
      [a](x0) = [-1 1 ]x0 + [-4]
     orientation:
      c#(c(x1)) = [1 1]x1 + [3] >= [1 1]x1 + [3] = c#(b(x1))
      
      c#(c(x1)) = [1 1]x1 + [3] >= [0 0]x1 = b#(x1)
      
      b#(b(x1)) = [1 1]x1 + [3] >= [1 0]x1 + [0] = c#(x1)
      
      b#(b(x1)) = [1 1]x1 + [3] >= [1 1]x1 + [3] = c#(c(x1))
      
      b#(b(x1)) = [1 1]x1 + [3] >= [1 1]x1 + [3] = c#(c(c(x1)))
      
      b#(b(x1)) = [1 1]x1 + [3] >= [1 1]x1 + [3] = c#(c(c(c(x1))))
      
                 [0 2]     [0 ]           
      a(a(x1)) = [0 2]x1 + [-1] >= x1 = x1
      
                 [0 0]     [2]    [0 0]     [2]                 
      b(b(x1)) = [2 2]x1 + [4] >= [0 0]x1 + [2] = c(c(c(c(x1))))
      
                 [0 0]     [2]    [0 0]     [2]              
      c(c(x1)) = [0 0]x1 + [2] >= [0 0]x1 + [2] = a(c(b(x1)))
     problem:
      DPs:
       c#(c(x1)) -> c#(b(x1))
       b#(b(x1)) -> c#(x1)
       b#(b(x1)) -> c#(c(x1))
       b#(b(x1)) -> c#(c(c(x1)))
       b#(b(x1)) -> c#(c(c(c(x1))))
      TRS:
       a(a(x1)) -> x1
       b(b(x1)) -> c(c(c(c(x1))))
       c(c(x1)) -> a(c(b(x1)))
     EDG Processor:
      DPs:
       c#(c(x1)) -> c#(b(x1))
       b#(b(x1)) -> c#(x1)
       b#(b(x1)) -> c#(c(x1))
       b#(b(x1)) -> c#(c(c(x1)))
       b#(b(x1)) -> c#(c(c(c(x1))))
      TRS:
       a(a(x1)) -> x1
       b(b(x1)) -> c(c(c(c(x1))))
       c(c(x1)) -> a(c(b(x1)))
      graph:
       c#(c(x1)) -> c#(b(x1)) -> c#(c(x1)) -> c#(b(x1))
       b#(b(x1)) -> c#(c(c(c(x1)))) -> c#(c(x1)) -> c#(b(x1))
       b#(b(x1)) -> c#(c(c(x1))) -> c#(c(x1)) -> c#(b(x1))
       b#(b(x1)) -> c#(c(x1)) -> c#(c(x1)) -> c#(b(x1))
       b#(b(x1)) -> c#(x1) -> c#(c(x1)) -> c#(b(x1))
      SCC Processor:
       #sccs: 1
       #rules: 1
       #arcs: 5/25
       DPs:
        c#(c(x1)) -> c#(b(x1))
       TRS:
        a(a(x1)) -> x1
        b(b(x1)) -> c(c(c(c(x1))))
        c(c(x1)) -> a(c(b(x1)))
       Arctic Interpretation Processor:
        dimension: 2
        interpretation:
         [c#](x0) = [-& 0 ]x0 + [0],
         
                   [-& 0 ]     [0]
         [c](x0) = [0  0 ]x0 + [1],
         
                   [1  1 ]     [2]
         [b](x0) = [-1 -4]x0 + [0],
         
                   [1  -1]     [0]
         [a](x0) = [1  -&]x0 + [0]
        orientation:
         c#(c(x1)) = [0 0]x1 + [1] >= [-1 -4]x1 + [0] = c#(b(x1))
         
                    [2 0]     [1]           
         a(a(x1)) = [2 0]x1 + [1] >= x1 = x1
         
                    [2 2]     [3]    [0 0]     [1]                 
         b(b(x1)) = [0 0]x1 + [1] >= [0 0]x1 + [1] = c(c(c(c(x1))))
         
                    [0 0]     [1]    [0  0 ]     [1]              
         c(c(x1)) = [0 0]x1 + [1] >= [0  -3]x1 + [1] = a(c(b(x1)))
        problem:
         DPs:
          
         TRS:
          a(a(x1)) -> x1
          b(b(x1)) -> c(c(c(c(x1))))
          c(c(x1)) -> a(c(b(x1)))
        Qed