YES

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

Proof:
 DP Processor:
  DPs:
   a#(a(x1)) -> c#(x1)
   a#(a(x1)) -> b#(c(x1))
   b#(b(x1)) -> c#(x1)
   b#(b(x1)) -> c#(c(x1))
   b#(b(x1)) -> c#(c(c(x1)))
   b#(b(x1)) -> a#(c(c(c(x1))))
   c#(c(x1)) -> b#(x1)
   c#(c(x1)) -> a#(b(x1))
  TRS:
   a(a(x1)) -> b(c(x1))
   b(b(x1)) -> a(c(c(c(x1))))
   c(c(x1)) -> a(b(x1))
  TDG Processor:
   DPs:
    a#(a(x1)) -> c#(x1)
    a#(a(x1)) -> b#(c(x1))
    b#(b(x1)) -> c#(x1)
    b#(b(x1)) -> c#(c(x1))
    b#(b(x1)) -> c#(c(c(x1)))
    b#(b(x1)) -> a#(c(c(c(x1))))
    c#(c(x1)) -> b#(x1)
    c#(c(x1)) -> a#(b(x1))
   TRS:
    a(a(x1)) -> b(c(x1))
    b(b(x1)) -> a(c(c(c(x1))))
    c(c(x1)) -> a(b(x1))
   graph:
    b#(b(x1)) -> c#(c(c(x1))) -> c#(c(x1)) -> a#(b(x1))
    b#(b(x1)) -> c#(c(c(x1))) -> c#(c(x1)) -> b#(x1)
    b#(b(x1)) -> c#(c(x1)) -> c#(c(x1)) -> a#(b(x1))
    b#(b(x1)) -> c#(c(x1)) -> c#(c(x1)) -> b#(x1)
    b#(b(x1)) -> c#(x1) -> c#(c(x1)) -> a#(b(x1))
    b#(b(x1)) -> c#(x1) -> c#(c(x1)) -> b#(x1)
    b#(b(x1)) -> a#(c(c(c(x1)))) -> a#(a(x1)) -> b#(c(x1))
    b#(b(x1)) -> a#(c(c(c(x1)))) -> a#(a(x1)) -> c#(x1)
    c#(c(x1)) -> b#(x1) -> b#(b(x1)) -> a#(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)
    c#(c(x1)) -> a#(b(x1)) -> a#(a(x1)) -> b#(c(x1))
    c#(c(x1)) -> a#(b(x1)) -> a#(a(x1)) -> c#(x1)
    a#(a(x1)) -> b#(c(x1)) -> b#(b(x1)) -> a#(c(c(c(x1))))
    a#(a(x1)) -> b#(c(x1)) -> b#(b(x1)) -> c#(c(c(x1)))
    a#(a(x1)) -> b#(c(x1)) -> b#(b(x1)) -> c#(c(x1))
    a#(a(x1)) -> b#(c(x1)) -> b#(b(x1)) -> c#(x1)
    a#(a(x1)) -> c#(x1) -> c#(c(x1)) -> a#(b(x1))
    a#(a(x1)) -> c#(x1) -> c#(c(x1)) -> b#(x1)
   Arctic Interpretation Processor:
    dimension: 2
    interpretation:
     [b#](x0) = [1  -&]x0 + [0],
     
     [c#](x0) = [1  -&]x0 + [0],
     
     [a#](x0) = [-& 0 ]x0 + [0],
     
               [1  -&]     [3]
     [b](x0) = [0  -&]x0 + [0],
     
               [0  -&]     [2]
     [c](x0) = [2  -&]x0 + [0],
     
               [-1 0 ]     [2]
     [a](x0) = [1  -&]x0 + [3]
    orientation:
     a#(a(x1)) = [1  -&]x1 + [3] >= [1  -&]x1 + [0] = c#(x1)
     
     a#(a(x1)) = [1  -&]x1 + [3] >= [1  -&]x1 + [3] = b#(c(x1))
     
     b#(b(x1)) = [2  -&]x1 + [4] >= [1  -&]x1 + [0] = c#(x1)
     
     b#(b(x1)) = [2  -&]x1 + [4] >= [1  -&]x1 + [3] = c#(c(x1))
     
     b#(b(x1)) = [2  -&]x1 + [4] >= [1  -&]x1 + [3] = c#(c(c(x1)))
     
     b#(b(x1)) = [2  -&]x1 + [4] >= [2  -&]x1 + [4] = a#(c(c(c(x1))))
     
     c#(c(x1)) = [1  -&]x1 + [3] >= [1  -&]x1 + [0] = b#(x1)
     
     c#(c(x1)) = [1  -&]x1 + [3] >= [0  -&]x1 + [0] = a#(b(x1))
     
                [1  -1]     [3]    [1  -&]     [3]           
     a(a(x1)) = [0  1 ]x1 + [3] >= [0  -&]x1 + [2] = b(c(x1))
     
                [2  -&]     [4]    [2  -&]     [4]                 
     b(b(x1)) = [1  -&]x1 + [3] >= [1  -&]x1 + [3] = a(c(c(c(x1))))
     
                [0  -&]     [2]    [0  -&]     [2]           
     c(c(x1)) = [2  -&]x1 + [4] >= [2  -&]x1 + [4] = a(b(x1))
    problem:
     DPs:
      a#(a(x1)) -> c#(x1)
      a#(a(x1)) -> b#(c(x1))
      b#(b(x1)) -> a#(c(c(c(x1))))
      c#(c(x1)) -> b#(x1)
     TRS:
      a(a(x1)) -> b(c(x1))
      b(b(x1)) -> a(c(c(c(x1))))
      c(c(x1)) -> a(b(x1))
    EDG Processor:
     DPs:
      a#(a(x1)) -> c#(x1)
      a#(a(x1)) -> b#(c(x1))
      b#(b(x1)) -> a#(c(c(c(x1))))
      c#(c(x1)) -> b#(x1)
     TRS:
      a(a(x1)) -> b(c(x1))
      b(b(x1)) -> a(c(c(c(x1))))
      c(c(x1)) -> a(b(x1))
     graph:
      b#(b(x1)) -> a#(c(c(c(x1)))) -> a#(a(x1)) -> c#(x1)
      b#(b(x1)) -> a#(c(c(c(x1)))) -> a#(a(x1)) -> b#(c(x1))
      c#(c(x1)) -> b#(x1) -> b#(b(x1)) -> a#(c(c(c(x1))))
      a#(a(x1)) -> b#(c(x1)) -> b#(b(x1)) -> a#(c(c(c(x1))))
      a#(a(x1)) -> c#(x1) -> c#(c(x1)) -> b#(x1)
     Arctic Interpretation Processor:
      dimension: 2
      interpretation:
       [b#](x0) = [-& 1 ]x0 + [0],
       
       [c#](x0) = [0 0]x0 + [3],
       
       [a#](x0) = [0 0]x0 + [3],
       
                 [-& 0 ]     [0]
       [b](x0) = [-2 1 ]x0 + [2],
       
                 [-4 2 ]     [2]
       [c](x0) = [-2 -1]x0 + [1],
       
                 [-4 0 ]     [2 ]
       [a](x0) = [0  -1]x0 + [-2]
      orientation:
       a#(a(x1)) = [0 0]x1 + [3] >= [0 0]x1 + [3] = c#(x1)
       
       a#(a(x1)) = [0 0]x1 + [3] >= [-1 0 ]x1 + [2] = b#(c(x1))
       
       b#(b(x1)) = [-1 2 ]x1 + [3] >= [-1 2 ]x1 + [3] = a#(c(c(c(x1))))
       
       c#(c(x1)) = [-2 2 ]x1 + [3] >= [-& 1 ]x1 + [0] = b#(x1)
       
                  [0  -1]     [2]    [-2 -1]     [1]           
       a(a(x1)) = [-1 0 ]x1 + [2] >= [-1 0 ]x1 + [2] = b(c(x1))
       
                  [-2 1 ]     [2]    [-2 -1]     [2]                 
       b(b(x1)) = [-1 2 ]x1 + [3] >= [-1 2 ]x1 + [3] = a(c(c(c(x1))))
       
                  [0  1 ]     [3]    [-2 1 ]     [2]           
       c(c(x1)) = [-3 0 ]x1 + [1] >= [-3 0 ]x1 + [1] = a(b(x1))
      problem:
       DPs:
        a#(a(x1)) -> c#(x1)
        a#(a(x1)) -> b#(c(x1))
        b#(b(x1)) -> a#(c(c(c(x1))))
       TRS:
        a(a(x1)) -> b(c(x1))
        b(b(x1)) -> a(c(c(c(x1))))
        c(c(x1)) -> a(b(x1))
      EDG Processor:
       DPs:
        a#(a(x1)) -> c#(x1)
        a#(a(x1)) -> b#(c(x1))
        b#(b(x1)) -> a#(c(c(c(x1))))
       TRS:
        a(a(x1)) -> b(c(x1))
        b(b(x1)) -> a(c(c(c(x1))))
        c(c(x1)) -> a(b(x1))
       graph:
        b#(b(x1)) -> a#(c(c(c(x1)))) -> a#(a(x1)) -> b#(c(x1))
        b#(b(x1)) -> a#(c(c(c(x1)))) -> a#(a(x1)) -> c#(x1)
        a#(a(x1)) -> b#(c(x1)) -> b#(b(x1)) -> a#(c(c(c(x1))))
       SCC Processor:
        #sccs: 1
        #rules: 2
        #arcs: 3/9
        DPs:
         b#(b(x1)) -> a#(c(c(c(x1))))
         a#(a(x1)) -> b#(c(x1))
        TRS:
         a(a(x1)) -> b(c(x1))
         b(b(x1)) -> a(c(c(c(x1))))
         c(c(x1)) -> a(b(x1))
        Matrix Interpretation Processor: dim=3
         
         interpretation:
          [b#](x0) = [0 0 1]x0 + [1],
          
          [a#](x0) = [0 0 1]x0 + [1],
          
                    [0 0 0]     [0]
          [b](x0) = [0 0 1]x0 + [0]
                    [1 0 2]     [1],
          
                    [0 0 1]     [0]
          [c](x0) = [2 0 2]x0 + [2]
                    [1 0 0]     [0],
          
                    [0 0 0]     [0]
          [a](x0) = [2 0 1]x0 + [1]
                    [1 1 0]     [0]
         orientation:
          b#(b(x1)) = [1 0 2]x1 + [2] >= [1 0 0]x1 + [1] = a#(c(c(c(x1))))
          
          a#(a(x1)) = [1 1 0]x1 + [1] >= [1 0 0]x1 + [1] = b#(c(x1))
          
                     [0 0 0]     [0]    [0 0 0]     [0]           
          a(a(x1)) = [1 1 0]x1 + [1] >= [1 0 0]x1 + [0] = b(c(x1))
                     [2 0 1]     [1]    [2 0 1]     [1]           
          
                     [0 0 0]     [0]    [0 0 0]     [0]                 
          b(b(x1)) = [1 0 2]x1 + [1] >= [1 0 2]x1 + [1] = a(c(c(c(x1))))
                     [2 0 4]     [3]    [2 0 3]     [2]                 
          
                     [1 0 0]     [0]    [0 0 0]     [0]           
          c(c(x1)) = [2 0 2]x1 + [2] >= [1 0 2]x1 + [2] = a(b(x1))
                     [0 0 1]     [0]    [0 0 1]     [0]           
         problem:
          DPs:
           a#(a(x1)) -> b#(c(x1))
          TRS:
           a(a(x1)) -> b(c(x1))
           b(b(x1)) -> a(c(c(c(x1))))
           c(c(x1)) -> a(b(x1))
         EDG Processor:
          DPs:
           a#(a(x1)) -> b#(c(x1))
          TRS:
           a(a(x1)) -> b(c(x1))
           b(b(x1)) -> a(c(c(c(x1))))
           c(c(x1)) -> a(b(x1))
          graph:
           
          Qed