YES

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

Proof:
 DP Processor:
  DPs:
   a#(a(x)) -> c#(x)
   a#(a(x)) -> b#(c(x))
   b#(b(x)) -> c#(x)
   b#(b(x)) -> a#(c(x))
   c#(c(x)) -> b#(x)
   c#(c(x)) -> a#(b(x))
  TRS:
   a(a(x)) -> b(c(x))
   b(b(x)) -> a(c(x))
   c(c(x)) -> a(b(x))
  TDG Processor:
   DPs:
    a#(a(x)) -> c#(x)
    a#(a(x)) -> b#(c(x))
    b#(b(x)) -> c#(x)
    b#(b(x)) -> a#(c(x))
    c#(c(x)) -> b#(x)
    c#(c(x)) -> a#(b(x))
   TRS:
    a(a(x)) -> b(c(x))
    b(b(x)) -> a(c(x))
    c(c(x)) -> a(b(x))
   graph:
    b#(b(x)) -> c#(x) -> c#(c(x)) -> a#(b(x))
    b#(b(x)) -> c#(x) -> c#(c(x)) -> b#(x)
    b#(b(x)) -> a#(c(x)) -> a#(a(x)) -> b#(c(x))
    b#(b(x)) -> a#(c(x)) -> a#(a(x)) -> c#(x)
    c#(c(x)) -> b#(x) -> b#(b(x)) -> a#(c(x))
    c#(c(x)) -> b#(x) -> b#(b(x)) -> c#(x)
    c#(c(x)) -> a#(b(x)) -> a#(a(x)) -> b#(c(x))
    c#(c(x)) -> a#(b(x)) -> a#(a(x)) -> c#(x)
    a#(a(x)) -> b#(c(x)) -> b#(b(x)) -> a#(c(x))
    a#(a(x)) -> b#(c(x)) -> b#(b(x)) -> c#(x)
    a#(a(x)) -> c#(x) -> c#(c(x)) -> a#(b(x))
    a#(a(x)) -> c#(x) -> c#(c(x)) -> b#(x)
   Arctic Interpretation Processor:
    dimension: 1
    interpretation:
     [b#](x0) = x0 + 0,
     
     [c#](x0) = x0 + 0,
     
     [a#](x0) = x0 + 0,
     
     [b](x0) = 1x0 + 0,
     
     [c](x0) = 1x0 + 0,
     
     [a](x0) = 1x0 + 1
    orientation:
     a#(a(x)) = 1x + 1 >= x + 0 = c#(x)
     
     a#(a(x)) = 1x + 1 >= 1x + 0 = b#(c(x))
     
     b#(b(x)) = 1x + 0 >= x + 0 = c#(x)
     
     b#(b(x)) = 1x + 0 >= 1x + 0 = a#(c(x))
     
     c#(c(x)) = 1x + 0 >= x + 0 = b#(x)
     
     c#(c(x)) = 1x + 0 >= 1x + 0 = a#(b(x))
     
     a(a(x)) = 2x + 2 >= 2x + 1 = b(c(x))
     
     b(b(x)) = 2x + 1 >= 2x + 1 = a(c(x))
     
     c(c(x)) = 2x + 1 >= 2x + 1 = a(b(x))
    problem:
     DPs:
      a#(a(x)) -> b#(c(x))
      b#(b(x)) -> c#(x)
      b#(b(x)) -> a#(c(x))
      c#(c(x)) -> b#(x)
      c#(c(x)) -> a#(b(x))
     TRS:
      a(a(x)) -> b(c(x))
      b(b(x)) -> a(c(x))
      c(c(x)) -> a(b(x))
    EDG Processor:
     DPs:
      a#(a(x)) -> b#(c(x))
      b#(b(x)) -> c#(x)
      b#(b(x)) -> a#(c(x))
      c#(c(x)) -> b#(x)
      c#(c(x)) -> a#(b(x))
     TRS:
      a(a(x)) -> b(c(x))
      b(b(x)) -> a(c(x))
      c(c(x)) -> a(b(x))
     graph:
      b#(b(x)) -> c#(x) -> c#(c(x)) -> b#(x)
      b#(b(x)) -> c#(x) -> c#(c(x)) -> a#(b(x))
      b#(b(x)) -> a#(c(x)) -> a#(a(x)) -> b#(c(x))
      c#(c(x)) -> b#(x) -> b#(b(x)) -> c#(x)
      c#(c(x)) -> b#(x) -> b#(b(x)) -> a#(c(x))
      c#(c(x)) -> a#(b(x)) -> a#(a(x)) -> b#(c(x))
      a#(a(x)) -> b#(c(x)) -> b#(b(x)) -> c#(x)
      a#(a(x)) -> b#(c(x)) -> b#(b(x)) -> a#(c(x))
     Arctic Interpretation Processor:
      dimension: 1
      interpretation:
       [b#](x0) = 1x0,
       
       [c#](x0) = 9x0,
       
       [a#](x0) = 1x0,
       
       [b](x0) = 8x0,
       
       [c](x0) = 8x0,
       
       [a](x0) = 8x0
      orientation:
       a#(a(x)) = 9x >= 9x = b#(c(x))
       
       b#(b(x)) = 9x >= 9x = c#(x)
       
       b#(b(x)) = 9x >= 9x = a#(c(x))
       
       c#(c(x)) = 17x >= 1x = b#(x)
       
       c#(c(x)) = 17x >= 9x = a#(b(x))
       
       a(a(x)) = 16x >= 16x = b(c(x))
       
       b(b(x)) = 16x >= 16x = a(c(x))
       
       c(c(x)) = 16x >= 16x = a(b(x))
      problem:
       DPs:
        a#(a(x)) -> b#(c(x))
        b#(b(x)) -> c#(x)
        b#(b(x)) -> a#(c(x))
       TRS:
        a(a(x)) -> b(c(x))
        b(b(x)) -> a(c(x))
        c(c(x)) -> a(b(x))
      EDG Processor:
       DPs:
        a#(a(x)) -> b#(c(x))
        b#(b(x)) -> c#(x)
        b#(b(x)) -> a#(c(x))
       TRS:
        a(a(x)) -> b(c(x))
        b(b(x)) -> a(c(x))
        c(c(x)) -> a(b(x))
       graph:
        b#(b(x)) -> a#(c(x)) -> a#(a(x)) -> b#(c(x))
        a#(a(x)) -> b#(c(x)) -> b#(b(x)) -> a#(c(x))
        a#(a(x)) -> b#(c(x)) -> b#(b(x)) -> c#(x)
       SCC Processor:
        #sccs: 1
        #rules: 2
        #arcs: 3/9
        DPs:
         b#(b(x)) -> a#(c(x))
         a#(a(x)) -> b#(c(x))
        TRS:
         a(a(x)) -> b(c(x))
         b(b(x)) -> a(c(x))
         c(c(x)) -> a(b(x))
        Arctic Interpretation Processor:
         dimension: 2
         interpretation:
          [b#](x0) = [3 0]x0 + [0],
          
          [a#](x0) = [2 1]x0 + [0],
          
                    [1  0 ]     [3]
          [b](x0) = [0  -&]x0 + [0],
          
                    [-& 0 ]     [0]
          [c](x0) = [1  0 ]x0 + [3],
          
                    [-& 1 ]     [2]
          [a](x0) = [0  0 ]x0 + [3]
         orientation:
          b#(b(x)) = [4 3]x + [6] >= [2 2]x + [4] = a#(c(x))
          
          a#(a(x)) = [1 3]x + [4] >= [1 3]x + [3] = b#(c(x))
          
                    [1 1]    [4]    [1  1 ]    [3]          
          a(a(x)) = [0 1]x + [3] >= [-& 0 ]x + [0] = b(c(x))
          
                    [2 1]    [4]    [2 1]    [4]          
          b(b(x)) = [1 0]x + [3] >= [1 0]x + [3] = a(c(x))
          
                    [1 0]    [3]    [1  -&]    [2]          
          c(c(x)) = [1 1]x + [3] >= [1  0 ]x + [3] = a(b(x))
         problem:
          DPs:
           a#(a(x)) -> b#(c(x))
          TRS:
           a(a(x)) -> b(c(x))
           b(b(x)) -> a(c(x))
           c(c(x)) -> a(b(x))
         EDG Processor:
          DPs:
           a#(a(x)) -> b#(c(x))
          TRS:
           a(a(x)) -> b(c(x))
           b(b(x)) -> a(c(x))
           c(c(x)) -> a(b(x))
          graph:
           
          Qed