YES

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

Proof:
 DP Processor:
  DPs:
   a#(b(x1)) -> b#(b(x1))
   a#(b(x1)) -> b#(b(b(x1)))
   a#(c(x1)) -> b#(x1)
   a#(c(x1)) -> b#(b(x1))
   a#(c(x1)) -> c#(b(b(x1)))
   b#(a(x1)) -> a#(a(x1))
   b#(a(x1)) -> a#(a(a(x1)))
   b#(c(x1)) -> c#(c(x1))
   b#(c(x1)) -> c#(c(c(x1)))
  TRS:
   a(b(x1)) -> b(b(b(x1)))
   a(c(x1)) -> c(b(b(x1)))
   b(a(x1)) -> a(a(a(x1)))
   b(c(x1)) -> c(c(c(x1)))
   a(x1) -> x1
   b(x1) -> x1
   c(x1) -> x1
  TDG Processor:
   DPs:
    a#(b(x1)) -> b#(b(x1))
    a#(b(x1)) -> b#(b(b(x1)))
    a#(c(x1)) -> b#(x1)
    a#(c(x1)) -> b#(b(x1))
    a#(c(x1)) -> c#(b(b(x1)))
    b#(a(x1)) -> a#(a(x1))
    b#(a(x1)) -> a#(a(a(x1)))
    b#(c(x1)) -> c#(c(x1))
    b#(c(x1)) -> c#(c(c(x1)))
   TRS:
    a(b(x1)) -> b(b(b(x1)))
    a(c(x1)) -> c(b(b(x1)))
    b(a(x1)) -> a(a(a(x1)))
    b(c(x1)) -> c(c(c(x1)))
    a(x1) -> x1
    b(x1) -> x1
    c(x1) -> x1
   graph:
    b#(a(x1)) -> a#(a(a(x1))) -> a#(c(x1)) -> c#(b(b(x1)))
    b#(a(x1)) -> a#(a(a(x1))) -> a#(c(x1)) -> b#(b(x1))
    b#(a(x1)) -> a#(a(a(x1))) -> a#(c(x1)) -> b#(x1)
    b#(a(x1)) -> a#(a(a(x1))) -> a#(b(x1)) -> b#(b(b(x1)))
    b#(a(x1)) -> a#(a(a(x1))) -> a#(b(x1)) -> b#(b(x1))
    b#(a(x1)) -> a#(a(x1)) -> a#(c(x1)) -> c#(b(b(x1)))
    b#(a(x1)) -> a#(a(x1)) -> a#(c(x1)) -> b#(b(x1))
    b#(a(x1)) -> a#(a(x1)) -> a#(c(x1)) -> b#(x1)
    b#(a(x1)) -> a#(a(x1)) -> a#(b(x1)) -> b#(b(b(x1)))
    b#(a(x1)) -> a#(a(x1)) -> a#(b(x1)) -> b#(b(x1))
    a#(c(x1)) -> b#(b(x1)) -> b#(c(x1)) -> c#(c(c(x1)))
    a#(c(x1)) -> b#(b(x1)) -> b#(c(x1)) -> c#(c(x1))
    a#(c(x1)) -> b#(b(x1)) -> b#(a(x1)) -> a#(a(a(x1)))
    a#(c(x1)) -> b#(b(x1)) -> b#(a(x1)) -> a#(a(x1))
    a#(c(x1)) -> b#(x1) -> b#(c(x1)) -> c#(c(c(x1)))
    a#(c(x1)) -> b#(x1) -> b#(c(x1)) -> c#(c(x1))
    a#(c(x1)) -> b#(x1) -> b#(a(x1)) -> a#(a(a(x1)))
    a#(c(x1)) -> b#(x1) -> b#(a(x1)) -> a#(a(x1))
    a#(b(x1)) -> b#(b(b(x1))) -> b#(c(x1)) -> c#(c(c(x1)))
    a#(b(x1)) -> b#(b(b(x1))) -> b#(c(x1)) -> c#(c(x1))
    a#(b(x1)) -> b#(b(b(x1))) -> b#(a(x1)) -> a#(a(a(x1)))
    a#(b(x1)) -> b#(b(b(x1))) -> b#(a(x1)) -> a#(a(x1))
    a#(b(x1)) -> b#(b(x1)) -> b#(c(x1)) -> c#(c(c(x1)))
    a#(b(x1)) -> b#(b(x1)) -> b#(c(x1)) -> c#(c(x1))
    a#(b(x1)) -> b#(b(x1)) -> b#(a(x1)) -> a#(a(a(x1)))
    a#(b(x1)) -> b#(b(x1)) -> b#(a(x1)) -> a#(a(x1))
   SCC Processor:
    #sccs: 1
    #rules: 6
    #arcs: 26/81
    DPs:
     b#(a(x1)) -> a#(a(a(x1)))
     a#(b(x1)) -> b#(b(x1))
     b#(a(x1)) -> a#(a(x1))
     a#(b(x1)) -> b#(b(b(x1)))
     a#(c(x1)) -> b#(x1)
     a#(c(x1)) -> b#(b(x1))
    TRS:
     a(b(x1)) -> b(b(b(x1)))
     a(c(x1)) -> c(b(b(x1)))
     b(a(x1)) -> a(a(a(x1)))
     b(c(x1)) -> c(c(c(x1)))
     a(x1) -> x1
     b(x1) -> x1
     c(x1) -> x1
    Arctic Interpretation Processor:
     dimension: 2
     interpretation:
      [b#](x0) = [-& 0 ]x0 + [0],
      
      [a#](x0) = [0 0]x0 + [0],
      
                [0  1 ]     [1]
      [c](x0) = [-& 0 ]x0 + [0],
      
                [0  -&]     [0]
      [a](x0) = [1  0 ]x0 + [1],
      
                [0  0 ]     [1 ]
      [b](x0) = [-& 0 ]x0 + [-&]
     orientation:
      b#(a(x1)) = [1 0]x1 + [1] >= [1 0]x1 + [1] = a#(a(a(x1)))
      
      a#(b(x1)) = [0 0]x1 + [1] >= [-& 0 ]x1 + [0] = b#(b(x1))
      
      b#(a(x1)) = [1 0]x1 + [1] >= [1 0]x1 + [1] = a#(a(x1))
      
      a#(b(x1)) = [0 0]x1 + [1] >= [-& 0 ]x1 + [0] = b#(b(b(x1)))
      
      a#(c(x1)) = [0 1]x1 + [1] >= [-& 0 ]x1 + [0] = b#(x1)
      
      a#(c(x1)) = [0 1]x1 + [1] >= [-& 0 ]x1 + [0] = b#(b(x1))
      
                 [0 0]     [1]    [0  0 ]     [1 ]              
      a(b(x1)) = [1 1]x1 + [2] >= [-& 0 ]x1 + [-&] = b(b(b(x1)))
      
                 [0 1]     [1]    [0  1 ]     [1]              
      a(c(x1)) = [1 2]x1 + [2] >= [-& 0 ]x1 + [0] = c(b(b(x1)))
      
                 [1 0]     [1]    [0  -&]     [0]              
      b(a(x1)) = [1 0]x1 + [1] >= [1  0 ]x1 + [1] = a(a(a(x1)))
      
                 [0  1 ]     [1]    [0  1 ]     [1]              
      b(c(x1)) = [-& 0 ]x1 + [0] >= [-& 0 ]x1 + [0] = c(c(c(x1)))
      
              [0  -&]     [0]           
      a(x1) = [1  0 ]x1 + [1] >= x1 = x1
      
              [0  0 ]     [1 ]           
      b(x1) = [-& 0 ]x1 + [-&] >= x1 = x1
      
              [0  1 ]     [1]           
      c(x1) = [-& 0 ]x1 + [0] >= x1 = x1
     problem:
      DPs:
       b#(a(x1)) -> a#(a(a(x1)))
       a#(b(x1)) -> b#(b(x1))
       b#(a(x1)) -> a#(a(x1))
       a#(b(x1)) -> b#(b(b(x1)))
      TRS:
       a(b(x1)) -> b(b(b(x1)))
       a(c(x1)) -> c(b(b(x1)))
       b(a(x1)) -> a(a(a(x1)))
       b(c(x1)) -> c(c(c(x1)))
       a(x1) -> x1
       b(x1) -> x1
       c(x1) -> x1
     EDG Processor:
      DPs:
       b#(a(x1)) -> a#(a(a(x1)))
       a#(b(x1)) -> b#(b(x1))
       b#(a(x1)) -> a#(a(x1))
       a#(b(x1)) -> b#(b(b(x1)))
      TRS:
       a(b(x1)) -> b(b(b(x1)))
       a(c(x1)) -> c(b(b(x1)))
       b(a(x1)) -> a(a(a(x1)))
       b(c(x1)) -> c(c(c(x1)))
       a(x1) -> x1
       b(x1) -> x1
       c(x1) -> x1
      graph:
       b#(a(x1)) -> a#(a(a(x1))) -> a#(b(x1)) -> b#(b(x1))
       b#(a(x1)) -> a#(a(a(x1))) -> a#(b(x1)) -> b#(b(b(x1)))
       b#(a(x1)) -> a#(a(x1)) -> a#(b(x1)) -> b#(b(x1))
       b#(a(x1)) -> a#(a(x1)) -> a#(b(x1)) -> b#(b(b(x1)))
       a#(b(x1)) -> b#(b(b(x1))) -> b#(a(x1)) -> a#(a(x1))
       a#(b(x1)) -> b#(b(b(x1))) -> b#(a(x1)) -> a#(a(a(x1)))
       a#(b(x1)) -> b#(b(x1)) -> b#(a(x1)) -> a#(a(x1))
       a#(b(x1)) -> b#(b(x1)) -> b#(a(x1)) -> a#(a(a(x1)))
      Arctic Interpretation Processor:
       dimension: 2
       interpretation:
        [b#](x0) = [0 0]x0 + [0],
        
        [a#](x0) = [-& 0 ]x0 + [0],
        
                  [0 0]     [1]
        [c](x0) = [0 0]x0 + [1],
        
                  [0  2 ]     [2]
        [a](x0) = [-& 0 ]x0 + [0],
        
                  [0  -&]     [0]
        [b](x0) = [0  0 ]x0 + [0]
       orientation:
        b#(a(x1)) = [0 2]x1 + [2] >= [-& 0 ]x1 + [0] = a#(a(a(x1)))
        
        a#(b(x1)) = [0 0]x1 + [0] >= [0 0]x1 + [0] = b#(b(x1))
        
        b#(a(x1)) = [0 2]x1 + [2] >= [-& 0 ]x1 + [0] = a#(a(x1))
        
        a#(b(x1)) = [0 0]x1 + [0] >= [0 0]x1 + [0] = b#(b(b(x1)))
        
                   [2 2]     [2]    [0  -&]     [0]              
        a(b(x1)) = [0 0]x1 + [0] >= [0  0 ]x1 + [0] = b(b(b(x1)))
        
                   [2 2]     [3]    [0 0]     [1]              
        a(c(x1)) = [0 0]x1 + [1] >= [0 0]x1 + [1] = c(b(b(x1)))
        
                   [0 2]     [2]    [0  2 ]     [2]              
        b(a(x1)) = [0 2]x1 + [2] >= [-& 0 ]x1 + [0] = a(a(a(x1)))
        
                   [0 0]     [1]    [0 0]     [1]              
        b(c(x1)) = [0 0]x1 + [1] >= [0 0]x1 + [1] = c(c(c(x1)))
        
                [0  2 ]     [2]           
        a(x1) = [-& 0 ]x1 + [0] >= x1 = x1
        
                [0  -&]     [0]           
        b(x1) = [0  0 ]x1 + [0] >= x1 = x1
        
                [0 0]     [1]           
        c(x1) = [0 0]x1 + [1] >= x1 = x1
       problem:
        DPs:
         a#(b(x1)) -> b#(b(x1))
         a#(b(x1)) -> b#(b(b(x1)))
        TRS:
         a(b(x1)) -> b(b(b(x1)))
         a(c(x1)) -> c(b(b(x1)))
         b(a(x1)) -> a(a(a(x1)))
         b(c(x1)) -> c(c(c(x1)))
         a(x1) -> x1
         b(x1) -> x1
         c(x1) -> x1
       EDG Processor:
        DPs:
         a#(b(x1)) -> b#(b(x1))
         a#(b(x1)) -> b#(b(b(x1)))
        TRS:
         a(b(x1)) -> b(b(b(x1)))
         a(c(x1)) -> c(b(b(x1)))
         b(a(x1)) -> a(a(a(x1)))
         b(c(x1)) -> c(c(c(x1)))
         a(x1) -> x1
         b(x1) -> x1
         c(x1) -> x1
        graph:
         
        Qed