YES

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

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