YES

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

Proof:
 DP Processor:
  DPs:
   a#(a(x1)) -> b#(x1)
   a#(a(x1)) -> c#(b(x1))
   a#(a(x1)) -> a#(c(b(x1)))
   a#(a(x1)) -> b#(a(c(b(x1))))
   b#(b(x1)) -> a#(x1)
   b#(b(x1)) -> a#(a(x1))
  TRS:
   a(a(x1)) -> b(a(c(b(x1))))
   b(b(x1)) -> a(a(x1))
   c(a(x1)) -> x1
  TDG Processor:
   DPs:
    a#(a(x1)) -> b#(x1)
    a#(a(x1)) -> c#(b(x1))
    a#(a(x1)) -> a#(c(b(x1)))
    a#(a(x1)) -> b#(a(c(b(x1))))
    b#(b(x1)) -> a#(x1)
    b#(b(x1)) -> a#(a(x1))
   TRS:
    a(a(x1)) -> b(a(c(b(x1))))
    b(b(x1)) -> a(a(x1))
    c(a(x1)) -> x1
   graph:
    b#(b(x1)) -> a#(a(x1)) -> a#(a(x1)) -> b#(a(c(b(x1))))
    b#(b(x1)) -> a#(a(x1)) -> a#(a(x1)) -> a#(c(b(x1)))
    b#(b(x1)) -> a#(a(x1)) -> a#(a(x1)) -> c#(b(x1))
    b#(b(x1)) -> a#(a(x1)) -> a#(a(x1)) -> b#(x1)
    b#(b(x1)) -> a#(x1) -> a#(a(x1)) -> b#(a(c(b(x1))))
    b#(b(x1)) -> a#(x1) -> a#(a(x1)) -> a#(c(b(x1)))
    b#(b(x1)) -> a#(x1) -> a#(a(x1)) -> c#(b(x1))
    b#(b(x1)) -> a#(x1) -> a#(a(x1)) -> b#(x1)
    a#(a(x1)) -> b#(a(c(b(x1)))) -> b#(b(x1)) -> a#(a(x1))
    a#(a(x1)) -> b#(a(c(b(x1)))) -> b#(b(x1)) -> a#(x1)
    a#(a(x1)) -> b#(x1) -> b#(b(x1)) -> a#(a(x1))
    a#(a(x1)) -> b#(x1) -> b#(b(x1)) -> a#(x1)
    a#(a(x1)) -> a#(c(b(x1))) -> a#(a(x1)) -> b#(a(c(b(x1))))
    a#(a(x1)) -> a#(c(b(x1))) -> a#(a(x1)) -> a#(c(b(x1)))
    a#(a(x1)) -> a#(c(b(x1))) -> a#(a(x1)) -> c#(b(x1))
    a#(a(x1)) -> a#(c(b(x1))) -> a#(a(x1)) -> b#(x1)
   SCC Processor:
    #sccs: 1
    #rules: 5
    #arcs: 16/36
    DPs:
     b#(b(x1)) -> a#(a(x1))
     a#(a(x1)) -> b#(x1)
     b#(b(x1)) -> a#(x1)
     a#(a(x1)) -> a#(c(b(x1)))
     a#(a(x1)) -> b#(a(c(b(x1))))
    TRS:
     a(a(x1)) -> b(a(c(b(x1))))
     b(b(x1)) -> a(a(x1))
     c(a(x1)) -> x1
    Arctic Interpretation Processor:
     dimension: 1
     interpretation:
      [b#](x0) = x0,
      
      [a#](x0) = x0 + 0,
      
      [c](x0) = -2x0 + 0,
      
      [b](x0) = 2x0 + 2,
      
      [a](x0) = 2x0 + 2
     orientation:
      b#(b(x1)) = 2x1 + 2 >= 2x1 + 2 = a#(a(x1))
      
      a#(a(x1)) = 2x1 + 2 >= x1 = b#(x1)
      
      b#(b(x1)) = 2x1 + 2 >= x1 + 0 = a#(x1)
      
      a#(a(x1)) = 2x1 + 2 >= x1 + 0 = a#(c(b(x1)))
      
      a#(a(x1)) = 2x1 + 2 >= 2x1 + 2 = b#(a(c(b(x1))))
      
      a(a(x1)) = 4x1 + 4 >= 4x1 + 4 = b(a(c(b(x1))))
      
      b(b(x1)) = 4x1 + 4 >= 4x1 + 4 = a(a(x1))
      
      c(a(x1)) = x1 + 0 >= x1 = x1
     problem:
      DPs:
       b#(b(x1)) -> a#(a(x1))
       a#(a(x1)) -> b#(a(c(b(x1))))
      TRS:
       a(a(x1)) -> b(a(c(b(x1))))
       b(b(x1)) -> a(a(x1))
       c(a(x1)) -> x1
     EDG Processor:
      DPs:
       b#(b(x1)) -> a#(a(x1))
       a#(a(x1)) -> b#(a(c(b(x1))))
      TRS:
       a(a(x1)) -> b(a(c(b(x1))))
       b(b(x1)) -> a(a(x1))
       c(a(x1)) -> x1
      graph:
       b#(b(x1)) -> a#(a(x1)) -> a#(a(x1)) -> b#(a(c(b(x1))))
       a#(a(x1)) -> b#(a(c(b(x1)))) -> b#(b(x1)) -> a#(a(x1))
      Arctic Interpretation Processor:
       dimension: 3
       interpretation:
        [b#](x0) = [-& 0  0 ]x0 + [0],
        
        [a#](x0) = [0 0 0]x0 + [0],
        
                  [-& -& 0 ]     [0]
        [c](x0) = [-& -& 0 ]x0 + [0]
                  [0  0  0 ]     [0],
        
                  [-& 1  0 ]     [0]
        [b](x0) = [0  1  0 ]x0 + [1]
                  [-& 0  -&]     [0],
        
                  [0  1  0 ]     [1]
        [a](x0) = [0  0  -&]x0 + [0]
                  [0  0  -&]     [0]
       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(b(x1))))
        
                   [1 1 0]     [1]    [-& 1  -&]     [1]                 
        a(a(x1)) = [0 1 0]x1 + [1] >= [0  1  0 ]x1 + [1] = b(a(c(b(x1))))
                   [0 1 0]     [1]    [-& 0  -&]     [0]                 
        
                   [1 2 1]     [2]    [1 1 0]     [1]           
        b(b(x1)) = [1 2 1]x1 + [2] >= [0 1 0]x1 + [1] = a(a(x1))
                   [0 1 0]     [1]    [0 1 0]     [1]           
        
                   [0  0  -&]     [0]           
        c(a(x1)) = [0  0  -&]x1 + [0] >= x1 = x1
                   [0  1  0 ]     [1]           
       problem:
        DPs:
         b#(b(x1)) -> a#(a(x1))
        TRS:
         a(a(x1)) -> b(a(c(b(x1))))
         b(b(x1)) -> a(a(x1))
         c(a(x1)) -> x1
       EDG Processor:
        DPs:
         b#(b(x1)) -> a#(a(x1))
        TRS:
         a(a(x1)) -> b(a(c(b(x1))))
         b(b(x1)) -> a(a(x1))
         c(a(x1)) -> x1
        graph:
         
        Qed