YES

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

Proof:
 DP Processor:
  DPs:
   a#(a(x1)) -> b#(x1)
   a#(a(x1)) -> a#(b(x1))
   a#(c(x1)) -> b#(x1)
   b#(b(x1)) -> a#(x1)
   b#(b(x1)) -> b#(a(x1))
  TRS:
   a(a(x1)) -> a(b(x1))
   a(c(x1)) -> b(x1)
   b(b(x1)) -> c(b(a(x1)))
  TDG Processor:
   DPs:
    a#(a(x1)) -> b#(x1)
    a#(a(x1)) -> a#(b(x1))
    a#(c(x1)) -> b#(x1)
    b#(b(x1)) -> a#(x1)
    b#(b(x1)) -> b#(a(x1))
   TRS:
    a(a(x1)) -> a(b(x1))
    a(c(x1)) -> b(x1)
    b(b(x1)) -> c(b(a(x1)))
   graph:
    b#(b(x1)) -> b#(a(x1)) -> b#(b(x1)) -> b#(a(x1))
    b#(b(x1)) -> b#(a(x1)) -> b#(b(x1)) -> a#(x1)
    b#(b(x1)) -> a#(x1) -> a#(c(x1)) -> b#(x1)
    b#(b(x1)) -> a#(x1) -> a#(a(x1)) -> a#(b(x1))
    b#(b(x1)) -> a#(x1) -> a#(a(x1)) -> b#(x1)
    a#(c(x1)) -> b#(x1) -> b#(b(x1)) -> b#(a(x1))
    a#(c(x1)) -> b#(x1) -> b#(b(x1)) -> a#(x1)
    a#(a(x1)) -> b#(x1) -> b#(b(x1)) -> b#(a(x1))
    a#(a(x1)) -> b#(x1) -> b#(b(x1)) -> a#(x1)
    a#(a(x1)) -> a#(b(x1)) -> a#(c(x1)) -> b#(x1)
    a#(a(x1)) -> a#(b(x1)) -> a#(a(x1)) -> a#(b(x1))
    a#(a(x1)) -> a#(b(x1)) -> a#(a(x1)) -> b#(x1)
   Arctic Interpretation Processor:
    dimension: 1
    interpretation:
     [b#](x0) = -16x0 + 0,
     
     [a#](x0) = -8x0 + 0,
     
     [c](x0) = x0 + -16,
     
     [b](x0) = 8x0 + 12,
     
     [a](x0) = 8x0 + 12
    orientation:
     a#(a(x1)) = x1 + 4 >= -16x1 + 0 = b#(x1)
     
     a#(a(x1)) = x1 + 4 >= x1 + 4 = a#(b(x1))
     
     a#(c(x1)) = -8x1 + 0 >= -16x1 + 0 = b#(x1)
     
     b#(b(x1)) = -8x1 + 0 >= -8x1 + 0 = a#(x1)
     
     b#(b(x1)) = -8x1 + 0 >= -8x1 + 0 = b#(a(x1))
     
     a(a(x1)) = 16x1 + 20 >= 16x1 + 20 = a(b(x1))
     
     a(c(x1)) = 8x1 + 12 >= 8x1 + 12 = b(x1)
     
     b(b(x1)) = 16x1 + 20 >= 16x1 + 20 = c(b(a(x1)))
    problem:
     DPs:
      a#(a(x1)) -> a#(b(x1))
      a#(c(x1)) -> b#(x1)
      b#(b(x1)) -> a#(x1)
      b#(b(x1)) -> b#(a(x1))
     TRS:
      a(a(x1)) -> a(b(x1))
      a(c(x1)) -> b(x1)
      b(b(x1)) -> c(b(a(x1)))
    EDG Processor:
     DPs:
      a#(a(x1)) -> a#(b(x1))
      a#(c(x1)) -> b#(x1)
      b#(b(x1)) -> a#(x1)
      b#(b(x1)) -> b#(a(x1))
     TRS:
      a(a(x1)) -> a(b(x1))
      a(c(x1)) -> b(x1)
      b(b(x1)) -> c(b(a(x1)))
     graph:
      b#(b(x1)) -> b#(a(x1)) -> b#(b(x1)) -> a#(x1)
      b#(b(x1)) -> b#(a(x1)) -> b#(b(x1)) -> b#(a(x1))
      b#(b(x1)) -> a#(x1) -> a#(a(x1)) -> a#(b(x1))
      b#(b(x1)) -> a#(x1) -> a#(c(x1)) -> b#(x1)
      a#(c(x1)) -> b#(x1) -> b#(b(x1)) -> a#(x1)
      a#(c(x1)) -> b#(x1) -> b#(b(x1)) -> b#(a(x1))
      a#(a(x1)) -> a#(b(x1)) -> a#(a(x1)) -> a#(b(x1))
      a#(a(x1)) -> a#(b(x1)) -> a#(c(x1)) -> b#(x1)
     Arctic Interpretation Processor:
      dimension: 1
      interpretation:
       [b#](x0) = 8x0,
       
       [a#](x0) = 8x0,
       
       [c](x0) = x0,
       
       [b](x0) = 1x0,
       
       [a](x0) = 1x0
      orientation:
       a#(a(x1)) = 9x1 >= 9x1 = a#(b(x1))
       
       a#(c(x1)) = 8x1 >= 8x1 = b#(x1)
       
       b#(b(x1)) = 9x1 >= 8x1 = a#(x1)
       
       b#(b(x1)) = 9x1 >= 9x1 = b#(a(x1))
       
       a(a(x1)) = 2x1 >= 2x1 = a(b(x1))
       
       a(c(x1)) = 1x1 >= 1x1 = b(x1)
       
       b(b(x1)) = 2x1 >= 2x1 = c(b(a(x1)))
      problem:
       DPs:
        a#(a(x1)) -> a#(b(x1))
        a#(c(x1)) -> b#(x1)
        b#(b(x1)) -> b#(a(x1))
       TRS:
        a(a(x1)) -> a(b(x1))
        a(c(x1)) -> b(x1)
        b(b(x1)) -> c(b(a(x1)))
      EDG Processor:
       DPs:
        a#(a(x1)) -> a#(b(x1))
        a#(c(x1)) -> b#(x1)
        b#(b(x1)) -> b#(a(x1))
       TRS:
        a(a(x1)) -> a(b(x1))
        a(c(x1)) -> b(x1)
        b(b(x1)) -> c(b(a(x1)))
       graph:
        b#(b(x1)) -> b#(a(x1)) -> b#(b(x1)) -> b#(a(x1))
        a#(c(x1)) -> b#(x1) -> b#(b(x1)) -> b#(a(x1))
        a#(a(x1)) -> a#(b(x1)) -> a#(c(x1)) -> b#(x1)
        a#(a(x1)) -> a#(b(x1)) -> a#(a(x1)) -> a#(b(x1))
       SCC Processor:
        #sccs: 2
        #rules: 2
        #arcs: 4/9
        DPs:
         a#(a(x1)) -> a#(b(x1))
        TRS:
         a(a(x1)) -> a(b(x1))
         a(c(x1)) -> b(x1)
         b(b(x1)) -> c(b(a(x1)))
        Arctic Interpretation Processor:
         dimension: 1
         interpretation:
          [a#](x0) = x0 + 0,
          
          [c](x0) = x0 + -15,
          
          [b](x0) = 0,
          
          [a](x0) = x0 + 1
         orientation:
          a#(a(x1)) = x1 + 1 >= 0 = a#(b(x1))
          
          a(a(x1)) = x1 + 1 >= 1 = a(b(x1))
          
          a(c(x1)) = x1 + 1 >= 0 = b(x1)
          
          b(b(x1)) = 0 >= 0 = c(b(a(x1)))
         problem:
          DPs:
           
          TRS:
           a(a(x1)) -> a(b(x1))
           a(c(x1)) -> b(x1)
           b(b(x1)) -> c(b(a(x1)))
         Qed
        
        DPs:
         b#(b(x1)) -> b#(a(x1))
        TRS:
         a(a(x1)) -> a(b(x1))
         a(c(x1)) -> b(x1)
         b(b(x1)) -> c(b(a(x1)))
        Arctic Interpretation Processor:
         dimension: 3
         interpretation:
          [b#](x0) = [0  -& -&]x0 + [0],
          
                    [-& -& -&]     [0]
          [c](x0) = [-& 1  -&]x0 + [1]
                    [0  1  0 ]     [0],
          
                    [-& 1  -&]     [1]
          [b](x0) = [0  -& -&]x0 + [0]
                    [0  0  1 ]     [1],
          
                    [-& 0  -&]     [0]
          [a](x0) = [0  0  1 ]x0 + [0]
                    [0  0  1 ]     [1]
         orientation:
          b#(b(x1)) = [-& 1  -&]x1 + [1] >= [-& 0  -&]x1 + [0] = b#(a(x1))
          
                     [0 0 1]     [0]    [0  -& -&]     [0]           
          a(a(x1)) = [1 1 2]x1 + [2] >= [1  1  2 ]x1 + [2] = a(b(x1))
                     [1 1 2]     [2]    [1  1  2 ]     [2]           
          
                     [-& 1  -&]     [1]    [-& 1  -&]     [1]        
          a(c(x1)) = [1  2  1 ]x1 + [1] >= [0  -& -&]x1 + [0] = b(x1)
                     [1  2  1 ]     [1]    [0  0  1 ]     [1]        
          
                     [1  -& -&]     [1]    [-& -& -&]     [0]              
          b(b(x1)) = [-& 1  -&]x1 + [1] >= [-& 1  -&]x1 + [1] = c(b(a(x1)))
                     [1  1  2 ]     [2]    [1  1  2 ]     [2]              
         problem:
          DPs:
           
          TRS:
           a(a(x1)) -> a(b(x1))
           a(c(x1)) -> b(x1)
           b(b(x1)) -> c(b(a(x1)))
         Qed