YES

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

Proof:
 DP Processor:
  DPs:
   a#(c(x1)) -> b#(x1)
   a#(x1) -> b#(x1)
   a#(x1) -> b#(b(x1))
   a#(x1) -> b#(b(b(x1)))
   b#(c(b(x1))) -> a#(c(x1))
  TRS:
   a(c(x1)) -> c(b(x1))
   a(x1) -> b(b(b(x1)))
   b(c(b(x1))) -> a(c(x1))
  TDG Processor:
   DPs:
    a#(c(x1)) -> b#(x1)
    a#(x1) -> b#(x1)
    a#(x1) -> b#(b(x1))
    a#(x1) -> b#(b(b(x1)))
    b#(c(b(x1))) -> a#(c(x1))
   TRS:
    a(c(x1)) -> c(b(x1))
    a(x1) -> b(b(b(x1)))
    b(c(b(x1))) -> a(c(x1))
   graph:
    b#(c(b(x1))) -> a#(c(x1)) -> a#(x1) -> b#(b(b(x1)))
    b#(c(b(x1))) -> a#(c(x1)) -> a#(x1) -> b#(b(x1))
    b#(c(b(x1))) -> a#(c(x1)) -> a#(x1) -> b#(x1)
    b#(c(b(x1))) -> a#(c(x1)) -> a#(c(x1)) -> b#(x1)
    a#(c(x1)) -> b#(x1) -> b#(c(b(x1))) -> a#(c(x1))
    a#(x1) -> b#(b(b(x1))) -> b#(c(b(x1))) -> a#(c(x1))
    a#(x1) -> b#(b(x1)) -> b#(c(b(x1))) -> a#(c(x1))
    a#(x1) -> b#(x1) -> b#(c(b(x1))) -> a#(c(x1))
   Arctic Interpretation Processor:
    dimension: 1
    interpretation:
     [b#](x0) = x0 + 0,
     
     [a#](x0) = x0 + 0,
     
     [b](x0) = x0,
     
     [a](x0) = x0 + 0,
     
     [c](x0) = 1x0 + 8
    orientation:
     a#(c(x1)) = 1x1 + 8 >= x1 + 0 = b#(x1)
     
     a#(x1) = x1 + 0 >= x1 + 0 = b#(x1)
     
     a#(x1) = x1 + 0 >= x1 + 0 = b#(b(x1))
     
     a#(x1) = x1 + 0 >= x1 + 0 = b#(b(b(x1)))
     
     b#(c(b(x1))) = 1x1 + 8 >= 1x1 + 8 = a#(c(x1))
     
     a(c(x1)) = 1x1 + 8 >= 1x1 + 8 = c(b(x1))
     
     a(x1) = x1 + 0 >= x1 = b(b(b(x1)))
     
     b(c(b(x1))) = 1x1 + 8 >= 1x1 + 8 = a(c(x1))
    problem:
     DPs:
      a#(x1) -> b#(x1)
      a#(x1) -> b#(b(x1))
      a#(x1) -> b#(b(b(x1)))
      b#(c(b(x1))) -> a#(c(x1))
     TRS:
      a(c(x1)) -> c(b(x1))
      a(x1) -> b(b(b(x1)))
      b(c(b(x1))) -> a(c(x1))
    EDG Processor:
     DPs:
      a#(x1) -> b#(x1)
      a#(x1) -> b#(b(x1))
      a#(x1) -> b#(b(b(x1)))
      b#(c(b(x1))) -> a#(c(x1))
     TRS:
      a(c(x1)) -> c(b(x1))
      a(x1) -> b(b(b(x1)))
      b(c(b(x1))) -> a(c(x1))
     graph:
      b#(c(b(x1))) -> a#(c(x1)) -> a#(x1) -> b#(x1)
      b#(c(b(x1))) -> a#(c(x1)) -> a#(x1) -> b#(b(x1))
      b#(c(b(x1))) -> a#(c(x1)) -> a#(x1) -> b#(b(b(x1)))
      a#(x1) -> b#(b(b(x1))) -> b#(c(b(x1))) -> a#(c(x1))
      a#(x1) -> b#(b(x1)) -> b#(c(b(x1))) -> a#(c(x1))
      a#(x1) -> b#(x1) -> b#(c(b(x1))) -> a#(c(x1))
     Matrix Interpretation Processor: dim=4
      
      interpretation:
       [b#](x0) = [1 0 1 0]x0,
       
       [a#](x0) = [1 0 1 0]x0 + [1],
       
                 [0 0 0 0]     [1]
                 [1 0 0 1]     [0]
       [b](x0) = [0 0 1 0]x0 + [0]
                 [0 1 0 0]     [0],
       
                 [0 0 0 0]     [1]
                 [1 0 0 1]     [1]
       [a](x0) = [1 0 1 0]x0 + [0]
                 [0 1 0 0]     [1],
       
                 [1 0 0 0]  
                 [1 1 0 1]  
       [c](x0) = [0 1 1 1]x0
                 [0 1 0 1]  
      orientation:
       a#(x1) = [1 0 1 0]x1 + [1] >= [1 0 1 0]x1 = b#(x1)
       
       a#(x1) = [1 0 1 0]x1 + [1] >= [0 0 1 0]x1 + [1] = b#(b(x1))
       
       a#(x1) = [1 0 1 0]x1 + [1] >= [0 0 1 0]x1 + [1] = b#(b(b(x1)))
       
       b#(c(b(x1))) = [1 1 1 1]x1 + [1] >= [1 1 1 1]x1 + [1] = a#(c(x1))
       
                  [0 0 0 0]     [1]    [0 0 0 0]     [1]           
                  [1 1 0 1]     [1]    [1 1 0 1]     [1]           
       a(c(x1)) = [1 1 1 1]x1 + [0] >= [1 1 1 1]x1 + [0] = c(b(x1))
                  [1 1 0 1]     [1]    [1 1 0 1]     [0]           
       
               [0 0 0 0]     [1]    [0 0 0 0]     [1]              
               [1 0 0 1]     [1]    [1 0 0 1]     [1]              
       a(x1) = [1 0 1 0]x1 + [0] >= [0 0 1 0]x1 + [0] = b(b(b(x1)))
               [0 1 0 0]     [1]    [0 1 0 0]     [1]              
       
                     [0 0 0 0]     [1]    [0 0 0 0]     [1]           
                     [1 1 0 1]     [1]    [1 1 0 1]     [1]           
       b(c(b(x1))) = [1 1 1 1]x1 + [0] >= [1 1 1 1]x1 + [0] = a(c(x1))
                     [1 1 0 1]     [1]    [1 1 0 1]     [1]           
      problem:
       DPs:
        a#(x1) -> b#(b(x1))
        a#(x1) -> b#(b(b(x1)))
        b#(c(b(x1))) -> a#(c(x1))
       TRS:
        a(c(x1)) -> c(b(x1))
        a(x1) -> b(b(b(x1)))
        b(c(b(x1))) -> a(c(x1))
      EDG Processor:
       DPs:
        a#(x1) -> b#(b(x1))
        a#(x1) -> b#(b(b(x1)))
        b#(c(b(x1))) -> a#(c(x1))
       TRS:
        a(c(x1)) -> c(b(x1))
        a(x1) -> b(b(b(x1)))
        b(c(b(x1))) -> a(c(x1))
       graph:
        b#(c(b(x1))) -> a#(c(x1)) -> a#(x1) -> b#(b(b(x1)))
        b#(c(b(x1))) -> a#(c(x1)) -> a#(x1) -> b#(b(x1))
        a#(x1) -> b#(b(b(x1))) -> b#(c(b(x1))) -> a#(c(x1))
        a#(x1) -> b#(b(x1)) -> b#(c(b(x1))) -> a#(c(x1))
       Matrix Interpretation Processor: dim=4
        
        interpretation:
         [b#](x0) = [0 0 1 0]x0,
         
         [a#](x0) = [0 0 1 0]x0 + [1],
         
                   [0 0 0 0]     [1]
                   [0 0 0 1]     [0]
         [b](x0) = [0 0 1 0]x0 + [0]
                   [1 1 0 0]     [0],
         
                   [0 0 0 0]     [1]
                   [0 0 0 1]     [1]
         [a](x0) = [1 0 1 0]x0 + [1]
                   [1 1 0 0]     [1],
         
                   [0 0 0 0]  
                   [1 1 0 1]  
         [c](x0) = [1 1 0 1]x0
                   [1 1 0 1]  
        orientation:
         a#(x1) = [0 0 1 0]x1 + [1] >= [0 0 1 0]x1 = b#(b(x1))
         
         a#(x1) = [0 0 1 0]x1 + [1] >= [0 0 1 0]x1 = b#(b(b(x1)))
         
         b#(c(b(x1))) = [1 1 0 1]x1 + [1] >= [1 1 0 1]x1 + [1] = a#(c(x1))
         
                    [0 0 0 0]     [1]    [0 0 0 0]     [0]           
                    [1 1 0 1]     [1]    [1 1 0 1]     [1]           
         a(c(x1)) = [1 1 0 1]x1 + [1] >= [1 1 0 1]x1 + [1] = c(b(x1))
                    [1 1 0 1]     [1]    [1 1 0 1]     [1]           
         
                 [0 0 0 0]     [1]    [0 0 0 0]     [1]              
                 [0 0 0 1]     [1]    [0 0 0 1]     [1]              
         a(x1) = [1 0 1 0]x1 + [1] >= [0 0 1 0]x1 + [0] = b(b(b(x1)))
                 [1 1 0 0]     [1]    [1 1 0 0]     [1]              
         
                       [0 0 0 0]     [1]    [0 0 0 0]     [1]           
                       [1 1 0 1]     [1]    [1 1 0 1]     [1]           
         b(c(b(x1))) = [1 1 0 1]x1 + [1] >= [1 1 0 1]x1 + [1] = a(c(x1))
                       [1 1 0 1]     [1]    [1 1 0 1]     [1]           
        problem:
         DPs:
          b#(c(b(x1))) -> a#(c(x1))
         TRS:
          a(c(x1)) -> c(b(x1))
          a(x1) -> b(b(b(x1)))
          b(c(b(x1))) -> a(c(x1))
        EDG Processor:
         DPs:
          b#(c(b(x1))) -> a#(c(x1))
         TRS:
          a(c(x1)) -> c(b(x1))
          a(x1) -> b(b(b(x1)))
          b(c(b(x1))) -> a(c(x1))
         graph:
          
         Qed