YES

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

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