YES

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

Proof:
 DP Processor:
  DPs:
   a#(a(b(b(x1)))) -> a#(x1)
   a#(a(b(b(x1)))) -> a#(a(x1))
   a#(a(b(b(x1)))) -> a#(a(a(x1)))
   a#(c(x1)) -> a#(x1)
   a#(c(x1)) -> c#(a(x1))
   c#(b(x1)) -> c#(x1)
  TRS:
   a(a(b(b(x1)))) -> b(b(b(a(a(a(x1))))))
   a(c(x1)) -> c(a(x1))
   c(b(x1)) -> b(c(x1))
  TDG Processor:
   DPs:
    a#(a(b(b(x1)))) -> a#(x1)
    a#(a(b(b(x1)))) -> a#(a(x1))
    a#(a(b(b(x1)))) -> a#(a(a(x1)))
    a#(c(x1)) -> a#(x1)
    a#(c(x1)) -> c#(a(x1))
    c#(b(x1)) -> c#(x1)
   TRS:
    a(a(b(b(x1)))) -> b(b(b(a(a(a(x1))))))
    a(c(x1)) -> c(a(x1))
    c(b(x1)) -> b(c(x1))
   graph:
    c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> c#(x1)
    a#(c(x1)) -> c#(a(x1)) -> c#(b(x1)) -> c#(x1)
    a#(c(x1)) -> a#(x1) -> a#(c(x1)) -> c#(a(x1))
    a#(c(x1)) -> a#(x1) -> a#(c(x1)) -> a#(x1)
    a#(c(x1)) -> a#(x1) -> a#(a(b(b(x1)))) -> a#(a(a(x1)))
    a#(c(x1)) -> a#(x1) -> a#(a(b(b(x1)))) -> a#(a(x1))
    a#(c(x1)) -> a#(x1) -> a#(a(b(b(x1)))) -> a#(x1)
    a#(a(b(b(x1)))) -> a#(a(a(x1))) -> a#(c(x1)) -> c#(a(x1))
    a#(a(b(b(x1)))) -> a#(a(a(x1))) -> a#(c(x1)) -> a#(x1)
    a#(a(b(b(x1)))) -> a#(a(a(x1))) ->
    a#(a(b(b(x1)))) -> a#(a(a(x1)))
    a#(a(b(b(x1)))) -> a#(a(a(x1))) -> a#(a(b(b(x1)))) -> a#(a(x1))
    a#(a(b(b(x1)))) -> a#(a(a(x1))) -> a#(a(b(b(x1)))) -> a#(x1)
    a#(a(b(b(x1)))) -> a#(a(x1)) -> a#(c(x1)) -> c#(a(x1))
    a#(a(b(b(x1)))) -> a#(a(x1)) -> a#(c(x1)) -> a#(x1)
    a#(a(b(b(x1)))) -> a#(a(x1)) -> a#(a(b(b(x1)))) -> a#(a(a(x1)))
    a#(a(b(b(x1)))) -> a#(a(x1)) -> a#(a(b(b(x1)))) -> a#(a(x1))
    a#(a(b(b(x1)))) -> a#(a(x1)) -> a#(a(b(b(x1)))) -> a#(x1)
    a#(a(b(b(x1)))) -> a#(x1) -> a#(c(x1)) -> c#(a(x1))
    a#(a(b(b(x1)))) -> a#(x1) -> a#(c(x1)) -> a#(x1)
    a#(a(b(b(x1)))) -> a#(x1) -> a#(a(b(b(x1)))) -> a#(a(a(x1)))
    a#(a(b(b(x1)))) -> a#(x1) -> a#(a(b(b(x1)))) -> a#(a(x1))
    a#(a(b(b(x1)))) -> a#(x1) -> a#(a(b(b(x1)))) -> a#(x1)
   CDG Processor:
    DPs:
     a#(a(b(b(x1)))) -> a#(x1)
     a#(a(b(b(x1)))) -> a#(a(x1))
     a#(a(b(b(x1)))) -> a#(a(a(x1)))
     a#(c(x1)) -> a#(x1)
     a#(c(x1)) -> c#(a(x1))
     c#(b(x1)) -> c#(x1)
    TRS:
     a(a(b(b(x1)))) -> b(b(b(a(a(a(x1))))))
     a(c(x1)) -> c(a(x1))
     c(b(x1)) -> b(c(x1))
    graph:
     a#(c(x1)) -> c#(a(x1)) -> c#(b(x1)) -> c#(x1)
     a#(c(x1)) -> a#(x1) -> a#(a(b(b(x1)))) -> a#(x1)
     a#(c(x1)) -> a#(x1) -> a#(a(b(b(x1)))) -> a#(a(x1))
     a#(c(x1)) -> a#(x1) -> a#(a(b(b(x1)))) -> a#(a(a(x1)))
     a#(c(x1)) -> a#(x1) -> a#(c(x1)) -> a#(x1)
     a#(c(x1)) -> a#(x1) -> a#(c(x1)) -> c#(a(x1))
     a#(a(b(b(x1)))) -> a#(a(a(x1))) -> a#(a(b(b(x1)))) -> a#(x1)
     a#(a(b(b(x1)))) -> a#(a(a(x1))) -> a#(a(b(b(x1)))) -> a#(a(x1))
     a#(a(b(b(x1)))) -> a#(a(a(x1))) ->
     a#(a(b(b(x1)))) -> a#(a(a(x1)))
     a#(a(b(b(x1)))) -> a#(a(a(x1))) -> a#(c(x1)) -> a#(x1)
     a#(a(b(b(x1)))) -> a#(a(a(x1))) -> a#(c(x1)) -> c#(a(x1))
     a#(a(b(b(x1)))) -> a#(a(x1)) -> a#(a(b(b(x1)))) -> a#(x1)
     a#(a(b(b(x1)))) -> a#(a(x1)) -> a#(a(b(b(x1)))) -> a#(a(x1))
     a#(a(b(b(x1)))) -> a#(a(x1)) -> a#(a(b(b(x1)))) -> a#(a(a(x1)))
     a#(a(b(b(x1)))) -> a#(a(x1)) -> a#(c(x1)) -> a#(x1)
     a#(a(b(b(x1)))) -> a#(a(x1)) -> a#(c(x1)) -> c#(a(x1))
     a#(a(b(b(x1)))) -> a#(x1) -> a#(a(b(b(x1)))) -> a#(x1)
     a#(a(b(b(x1)))) -> a#(x1) -> a#(a(b(b(x1)))) -> a#(a(x1))
     a#(a(b(b(x1)))) -> a#(x1) -> a#(a(b(b(x1)))) -> a#(a(a(x1)))
     a#(a(b(b(x1)))) -> a#(x1) -> a#(c(x1)) -> a#(x1)
     a#(a(b(b(x1)))) -> a#(x1) -> a#(c(x1)) -> c#(a(x1))
    SCC Processor:
     #sccs: 1
     #rules: 4
     #arcs: 21/36
     DPs:
      a#(c(x1)) -> a#(x1)
      a#(a(b(b(x1)))) -> a#(a(a(x1)))
      a#(a(b(b(x1)))) -> a#(a(x1))
      a#(a(b(b(x1)))) -> a#(x1)
     TRS:
      a(a(b(b(x1)))) -> b(b(b(a(a(a(x1))))))
      a(c(x1)) -> c(a(x1))
      c(b(x1)) -> b(c(x1))
     Matrix Interpretation Processor: dim=3
      
      interpretation:
       [a#](x0) = [0 2 0]x0 + [2],
       
                 [2 0 0]     [1]
       [c](x0) = [0 2 0]x0 + [1]
                 [0 0 2]     [1],
       
                 [0 2 0]  
       [a](x0) = [0 0 1]x0
                 [0 1 0]  ,
       
                 [0 2 1]     [2]
       [b](x0) = [0 0 0]x0 + [0]
                 [1 0 0]     [0]
      orientation:
       a#(c(x1)) = [0 4 0]x1 + [4] >= [0 2 0]x1 + [2] = a#(x1)
       
       a#(a(b(b(x1)))) = [0 4 2]x1 + [6] >= [0 2 0]x1 + [2] = a#(a(a(x1)))
       
       a#(a(b(b(x1)))) = [0 4 2]x1 + [6] >= [0 0 2]x1 + [2] = a#(a(x1))
       
       a#(a(b(b(x1)))) = [0 4 2]x1 + [6] >= [0 2 0]x1 + [2] = a#(x1)
       
                        [0 4 2]     [4]    [0 1 2]     [4]                       
       a(a(b(b(x1)))) = [0 0 0]x1 + [0] >= [0 0 0]x1 + [0] = b(b(b(a(a(a(x1))))))
                        [0 2 1]     [2]    [0 2 0]     [2]                       
       
                  [0 4 0]     [2]    [0 4 0]     [1]           
       a(c(x1)) = [0 0 2]x1 + [1] >= [0 0 2]x1 + [1] = c(a(x1))
                  [0 2 0]     [1]    [0 2 0]     [1]           
       
                  [0 4 2]     [5]    [0 4 2]     [5]           
       c(b(x1)) = [0 0 0]x1 + [1] >= [0 0 0]x1 + [0] = b(c(x1))
                  [2 0 0]     [1]    [2 0 0]     [1]           
      problem:
       DPs:
        
       TRS:
        a(a(b(b(x1)))) -> b(b(b(a(a(a(x1))))))
        a(c(x1)) -> c(a(x1))
        c(b(x1)) -> b(c(x1))
      Qed