YES

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

Proof:
 String Reversal Processor:
  a(x1) -> x1
  a(x1) -> b(x1)
  c(a(x1)) -> b(a(c(c(x1))))
  b(b(x1)) -> a(x1)
  DP Processor:
   DPs:
    a#(x1) -> b#(x1)
    c#(a(x1)) -> c#(x1)
    c#(a(x1)) -> c#(c(x1))
    c#(a(x1)) -> a#(c(c(x1)))
    c#(a(x1)) -> b#(a(c(c(x1))))
    b#(b(x1)) -> a#(x1)
   TRS:
    a(x1) -> x1
    a(x1) -> b(x1)
    c(a(x1)) -> b(a(c(c(x1))))
    b(b(x1)) -> a(x1)
   TDG Processor:
    DPs:
     a#(x1) -> b#(x1)
     c#(a(x1)) -> c#(x1)
     c#(a(x1)) -> c#(c(x1))
     c#(a(x1)) -> a#(c(c(x1)))
     c#(a(x1)) -> b#(a(c(c(x1))))
     b#(b(x1)) -> a#(x1)
    TRS:
     a(x1) -> x1
     a(x1) -> b(x1)
     c(a(x1)) -> b(a(c(c(x1))))
     b(b(x1)) -> a(x1)
    graph:
     c#(a(x1)) -> c#(c(x1)) -> c#(a(x1)) -> b#(a(c(c(x1))))
     c#(a(x1)) -> c#(c(x1)) -> c#(a(x1)) -> a#(c(c(x1)))
     c#(a(x1)) -> c#(c(x1)) -> c#(a(x1)) -> c#(c(x1))
     c#(a(x1)) -> c#(c(x1)) -> c#(a(x1)) -> c#(x1)
     c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> b#(a(c(c(x1))))
     c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> a#(c(c(x1)))
     c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> c#(c(x1))
     c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> c#(x1)
     c#(a(x1)) -> b#(a(c(c(x1)))) -> b#(b(x1)) -> a#(x1)
     c#(a(x1)) -> a#(c(c(x1))) -> a#(x1) -> b#(x1)
     b#(b(x1)) -> a#(x1) -> a#(x1) -> b#(x1)
     a#(x1) -> b#(x1) -> b#(b(x1)) -> a#(x1)
    CDG Processor:
     DPs:
      a#(x1) -> b#(x1)
      c#(a(x1)) -> c#(x1)
      c#(a(x1)) -> c#(c(x1))
      c#(a(x1)) -> a#(c(c(x1)))
      c#(a(x1)) -> b#(a(c(c(x1))))
      b#(b(x1)) -> a#(x1)
     TRS:
      a(x1) -> x1
      a(x1) -> b(x1)
      c(a(x1)) -> b(a(c(c(x1))))
      b(b(x1)) -> a(x1)
     graph:
      c#(a(x1)) -> c#(c(x1)) -> c#(a(x1)) -> c#(x1)
      c#(a(x1)) -> c#(c(x1)) -> c#(a(x1)) -> c#(c(x1))
      c#(a(x1)) -> c#(c(x1)) -> c#(a(x1)) -> a#(c(c(x1)))
      c#(a(x1)) -> c#(c(x1)) -> c#(a(x1)) -> b#(a(c(c(x1))))
      c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> c#(x1)
      c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> c#(c(x1))
      c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> a#(c(c(x1)))
      c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> b#(a(c(c(x1))))
      c#(a(x1)) -> b#(a(c(c(x1)))) -> b#(b(x1)) -> a#(x1)
      c#(a(x1)) -> a#(c(c(x1))) -> a#(x1) -> b#(x1)
      b#(b(x1)) -> a#(x1) -> a#(x1) -> b#(x1)
     SCC Processor:
      #sccs: 1
      #rules: 2
      #arcs: 11/36
      DPs:
       c#(a(x1)) -> c#(c(x1))
       c#(a(x1)) -> c#(x1)
      TRS:
       a(x1) -> x1
       a(x1) -> b(x1)
       c(a(x1)) -> b(a(c(c(x1))))
       b(b(x1)) -> a(x1)
      Arctic Interpretation Processor:
       dimension: 3
       usable rules:
        a(x1) -> x1
        a(x1) -> b(x1)
        c(a(x1)) -> b(a(c(c(x1))))
        b(b(x1)) -> a(x1)
       interpretation:
        [c#](x0) = [-& 0  1 ]x0 + [0],
        
                  [0  -& 0 ]     [0]
        [c](x0) = [1  -& 1 ]x0 + [0]
                  [0  -& 0 ]     [0],
        
                  [0  -& 0 ]     [0 ]
        [b](x0) = [0  0  0 ]x0 + [-&]
                  [1  0  0 ]     [1 ],
        
                  [0  -& 0 ]     [0 ]
        [a](x0) = [0  0  0 ]x0 + [-&]
                  [1  0  1 ]     [1 ]
       orientation:
        c#(a(x1)) = [2 1 2]x1 + [2] >= [1  -& 1 ]x1 + [1] = c#(c(x1))
        
        c#(a(x1)) = [2 1 2]x1 + [2] >= [-& 0  1 ]x1 + [0] = c#(x1)
        
                [0  -& 0 ]     [0 ]           
        a(x1) = [0  0  0 ]x1 + [-&] >= x1 = x1
                [1  0  1 ]     [1 ]           
        
                [0  -& 0 ]     [0 ]    [0  -& 0 ]     [0 ]        
        a(x1) = [0  0  0 ]x1 + [-&] >= [0  0  0 ]x1 + [-&] = b(x1)
                [1  0  1 ]     [1 ]    [1  0  0 ]     [1 ]        
        
                   [1 0 1]     [1]    [1  -& 1 ]     [1]                 
        c(a(x1)) = [2 1 2]x1 + [2] >= [1  -& 1 ]x1 + [1] = b(a(c(c(x1))))
                   [1 0 1]     [1]    [1  -& 1 ]     [1]                 
        
                   [1 0 0]     [1]    [0  -& 0 ]     [0 ]        
        b(b(x1)) = [1 0 0]x1 + [1] >= [0  0  0 ]x1 + [-&] = a(x1)
                   [1 0 1]     [1]    [1  0  1 ]     [1 ]        
       problem:
        DPs:
         
        TRS:
         a(x1) -> x1
         a(x1) -> b(x1)
         c(a(x1)) -> b(a(c(c(x1))))
         b(b(x1)) -> a(x1)
       Qed