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)
    SCC Processor:
     #sccs: 2
     #rules: 4
     #arcs: 12/36
     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: 2
      usable rules:
       a(x1) -> x1
       a(x1) -> b(x1)
       c(a(x1)) -> b(a(c(c(x1))))
       b(b(x1)) -> a(x1)
      interpretation:
                 [0 0]     [-&]
       [b](x0) = [1 0]x0 + [0 ],
       
       [c#](x0) = [0 0]x0,
       
                 [0 0]     [-&]
       [a](x0) = [1 1]x0 + [0 ],
       
                 [0  0 ]  
       [c](x0) = [-& 0 ]x0
      orientation:
       c#(a(x1)) = [1 1]x1 + [0] >= [0 0]x1 = c#(c(x1))
       
       c#(a(x1)) = [1 1]x1 + [0] >= [0 0]x1 = c#(x1)
       
               [0 0]     [-&]           
       a(x1) = [1 1]x1 + [0 ] >= x1 = x1
       
               [0 0]     [-&]    [0 0]     [-&]        
       a(x1) = [1 1]x1 + [0 ] >= [1 0]x1 + [0 ] = b(x1)
       
                  [1 1]     [0]    [1 1]     [0]                 
       c(a(x1)) = [1 1]x1 + [0] >= [1 1]x1 + [0] = b(a(c(c(x1))))
       
                  [1 0]     [0]    [0 0]     [-&]        
       b(b(x1)) = [1 1]x1 + [0] >= [1 1]x1 + [0 ] = a(x1)
      problem:
       DPs:
        
       TRS:
        a(x1) -> x1
        a(x1) -> b(x1)
        c(a(x1)) -> b(a(c(c(x1))))
        b(b(x1)) -> a(x1)
      Qed
     
     DPs:
      a#(x1) -> b#(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)
     Usable Rule Processor:
      DPs:
       a#(x1) -> b#(x1)
       b#(b(x1)) -> a#(x1)
      TRS:
       
      Arctic Interpretation Processor:
       dimension: 4
       usable rules:
        
       interpretation:
        [b#](x0) = [0  -& -& -&]x0,
        
                  [1 0 0 0]     [0]
                  [1 0 0 0]     [0]
        [b](x0) = [0 0 0 0]x0 + [0]
                  [0 0 0 0]     [0],
        
        [a#](x0) = [0  -& -& -&]x0
       orientation:
        a#(x1) = [0  -& -& -&]x1 >= [0  -& -& -&]x1 = b#(x1)
        
        b#(b(x1)) = [1 0 0 0]x1 + [0] >= [0  -& -& -&]x1 = a#(x1)
       problem:
        DPs:
         a#(x1) -> b#(x1)
        TRS:
         
       Restore Modifier:
        DPs:
         a#(x1) -> b#(x1)
        TRS:
         a(x1) -> x1
         a(x1) -> b(x1)
         c(a(x1)) -> b(a(c(c(x1))))
         b(b(x1)) -> a(x1)
        EDG Processor:
         DPs:
          a#(x1) -> b#(x1)
         TRS:
          a(x1) -> x1
          a(x1) -> b(x1)
          c(a(x1)) -> b(a(c(c(x1))))
          b(b(x1)) -> a(x1)
         graph:
          
         SCC Processor:
          #sccs: 0
          #rules: 0
          #arcs: 0/1