YES

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

Proof:
 String Reversal Processor:
  a(x1) -> x1
  a(a(x1)) -> b(x1)
  b(x1) -> a(x1)
  c(b(x1)) -> a(b(c(c(x1))))
  DP Processor:
   DPs:
    a#(a(x1)) -> b#(x1)
    b#(x1) -> a#(x1)
    c#(b(x1)) -> c#(x1)
    c#(b(x1)) -> c#(c(x1))
    c#(b(x1)) -> b#(c(c(x1)))
    c#(b(x1)) -> a#(b(c(c(x1))))
   TRS:
    a(x1) -> x1
    a(a(x1)) -> b(x1)
    b(x1) -> a(x1)
    c(b(x1)) -> a(b(c(c(x1))))
   TDG Processor:
    DPs:
     a#(a(x1)) -> b#(x1)
     b#(x1) -> a#(x1)
     c#(b(x1)) -> c#(x1)
     c#(b(x1)) -> c#(c(x1))
     c#(b(x1)) -> b#(c(c(x1)))
     c#(b(x1)) -> a#(b(c(c(x1))))
    TRS:
     a(x1) -> x1
     a(a(x1)) -> b(x1)
     b(x1) -> a(x1)
     c(b(x1)) -> a(b(c(c(x1))))
    graph:
     c#(b(x1)) -> c#(c(x1)) -> c#(b(x1)) -> a#(b(c(c(x1))))
     c#(b(x1)) -> c#(c(x1)) -> c#(b(x1)) -> b#(c(c(x1)))
     c#(b(x1)) -> c#(c(x1)) -> c#(b(x1)) -> c#(c(x1))
     c#(b(x1)) -> c#(c(x1)) -> c#(b(x1)) -> c#(x1)
     c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> a#(b(c(c(x1))))
     c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> b#(c(c(x1)))
     c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> c#(c(x1))
     c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> c#(x1)
     c#(b(x1)) -> b#(c(c(x1))) -> b#(x1) -> a#(x1)
     c#(b(x1)) -> a#(b(c(c(x1)))) -> a#(a(x1)) -> b#(x1)
     b#(x1) -> a#(x1) -> a#(a(x1)) -> b#(x1)
     a#(a(x1)) -> b#(x1) -> b#(x1) -> a#(x1)
    SCC Processor:
     #sccs: 2
     #rules: 4
     #arcs: 12/36
     DPs:
      c#(b(x1)) -> c#(c(x1))
      c#(b(x1)) -> c#(x1)
     TRS:
      a(x1) -> x1
      a(a(x1)) -> b(x1)
      b(x1) -> a(x1)
      c(b(x1)) -> a(b(c(c(x1))))
     Arctic Interpretation Processor:
      dimension: 2
      usable rules:
       a(x1) -> x1
       a(a(x1)) -> b(x1)
       b(x1) -> a(x1)
       c(b(x1)) -> a(b(c(c(x1))))
      interpretation:
                 [2 2]     [2 ]
       [b](x0) = [0 0]x0 + [-&],
       
       [c#](x0) = [0 0]x0,
       
                 [0 2]     [2 ]
       [a](x0) = [0 0]x0 + [-&],
       
                 [0 0]     [0]
       [c](x0) = [0 0]x0 + [0]
      orientation:
       c#(b(x1)) = [2 2]x1 + [2] >= [0 0]x1 + [0] = c#(c(x1))
       
       c#(b(x1)) = [2 2]x1 + [2] >= [0 0]x1 = c#(x1)
       
               [0 2]     [2 ]           
       a(x1) = [0 0]x1 + [-&] >= x1 = x1
       
                  [2 2]     [2]    [2 2]     [2 ]        
       a(a(x1)) = [0 2]x1 + [2] >= [0 0]x1 + [-&] = b(x1)
       
               [2 2]     [2 ]    [0 2]     [2 ]        
       b(x1) = [0 0]x1 + [-&] >= [0 0]x1 + [-&] = a(x1)
       
                  [2 2]     [2]    [2 2]     [2]                 
       c(b(x1)) = [2 2]x1 + [2] >= [2 2]x1 + [2] = a(b(c(c(x1))))
      problem:
       DPs:
        
       TRS:
        a(x1) -> x1
        a(a(x1)) -> b(x1)
        b(x1) -> a(x1)
        c(b(x1)) -> a(b(c(c(x1))))
      Qed
     
     DPs:
      b#(x1) -> a#(x1)
      a#(a(x1)) -> b#(x1)
     TRS:
      a(x1) -> x1
      a(a(x1)) -> b(x1)
      b(x1) -> a(x1)
      c(b(x1)) -> a(b(c(c(x1))))
     Usable Rule Processor:
      DPs:
       b#(x1) -> a#(x1)
       a#(a(x1)) -> b#(x1)
      TRS:
       
      Arctic Interpretation Processor:
       dimension: 1
       usable rules:
        
       interpretation:
        [b#](x0) = x0,
        
        [a#](x0) = x0,
        
        [a](x0) = 3x0 + 1
       orientation:
        b#(x1) = x1 >= x1 = a#(x1)
        
        a#(a(x1)) = 3x1 + 1 >= x1 = b#(x1)
       problem:
        DPs:
         b#(x1) -> a#(x1)
        TRS:
         
       Restore Modifier:
        DPs:
         b#(x1) -> a#(x1)
        TRS:
         a(x1) -> x1
         a(a(x1)) -> b(x1)
         b(x1) -> a(x1)
         c(b(x1)) -> a(b(c(c(x1))))
        EDG Processor:
         DPs:
          b#(x1) -> a#(x1)
         TRS:
          a(x1) -> x1
          a(a(x1)) -> b(x1)
          b(x1) -> a(x1)
          c(b(x1)) -> a(b(c(c(x1))))
         graph:
          
         SCC Processor:
          #sccs: 0
          #rules: 0
          #arcs: 0/1