YES

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

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