YES

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

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