YES

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

Proof:
 DP Processor:
  DPs:
   a#(a(b(x1))) -> a#(x1)
   a#(a(b(x1))) -> a#(a(x1))
   a#(a(b(x1))) -> b#(a(a(x1)))
   b#(c(x1)) -> b#(x1)
   b#(c(x1)) -> a#(b(x1))
  TRS:
   a(x1) -> x1
   a(a(b(x1))) -> c(b(a(a(x1))))
   b(c(x1)) -> a(b(x1))
  TDG Processor:
   DPs:
    a#(a(b(x1))) -> a#(x1)
    a#(a(b(x1))) -> a#(a(x1))
    a#(a(b(x1))) -> b#(a(a(x1)))
    b#(c(x1)) -> b#(x1)
    b#(c(x1)) -> a#(b(x1))
   TRS:
    a(x1) -> x1
    a(a(b(x1))) -> c(b(a(a(x1))))
    b(c(x1)) -> a(b(x1))
   graph:
    b#(c(x1)) -> b#(x1) -> b#(c(x1)) -> a#(b(x1))
    b#(c(x1)) -> b#(x1) -> b#(c(x1)) -> b#(x1)
    b#(c(x1)) -> a#(b(x1)) -> a#(a(b(x1))) -> b#(a(a(x1)))
    b#(c(x1)) -> a#(b(x1)) -> a#(a(b(x1))) -> a#(a(x1))
    b#(c(x1)) -> a#(b(x1)) -> a#(a(b(x1))) -> a#(x1)
    a#(a(b(x1))) -> b#(a(a(x1))) -> b#(c(x1)) -> a#(b(x1))
    a#(a(b(x1))) -> b#(a(a(x1))) -> b#(c(x1)) -> b#(x1)
    a#(a(b(x1))) -> a#(a(x1)) -> a#(a(b(x1))) -> b#(a(a(x1)))
    a#(a(b(x1))) -> a#(a(x1)) -> a#(a(b(x1))) -> a#(a(x1))
    a#(a(b(x1))) -> a#(a(x1)) -> a#(a(b(x1))) -> a#(x1)
    a#(a(b(x1))) -> a#(x1) -> a#(a(b(x1))) -> b#(a(a(x1)))
    a#(a(b(x1))) -> a#(x1) -> a#(a(b(x1))) -> a#(a(x1))
    a#(a(b(x1))) -> a#(x1) -> a#(a(b(x1))) -> a#(x1)
   Arctic Interpretation Processor:
    dimension: 2
    usable rules:
     a(x1) -> x1
     a(a(b(x1))) -> c(b(a(a(x1))))
     b(c(x1)) -> a(b(x1))
    interpretation:
     [b#](x0) = [3 3]x0 + [0],
     
     [a#](x0) = [2 0]x0,
     
               [0  0 ]     [3]
     [c](x0) = [-& 0 ]x0 + [0],
     
               [1 1]     [3 ]
     [b](x0) = [0 0]x0 + [-&],
     
               [0  0 ]     [0]
     [a](x0) = [-& 0 ]x0 + [0]
    orientation:
     a#(a(b(x1))) = [3 3]x1 + [5] >= [2 0]x1 = a#(x1)
     
     a#(a(b(x1))) = [3 3]x1 + [5] >= [2 2]x1 + [2] = a#(a(x1))
     
     a#(a(b(x1))) = [3 3]x1 + [5] >= [3 3]x1 + [3] = b#(a(a(x1)))
     
     b#(c(x1)) = [3 3]x1 + [6] >= [3 3]x1 + [0] = b#(x1)
     
     b#(c(x1)) = [3 3]x1 + [6] >= [3 3]x1 + [5] = a#(b(x1))
     
             [0  0 ]     [0]           
     a(x1) = [-& 0 ]x1 + [0] >= x1 = x1
     
                   [1 1]     [3]    [1 1]     [3]                 
     a(a(b(x1))) = [0 0]x1 + [0] >= [0 0]x1 + [0] = c(b(a(a(x1))))
     
                [1 1]     [4]    [1 1]     [3]           
     b(c(x1)) = [0 0]x1 + [3] >= [0 0]x1 + [0] = a(b(x1))
    problem:
     DPs:
      a#(a(b(x1))) -> b#(a(a(x1)))
      b#(c(x1)) -> b#(x1)
      b#(c(x1)) -> a#(b(x1))
     TRS:
      a(x1) -> x1
      a(a(b(x1))) -> c(b(a(a(x1))))
      b(c(x1)) -> a(b(x1))
    Restore Modifier:
     DPs:
      a#(a(b(x1))) -> b#(a(a(x1)))
      b#(c(x1)) -> b#(x1)
      b#(c(x1)) -> a#(b(x1))
     TRS:
      a(x1) -> x1
      a(a(b(x1))) -> c(b(a(a(x1))))
      b(c(x1)) -> a(b(x1))
     EDG Processor:
      DPs:
       a#(a(b(x1))) -> b#(a(a(x1)))
       b#(c(x1)) -> b#(x1)
       b#(c(x1)) -> a#(b(x1))
      TRS:
       a(x1) -> x1
       a(a(b(x1))) -> c(b(a(a(x1))))
       b(c(x1)) -> a(b(x1))
      graph:
       b#(c(x1)) -> b#(x1) -> b#(c(x1)) -> b#(x1)
       b#(c(x1)) -> b#(x1) -> b#(c(x1)) -> a#(b(x1))
       b#(c(x1)) -> a#(b(x1)) -> a#(a(b(x1))) -> b#(a(a(x1)))
       a#(a(b(x1))) -> b#(a(a(x1))) -> b#(c(x1)) -> b#(x1)
       a#(a(b(x1))) -> b#(a(a(x1))) -> b#(c(x1)) -> a#(b(x1))
      Arctic Interpretation Processor:
       dimension: 3
       usable rules:
        a(x1) -> x1
        a(a(b(x1))) -> c(b(a(a(x1))))
        b(c(x1)) -> a(b(x1))
       interpretation:
        [b#](x0) = [0 0 0]x0,
        
        [a#](x0) = [0  0  -&]x0 + [0],
        
                  [1 0 0]     [0]
        [c](x0) = [0 0 0]x0 + [0]
                  [1 0 0]     [0],
        
                  [0  -& -&]     [-&]
        [b](x0) = [1  0  -&]x0 + [0 ]
                  [1  1  1 ]     [0 ],
        
                  [0  0  -&]  
        [a](x0) = [0  0  0 ]x0
                  [0  0  0 ]  
       orientation:
        a#(a(b(x1))) = [1 1 1]x1 + [0] >= [0 0 0]x1 = b#(a(a(x1)))
        
        b#(c(x1)) = [1 0 0]x1 + [0] >= [0 0 0]x1 = b#(x1)
        
        b#(c(x1)) = [1 0 0]x1 + [0] >= [1  0  -&]x1 + [0] = a#(b(x1))
        
                [0  0  -&]             
        a(x1) = [0  0  0 ]x1 >= x1 = x1
                [0  0  0 ]             
        
                      [1 1 1]     [0]    [1 1 1]     [0]                 
        a(a(b(x1))) = [1 1 1]x1 + [0] >= [1 1 1]x1 + [0] = c(b(a(a(x1))))
                      [1 1 1]     [0]    [1 1 1]     [0]                 
        
                   [1 0 0]     [0]    [1  0  -&]     [0]           
        b(c(x1)) = [2 1 1]x1 + [1] >= [1  1  1 ]x1 + [0] = a(b(x1))
                   [2 1 1]     [1]    [1  1  1 ]     [0]           
       problem:
        DPs:
         b#(c(x1)) -> b#(x1)
         b#(c(x1)) -> a#(b(x1))
        TRS:
         a(x1) -> x1
         a(a(b(x1))) -> c(b(a(a(x1))))
         b(c(x1)) -> a(b(x1))
       Restore Modifier:
        DPs:
         b#(c(x1)) -> b#(x1)
         b#(c(x1)) -> a#(b(x1))
        TRS:
         a(x1) -> x1
         a(a(b(x1))) -> c(b(a(a(x1))))
         b(c(x1)) -> a(b(x1))
        EDG Processor:
         DPs:
          b#(c(x1)) -> b#(x1)
          b#(c(x1)) -> a#(b(x1))
         TRS:
          a(x1) -> x1
          a(a(b(x1))) -> c(b(a(a(x1))))
          b(c(x1)) -> a(b(x1))
         graph:
          b#(c(x1)) -> b#(x1) -> b#(c(x1)) -> a#(b(x1))
          b#(c(x1)) -> b#(x1) -> b#(c(x1)) -> b#(x1)
         SCC Processor:
          #sccs: 1
          #rules: 1
          #arcs: 2/4
          DPs:
           b#(c(x1)) -> b#(x1)
          TRS:
           a(x1) -> x1
           a(a(b(x1))) -> c(b(a(a(x1))))
           b(c(x1)) -> a(b(x1))
          Usable Rule Processor:
           DPs:
            b#(c(x1)) -> b#(x1)
           TRS:
            
           Arctic Interpretation Processor:
            dimension: 4
            usable rules:
             
            interpretation:
             [b#](x0) = [0  -& -& -&]x0,
             
                       [1 0 0 0]     [0]
                       [1 0 0 0]     [0]
             [c](x0) = [1 0 0 0]x0 + [0]
                       [0 0 0 0]     [0]
            orientation:
             b#(c(x1)) = [1 0 0 0]x1 + [0] >= [0  -& -& -&]x1 = b#(x1)
            problem:
             DPs:
              
             TRS:
              
            Qed