YES

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

Proof:
 DP Processor:
  DPs:
   a#(a(x1)) -> c#(x1)
   a#(a(x1)) -> b#(c(x1))
   a#(a(x1)) -> a#(b(c(x1)))
   c#(b(x1)) -> c#(x1)
   c#(b(x1)) -> a#(c(x1))
   c#(b(x1)) -> b#(a(c(x1)))
  TRS:
   a(x1) -> x1
   a(a(x1)) -> a(b(c(x1)))
   b(x1) -> x1
   c(b(x1)) -> b(a(c(x1)))
  TDG Processor:
   DPs:
    a#(a(x1)) -> c#(x1)
    a#(a(x1)) -> b#(c(x1))
    a#(a(x1)) -> a#(b(c(x1)))
    c#(b(x1)) -> c#(x1)
    c#(b(x1)) -> a#(c(x1))
    c#(b(x1)) -> b#(a(c(x1)))
   TRS:
    a(x1) -> x1
    a(a(x1)) -> a(b(c(x1)))
    b(x1) -> x1
    c(b(x1)) -> b(a(c(x1)))
   graph:
    c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> b#(a(c(x1)))
    c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> a#(c(x1))
    c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> c#(x1)
    c#(b(x1)) -> a#(c(x1)) -> a#(a(x1)) -> a#(b(c(x1)))
    c#(b(x1)) -> a#(c(x1)) -> a#(a(x1)) -> b#(c(x1))
    c#(b(x1)) -> a#(c(x1)) -> a#(a(x1)) -> c#(x1)
    a#(a(x1)) -> c#(x1) -> c#(b(x1)) -> b#(a(c(x1)))
    a#(a(x1)) -> c#(x1) -> c#(b(x1)) -> a#(c(x1))
    a#(a(x1)) -> c#(x1) -> c#(b(x1)) -> c#(x1)
    a#(a(x1)) -> a#(b(c(x1))) -> a#(a(x1)) -> a#(b(c(x1)))
    a#(a(x1)) -> a#(b(c(x1))) -> a#(a(x1)) -> b#(c(x1))
    a#(a(x1)) -> a#(b(c(x1))) -> a#(a(x1)) -> c#(x1)
   SCC Processor:
    #sccs: 1
    #rules: 4
    #arcs: 12/36
    DPs:
     c#(b(x1)) -> c#(x1)
     c#(b(x1)) -> a#(c(x1))
     a#(a(x1)) -> c#(x1)
     a#(a(x1)) -> a#(b(c(x1)))
    TRS:
     a(x1) -> x1
     a(a(x1)) -> a(b(c(x1)))
     b(x1) -> x1
     c(b(x1)) -> b(a(c(x1)))
    Arctic Interpretation Processor:
     dimension: 2
     usable rules:
      a(x1) -> x1
      a(a(x1)) -> a(b(c(x1)))
      b(x1) -> x1
      c(b(x1)) -> b(a(c(x1)))
     interpretation:
      [c#](x0) = [0 1]x0 + [0],
      
      [a#](x0) = [0 0]x0 + [0],
      
                [1 0]     [0]
      [b](x0) = [0 0]x0 + [0],
      
                [0  -&]  
      [c](x0) = [0  -&]x0,
      
                [0 0]     [-&]
      [a](x0) = [1 1]x0 + [0 ]
     orientation:
      c#(b(x1)) = [1 1]x1 + [1] >= [0 1]x1 + [0] = c#(x1)
      
      c#(b(x1)) = [1 1]x1 + [1] >= [0  -&]x1 + [0] = a#(c(x1))
      
      a#(a(x1)) = [1 1]x1 + [0] >= [0 1]x1 + [0] = c#(x1)
      
      a#(a(x1)) = [1 1]x1 + [0] >= [1  -&]x1 + [0] = a#(b(c(x1)))
      
              [0 0]     [-&]           
      a(x1) = [1 1]x1 + [0 ] >= x1 = x1
      
                 [1 1]     [0]    [1  -&]     [0]              
      a(a(x1)) = [2 2]x1 + [1] >= [2  -&]x1 + [1] = a(b(c(x1)))
      
              [1 0]     [0]           
      b(x1) = [0 0]x1 + [0] >= x1 = x1
      
                 [1 0]     [0]    [1  -&]     [0]              
      c(b(x1)) = [1 0]x1 + [0] >= [1  -&]x1 + [0] = b(a(c(x1)))
     problem:
      DPs:
       c#(b(x1)) -> c#(x1)
       a#(a(x1)) -> c#(x1)
       a#(a(x1)) -> a#(b(c(x1)))
      TRS:
       a(x1) -> x1
       a(a(x1)) -> a(b(c(x1)))
       b(x1) -> x1
       c(b(x1)) -> b(a(c(x1)))
     Restore Modifier:
      DPs:
       c#(b(x1)) -> c#(x1)
       a#(a(x1)) -> c#(x1)
       a#(a(x1)) -> a#(b(c(x1)))
      TRS:
       a(x1) -> x1
       a(a(x1)) -> a(b(c(x1)))
       b(x1) -> x1
       c(b(x1)) -> b(a(c(x1)))
      EDG Processor:
       DPs:
        c#(b(x1)) -> c#(x1)
        a#(a(x1)) -> c#(x1)
        a#(a(x1)) -> a#(b(c(x1)))
       TRS:
        a(x1) -> x1
        a(a(x1)) -> a(b(c(x1)))
        b(x1) -> x1
        c(b(x1)) -> b(a(c(x1)))
       graph:
        c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> c#(x1)
        a#(a(x1)) -> c#(x1) -> c#(b(x1)) -> c#(x1)
        a#(a(x1)) -> a#(b(c(x1))) -> a#(a(x1)) -> c#(x1)
        a#(a(x1)) -> a#(b(c(x1))) -> a#(a(x1)) -> a#(b(c(x1)))
       CDG Processor:
        DPs:
         c#(b(x1)) -> c#(x1)
         a#(a(x1)) -> c#(x1)
         a#(a(x1)) -> a#(b(c(x1)))
        TRS:
         a(x1) -> x1
         a(a(x1)) -> a(b(c(x1)))
         b(x1) -> x1
         c(b(x1)) -> b(a(c(x1)))
        graph:
         a#(a(x1)) -> a#(b(c(x1))) -> a#(a(x1)) -> a#(b(c(x1)))
         a#(a(x1)) -> a#(b(c(x1))) -> a#(a(x1)) -> c#(x1)
        SCC Processor:
         #sccs: 1
         #rules: 1
         #arcs: 2/9
         DPs:
          a#(a(x1)) -> a#(b(c(x1)))
         TRS:
          a(x1) -> x1
          a(a(x1)) -> a(b(c(x1)))
          b(x1) -> x1
          c(b(x1)) -> b(a(c(x1)))
         Arctic Interpretation Processor:
          dimension: 3
          usable rules:
           a(x1) -> x1
           a(a(x1)) -> a(b(c(x1)))
           b(x1) -> x1
           c(b(x1)) -> b(a(c(x1)))
          interpretation:
           [a#](x0) = [0 0 1]x0 + [0],
           
                     [0  -& -&]     [0]
           [b](x0) = [1  1  0 ]x0 + [1]
                     [0  0  0 ]     [0],
           
                     [-& -& -&]     [0]
           [c](x0) = [-& 0  -&]x0 + [0]
                     [0  0  -&]     [0],
           
                     [0  -& -&]     [0]
           [a](x0) = [-& 0  0 ]x0 + [0]
                     [1  1  1 ]     [1]
          orientation:
           a#(a(x1)) = [2 2 2]x1 + [2] >= [1  1  -&]x1 + [1] = a#(b(c(x1)))
           
                   [0  -& -&]     [0]           
           a(x1) = [-& 0  0 ]x1 + [0] >= x1 = x1
                   [1  1  1 ]     [1]           
           
                      [0  -& -&]     [0]    [-& -& -&]     [0]              
           a(a(x1)) = [1  1  1 ]x1 + [1] >= [0  1  -&]x1 + [1] = a(b(c(x1)))
                      [2  2  2 ]     [2]    [1  2  -&]     [2]              
           
                   [0  -& -&]     [0]           
           b(x1) = [1  1  0 ]x1 + [1] >= x1 = x1
                   [0  0  0 ]     [0]           
           
                      [-& -& -&]     [0]    [-& -& -&]     [0]              
           c(b(x1)) = [1  1  0 ]x1 + [1] >= [1  1  -&]x1 + [1] = b(a(c(x1)))
                      [1  1  0 ]     [1]    [1  1  -&]     [1]              
          problem:
           DPs:
            
           TRS:
            a(x1) -> x1
            a(a(x1)) -> a(b(c(x1)))
            b(x1) -> x1
            c(b(x1)) -> b(a(c(x1)))
          Qed