YES

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

Proof:
 DP Processor:
  DPs:
   a#(x1) -> c#(x1)
   a#(x1) -> b#(c(x1))
   c#(c(b(x1))) -> c#(x1)
   c#(c(b(x1))) -> a#(c(x1))
   c#(c(b(x1))) -> a#(a(c(x1)))
  TRS:
   a(x1) -> x1
   a(x1) -> b(c(x1))
   b(b(x1)) -> x1
   c(c(b(x1))) -> a(a(c(x1)))
  TDG Processor:
   DPs:
    a#(x1) -> c#(x1)
    a#(x1) -> b#(c(x1))
    c#(c(b(x1))) -> c#(x1)
    c#(c(b(x1))) -> a#(c(x1))
    c#(c(b(x1))) -> a#(a(c(x1)))
   TRS:
    a(x1) -> x1
    a(x1) -> b(c(x1))
    b(b(x1)) -> x1
    c(c(b(x1))) -> a(a(c(x1)))
   graph:
    c#(c(b(x1))) -> c#(x1) -> c#(c(b(x1))) -> a#(a(c(x1)))
    c#(c(b(x1))) -> c#(x1) -> c#(c(b(x1))) -> a#(c(x1))
    c#(c(b(x1))) -> c#(x1) -> c#(c(b(x1))) -> c#(x1)
    c#(c(b(x1))) -> a#(c(x1)) -> a#(x1) -> b#(c(x1))
    c#(c(b(x1))) -> a#(c(x1)) -> a#(x1) -> c#(x1)
    c#(c(b(x1))) -> a#(a(c(x1))) -> a#(x1) -> b#(c(x1))
    c#(c(b(x1))) -> a#(a(c(x1))) -> a#(x1) -> c#(x1)
    a#(x1) -> c#(x1) -> c#(c(b(x1))) -> a#(a(c(x1)))
    a#(x1) -> c#(x1) -> c#(c(b(x1))) -> a#(c(x1))
    a#(x1) -> c#(x1) -> c#(c(b(x1))) -> c#(x1)
   SCC Processor:
    #sccs: 1
    #rules: 4
    #arcs: 10/25
    DPs:
     c#(c(b(x1))) -> c#(x1)
     c#(c(b(x1))) -> a#(c(x1))
     a#(x1) -> c#(x1)
     c#(c(b(x1))) -> a#(a(c(x1)))
    TRS:
     a(x1) -> x1
     a(x1) -> b(c(x1))
     b(b(x1)) -> x1
     c(c(b(x1))) -> a(a(c(x1)))
    Arctic Interpretation Processor:
     dimension: 2
     usable rules:
      a(x1) -> x1
      a(x1) -> b(c(x1))
      b(b(x1)) -> x1
      c(c(b(x1))) -> a(a(c(x1)))
     interpretation:
      [c#](x0) = [1 0]x0,
      
                [-& 0 ]     [0]
      [c](x0) = [0  -&]x0 + [0],
      
      [a#](x0) = [1 0]x0 + [2],
      
                [0  -&]     [0]
      [a](x0) = [1  0 ]x0 + [2],
      
                [-& 0 ]     [0]
      [b](x0) = [0  1 ]x0 + [2]
     orientation:
      c#(c(b(x1))) = [1 2]x1 + [3] >= [1 0]x1 = c#(x1)
      
      c#(c(b(x1))) = [1 2]x1 + [3] >= [0 1]x1 + [2] = a#(c(x1))
      
      a#(x1) = [1 0]x1 + [2] >= [1 0]x1 = c#(x1)
      
      c#(c(b(x1))) = [1 2]x1 + [3] >= [0 1]x1 + [2] = a#(a(c(x1)))
      
              [0  -&]     [0]           
      a(x1) = [1  0 ]x1 + [2] >= x1 = x1
      
              [0  -&]     [0]    [0  -&]     [0]           
      a(x1) = [1  0 ]x1 + [2] >= [1  0 ]x1 + [2] = b(c(x1))
      
                 [0 1]     [2]           
      b(b(x1)) = [1 2]x1 + [3] >= x1 = x1
      
                    [-& 0 ]     [0]    [-& 0 ]     [0]              
      c(c(b(x1))) = [0  1 ]x1 + [2] >= [0  1 ]x1 + [2] = a(a(c(x1)))
     problem:
      DPs:
       c#(c(b(x1))) -> c#(x1)
       a#(x1) -> c#(x1)
      TRS:
       a(x1) -> x1
       a(x1) -> b(c(x1))
       b(b(x1)) -> x1
       c(c(b(x1))) -> a(a(c(x1)))
     Restore Modifier:
      DPs:
       c#(c(b(x1))) -> c#(x1)
       a#(x1) -> c#(x1)
      TRS:
       a(x1) -> x1
       a(x1) -> b(c(x1))
       b(b(x1)) -> x1
       c(c(b(x1))) -> a(a(c(x1)))
      EDG Processor:
       DPs:
        c#(c(b(x1))) -> c#(x1)
        a#(x1) -> c#(x1)
       TRS:
        a(x1) -> x1
        a(x1) -> b(c(x1))
        b(b(x1)) -> x1
        c(c(b(x1))) -> a(a(c(x1)))
       graph:
        c#(c(b(x1))) -> c#(x1) -> c#(c(b(x1))) -> c#(x1)
        a#(x1) -> c#(x1) -> c#(c(b(x1))) -> c#(x1)
       SCC Processor:
        #sccs: 1
        #rules: 1
        #arcs: 2/4
        DPs:
         c#(c(b(x1))) -> c#(x1)
        TRS:
         a(x1) -> x1
         a(x1) -> b(c(x1))
         b(b(x1)) -> x1
         c(c(b(x1))) -> a(a(c(x1)))
        Usable Rule Processor:
         DPs:
          c#(c(b(x1))) -> c#(x1)
         TRS:
          
         Arctic Interpretation Processor:
          dimension: 1
          usable rules:
           
          interpretation:
           [c#](x0) = x0,
           
           [c](x0) = x0 + 0,
           
           [b](x0) = 1x0 + 13
          orientation:
           c#(c(b(x1))) = 1x1 + 13 >= x1 = c#(x1)
          problem:
           DPs:
            
           TRS:
            
          Qed