YES

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

Proof:
 DP Processor:
  DPs:
   a#(x1) -> b#(x1)
   a#(x1) -> b#(b(x1))
   b#(a(c(x1))) -> a#(x1)
   b#(a(c(x1))) -> a#(a(x1))
   b#(a(c(x1))) -> c#(a(a(x1)))
   b#(a(c(x1))) -> c#(c(a(a(x1))))
  TRS:
   a(x1) -> x1
   a(x1) -> b(b(x1))
   b(a(c(x1))) -> c(c(a(a(x1))))
   c(x1) -> x1
  TDG Processor:
   DPs:
    a#(x1) -> b#(x1)
    a#(x1) -> b#(b(x1))
    b#(a(c(x1))) -> a#(x1)
    b#(a(c(x1))) -> a#(a(x1))
    b#(a(c(x1))) -> c#(a(a(x1)))
    b#(a(c(x1))) -> c#(c(a(a(x1))))
   TRS:
    a(x1) -> x1
    a(x1) -> b(b(x1))
    b(a(c(x1))) -> c(c(a(a(x1))))
    c(x1) -> x1
   graph:
    b#(a(c(x1))) -> a#(a(x1)) -> a#(x1) -> b#(b(x1))
    b#(a(c(x1))) -> a#(a(x1)) -> a#(x1) -> b#(x1)
    b#(a(c(x1))) -> a#(x1) -> a#(x1) -> b#(b(x1))
    b#(a(c(x1))) -> a#(x1) -> a#(x1) -> b#(x1)
    a#(x1) -> b#(b(x1)) -> b#(a(c(x1))) -> c#(c(a(a(x1))))
    a#(x1) -> b#(b(x1)) -> b#(a(c(x1))) -> c#(a(a(x1)))
    a#(x1) -> b#(b(x1)) -> b#(a(c(x1))) -> a#(a(x1))
    a#(x1) -> b#(b(x1)) -> b#(a(c(x1))) -> a#(x1)
    a#(x1) -> b#(x1) -> b#(a(c(x1))) -> c#(c(a(a(x1))))
    a#(x1) -> b#(x1) -> b#(a(c(x1))) -> c#(a(a(x1)))
    a#(x1) -> b#(x1) -> b#(a(c(x1))) -> a#(a(x1))
    a#(x1) -> b#(x1) -> b#(a(c(x1))) -> a#(x1)
   SCC Processor:
    #sccs: 1
    #rules: 4
    #arcs: 12/36
    DPs:
     b#(a(c(x1))) -> a#(a(x1))
     a#(x1) -> b#(x1)
     b#(a(c(x1))) -> a#(x1)
     a#(x1) -> b#(b(x1))
    TRS:
     a(x1) -> x1
     a(x1) -> b(b(x1))
     b(a(c(x1))) -> c(c(a(a(x1))))
     c(x1) -> x1
    Arctic Interpretation Processor:
     dimension: 2
     interpretation:
      [b#](x0) = [0 3]x0 + [0],
      
      [a#](x0) = [0 3]x0 + [1],
      
                [0  3 ]     [2]
      [c](x0) = [-& 0 ]x0 + [1],
      
                [0  3 ]  
      [b](x0) = [-& 0 ]x0,
      
                [0 3]     [-&]
      [a](x0) = [0 0]x0 + [3 ]
     orientation:
      b#(a(c(x1))) = [3 6]x1 + [6] >= [3 3]x1 + [6] = a#(a(x1))
      
      a#(x1) = [0 3]x1 + [1] >= [0 3]x1 + [0] = b#(x1)
      
      b#(a(c(x1))) = [3 6]x1 + [6] >= [0 3]x1 + [1] = a#(x1)
      
      a#(x1) = [0 3]x1 + [1] >= [0 3]x1 + [0] = b#(b(x1))
      
              [0 3]     [-&]           
      a(x1) = [0 0]x1 + [3 ] >= x1 = x1
      
              [0 3]     [-&]    [0  3 ]             
      a(x1) = [0 0]x1 + [3 ] >= [-& 0 ]x1 = b(b(x1))
      
                    [3 6]     [6]    [3 6]     [6]                 
      b(a(c(x1))) = [0 3]x1 + [3] >= [0 3]x1 + [3] = c(c(a(a(x1))))
      
              [0  3 ]     [2]           
      c(x1) = [-& 0 ]x1 + [1] >= x1 = x1
     problem:
      DPs:
       b#(a(c(x1))) -> a#(a(x1))
       a#(x1) -> b#(x1)
       a#(x1) -> b#(b(x1))
      TRS:
       a(x1) -> x1
       a(x1) -> b(b(x1))
       b(a(c(x1))) -> c(c(a(a(x1))))
       c(x1) -> x1
     EDG Processor:
      DPs:
       b#(a(c(x1))) -> a#(a(x1))
       a#(x1) -> b#(x1)
       a#(x1) -> b#(b(x1))
      TRS:
       a(x1) -> x1
       a(x1) -> b(b(x1))
       b(a(c(x1))) -> c(c(a(a(x1))))
       c(x1) -> x1
      graph:
       b#(a(c(x1))) -> a#(a(x1)) -> a#(x1) -> b#(x1)
       b#(a(c(x1))) -> a#(a(x1)) -> a#(x1) -> b#(b(x1))
       a#(x1) -> b#(b(x1)) -> b#(a(c(x1))) -> a#(a(x1))
       a#(x1) -> b#(x1) -> b#(a(c(x1))) -> a#(a(x1))
      Arctic Interpretation Processor:
       dimension: 2
       interpretation:
        [b#](x0) = [-& -2]x0 + [0],
        
        [a#](x0) = [-& -2]x0 + [1],
        
                  [1  2 ]     [3]
        [c](x0) = [-& 0 ]x0 + [0],
        
                  [-& 2 ]     [2 ]
        [b](x0) = [-& -1]x0 + [-4],
        
                  [0 1]     [2 ]
        [a](x0) = [2 0]x0 + [-4]
       orientation:
        b#(a(c(x1))) = [1 2]x1 + [3] >= [0  -2]x1 + [1] = a#(a(x1))
        
        a#(x1) = [-& -2]x1 + [1] >= [-& -2]x1 + [0] = b#(x1)
        
        a#(x1) = [-& -2]x1 + [1] >= [-& -3]x1 + [0] = b#(b(x1))
        
                [0 1]     [2 ]           
        a(x1) = [2 0]x1 + [-4] >= x1 = x1
        
                [0 1]     [2 ]    [-& 1 ]     [2 ]           
        a(x1) = [2 0]x1 + [-4] >= [-& -2]x1 + [-4] = b(b(x1))
        
                      [5 6]     [7]    [5 6]     [7]                 
        b(a(c(x1))) = [2 3]x1 + [4] >= [2 3]x1 + [4] = c(c(a(a(x1))))
        
                [1  2 ]     [3]           
        c(x1) = [-& 0 ]x1 + [0] >= x1 = x1
       problem:
        DPs:
         a#(x1) -> b#(x1)
        TRS:
         a(x1) -> x1
         a(x1) -> b(b(x1))
         b(a(c(x1))) -> c(c(a(a(x1))))
         c(x1) -> x1
       EDG Processor:
        DPs:
         a#(x1) -> b#(x1)
        TRS:
         a(x1) -> x1
         a(x1) -> b(b(x1))
         b(a(c(x1))) -> c(c(a(a(x1))))
         c(x1) -> x1
        graph:
         
        Qed