YES

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

Proof:
 DP Processor:
  DPs:
   a#(b(x1)) -> a#(x1)
   a#(b(x1)) -> c#(a(x1))
   a#(b(x1)) -> a#(c(a(x1)))
   a#(b(x1)) -> c#(a(c(a(x1))))
  TRS:
   a(x1) -> b(x1)
   a(b(x1)) -> c(a(c(a(x1))))
   c(b(c(x1))) -> b(x1)
  TDG Processor:
   DPs:
    a#(b(x1)) -> a#(x1)
    a#(b(x1)) -> c#(a(x1))
    a#(b(x1)) -> a#(c(a(x1)))
    a#(b(x1)) -> c#(a(c(a(x1))))
   TRS:
    a(x1) -> b(x1)
    a(b(x1)) -> c(a(c(a(x1))))
    c(b(c(x1))) -> b(x1)
   graph:
    a#(b(x1)) -> a#(c(a(x1))) -> a#(b(x1)) -> c#(a(c(a(x1))))
    a#(b(x1)) -> a#(c(a(x1))) -> a#(b(x1)) -> a#(c(a(x1)))
    a#(b(x1)) -> a#(c(a(x1))) -> a#(b(x1)) -> c#(a(x1))
    a#(b(x1)) -> a#(c(a(x1))) -> a#(b(x1)) -> a#(x1)
    a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> c#(a(c(a(x1))))
    a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> a#(c(a(x1)))
    a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> c#(a(x1))
    a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> a#(x1)
   SCC Processor:
    #sccs: 1
    #rules: 2
    #arcs: 8/16
    DPs:
     a#(b(x1)) -> a#(c(a(x1)))
     a#(b(x1)) -> a#(x1)
    TRS:
     a(x1) -> b(x1)
     a(b(x1)) -> c(a(c(a(x1))))
     c(b(c(x1))) -> b(x1)
    Arctic Interpretation Processor:
     dimension: 1
     interpretation:
      [a#](x0) = x0,
      
      [c](x0) = x0,
      
      [b](x0) = 10x0,
      
      [a](x0) = 10x0
     orientation:
      a#(b(x1)) = 10x1 >= 10x1 = a#(c(a(x1)))
      
      a#(b(x1)) = 10x1 >= x1 = a#(x1)
      
      a(x1) = 10x1 >= 10x1 = b(x1)
      
      a(b(x1)) = 20x1 >= 20x1 = c(a(c(a(x1))))
      
      c(b(c(x1))) = 10x1 >= 10x1 = b(x1)
     problem:
      DPs:
       a#(b(x1)) -> a#(c(a(x1)))
      TRS:
       a(x1) -> b(x1)
       a(b(x1)) -> c(a(c(a(x1))))
       c(b(c(x1))) -> b(x1)
     EDG Processor:
      DPs:
       a#(b(x1)) -> a#(c(a(x1)))
      TRS:
       a(x1) -> b(x1)
       a(b(x1)) -> c(a(c(a(x1))))
       c(b(c(x1))) -> b(x1)
      graph:
       a#(b(x1)) -> a#(c(a(x1))) -> a#(b(x1)) -> a#(c(a(x1)))
      Arctic Interpretation Processor:
       dimension: 2
       interpretation:
        [a#](x0) = [-& 0 ]x0 + [0],
        
                  [1  -&]     [2]
        [c](x0) = [-& -1]x0 + [1],
        
                  [-& 0 ]     [0]
        [b](x0) = [1  -&]x0 + [2],
        
                  [-& 0 ]     [1]
        [a](x0) = [1  -&]x0 + [2]
       orientation:
        a#(b(x1)) = [1  -&]x1 + [2] >= [0  -&]x1 + [1] = a#(c(a(x1)))
        
                [-& 0 ]     [1]    [-& 0 ]     [0]        
        a(x1) = [1  -&]x1 + [2] >= [1  -&]x1 + [2] = b(x1)
        
                   [1  -&]     [2]    [1  -&]     [2]                 
        a(b(x1)) = [-& 1 ]x1 + [2] >= [-& 1 ]x1 + [2] = c(a(c(a(x1))))
        
                      [-& 0 ]     [2]    [-& 0 ]     [0]        
        c(b(c(x1))) = [1  -&]x1 + [2] >= [1  -&]x1 + [2] = b(x1)
       problem:
        DPs:
         
        TRS:
         a(x1) -> b(x1)
         a(b(x1)) -> c(a(c(a(x1))))
         c(b(c(x1))) -> b(x1)
       Qed