YES

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

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