YES

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

Proof:
 DP Processor:
  DPs:
   a#(b(x1)) -> c#(x1)
   a#(b(x1)) -> a#(c(x1))
   a#(b(x1)) -> a#(a(c(x1)))
   a#(b(x1)) -> b#(a(a(c(x1))))
   a#(b(x1)) -> c#(b(a(a(c(x1)))))
   c#(c(x1)) -> b#(x1)
  TRS:
   a(x1) -> x1
   a(b(x1)) -> c(b(a(a(c(x1)))))
   b(x1) -> x1
   c(c(x1)) -> b(x1)
  TDG Processor:
   DPs:
    a#(b(x1)) -> c#(x1)
    a#(b(x1)) -> a#(c(x1))
    a#(b(x1)) -> a#(a(c(x1)))
    a#(b(x1)) -> b#(a(a(c(x1))))
    a#(b(x1)) -> c#(b(a(a(c(x1)))))
    c#(c(x1)) -> b#(x1)
   TRS:
    a(x1) -> x1
    a(b(x1)) -> c(b(a(a(c(x1)))))
    b(x1) -> x1
    c(c(x1)) -> b(x1)
   graph:
    a#(b(x1)) -> c#(b(a(a(c(x1))))) -> c#(c(x1)) -> b#(x1)
    a#(b(x1)) -> c#(x1) -> c#(c(x1)) -> b#(x1)
    a#(b(x1)) -> a#(c(x1)) -> a#(b(x1)) -> c#(b(a(a(c(x1)))))
    a#(b(x1)) -> a#(c(x1)) -> a#(b(x1)) -> b#(a(a(c(x1))))
    a#(b(x1)) -> a#(c(x1)) -> a#(b(x1)) -> a#(a(c(x1)))
    a#(b(x1)) -> a#(c(x1)) -> a#(b(x1)) -> a#(c(x1))
    a#(b(x1)) -> a#(c(x1)) -> a#(b(x1)) -> c#(x1)
    a#(b(x1)) -> a#(a(c(x1))) -> a#(b(x1)) -> c#(b(a(a(c(x1)))))
    a#(b(x1)) -> a#(a(c(x1))) -> a#(b(x1)) -> b#(a(a(c(x1))))
    a#(b(x1)) -> a#(a(c(x1))) -> a#(b(x1)) -> a#(a(c(x1)))
    a#(b(x1)) -> a#(a(c(x1))) -> a#(b(x1)) -> a#(c(x1))
    a#(b(x1)) -> a#(a(c(x1))) -> a#(b(x1)) -> c#(x1)
   CDG Processor:
    DPs:
     a#(b(x1)) -> c#(x1)
     a#(b(x1)) -> a#(c(x1))
     a#(b(x1)) -> a#(a(c(x1)))
     a#(b(x1)) -> b#(a(a(c(x1))))
     a#(b(x1)) -> c#(b(a(a(c(x1)))))
     c#(c(x1)) -> b#(x1)
    TRS:
     a(x1) -> x1
     a(b(x1)) -> c(b(a(a(c(x1)))))
     b(x1) -> x1
     c(c(x1)) -> b(x1)
    graph:
     a#(b(x1)) -> c#(b(a(a(c(x1))))) -> c#(c(x1)) -> b#(x1)
     a#(b(x1)) -> a#(c(x1)) -> a#(b(x1)) -> c#(x1)
     a#(b(x1)) -> a#(c(x1)) -> a#(b(x1)) -> a#(c(x1))
     a#(b(x1)) -> a#(c(x1)) -> a#(b(x1)) -> a#(a(c(x1)))
     a#(b(x1)) -> a#(c(x1)) -> a#(b(x1)) -> b#(a(a(c(x1))))
     a#(b(x1)) -> a#(c(x1)) -> a#(b(x1)) -> c#(b(a(a(c(x1)))))
     a#(b(x1)) -> a#(a(c(x1))) -> a#(b(x1)) -> c#(x1)
     a#(b(x1)) -> a#(a(c(x1))) -> a#(b(x1)) -> a#(c(x1))
     a#(b(x1)) -> a#(a(c(x1))) -> a#(b(x1)) -> a#(a(c(x1)))
     a#(b(x1)) -> a#(a(c(x1))) -> a#(b(x1)) -> b#(a(a(c(x1))))
     a#(b(x1)) -> a#(a(c(x1))) -> a#(b(x1)) -> c#(b(a(a(c(x1)))))
    SCC Processor:
     #sccs: 1
     #rules: 2
     #arcs: 11/36
     DPs:
      a#(b(x1)) -> a#(c(x1))
      a#(b(x1)) -> a#(a(c(x1)))
     TRS:
      a(x1) -> x1
      a(b(x1)) -> c(b(a(a(c(x1)))))
      b(x1) -> x1
      c(c(x1)) -> b(x1)
     Arctic Interpretation Processor:
      dimension: 2
      interpretation:
       [a#](x0) = [0 1]x0 + [0],
       
                 [2  0 ]     [3]
       [c](x0) = [0  -&]x0 + [0],
       
                 [0 0]     [0]
       [b](x0) = [2 0]x0 + [3],
       
                 [0  2 ]     [0]
       [a](x0) = [-& 0 ]x0 + [0]
      orientation:
       a#(b(x1)) = [3 1]x1 + [4] >= [2 0]x1 + [3] = a#(c(x1))
       
       a#(b(x1)) = [3 1]x1 + [4] >= [2 0]x1 + [3] = a#(a(c(x1)))
       
               [0  2 ]     [0]           
       a(x1) = [-& 0 ]x1 + [0] >= x1 = x1
       
                  [4 2]     [5]    [4 2]     [5]                    
       a(b(x1)) = [2 0]x1 + [3] >= [2 0]x1 + [3] = c(b(a(a(c(x1)))))
       
               [0 0]     [0]           
       b(x1) = [2 0]x1 + [3] >= x1 = x1
       
                  [4 2]     [5]    [0 0]     [0]        
       c(c(x1)) = [2 0]x1 + [3] >= [2 0]x1 + [3] = b(x1)
      problem:
       DPs:
        
       TRS:
        a(x1) -> x1
        a(b(x1)) -> c(b(a(a(c(x1)))))
        b(x1) -> x1
        c(c(x1)) -> b(x1)
      Qed