YES

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

Proof:
 DP Processor:
  DPs:
   a#(c(x1)) -> a#(x1)
   a#(c(x1)) -> a#(a(x1))
   a#(c(x1)) -> b#(c(c(a(a(x1)))))
   a#(c(x1)) -> b#(b(c(c(a(a(x1))))))
  TRS:
   a(b(x1)) -> x1
   a(c(x1)) -> b(b(c(c(a(a(x1))))))
   b(c(x1)) -> x1
  TDG Processor:
   DPs:
    a#(c(x1)) -> a#(x1)
    a#(c(x1)) -> a#(a(x1))
    a#(c(x1)) -> b#(c(c(a(a(x1)))))
    a#(c(x1)) -> b#(b(c(c(a(a(x1))))))
   TRS:
    a(b(x1)) -> x1
    a(c(x1)) -> b(b(c(c(a(a(x1))))))
    b(c(x1)) -> x1
   graph:
    a#(c(x1)) -> a#(a(x1)) -> a#(c(x1)) -> b#(b(c(c(a(a(x1))))))
    a#(c(x1)) -> a#(a(x1)) -> a#(c(x1)) -> b#(c(c(a(a(x1)))))
    a#(c(x1)) -> a#(a(x1)) -> a#(c(x1)) -> a#(a(x1))
    a#(c(x1)) -> a#(a(x1)) -> a#(c(x1)) -> a#(x1)
    a#(c(x1)) -> a#(x1) -> a#(c(x1)) -> b#(b(c(c(a(a(x1))))))
    a#(c(x1)) -> a#(x1) -> a#(c(x1)) -> b#(c(c(a(a(x1)))))
    a#(c(x1)) -> a#(x1) -> a#(c(x1)) -> a#(a(x1))
    a#(c(x1)) -> a#(x1) -> a#(c(x1)) -> a#(x1)
   SCC Processor:
    #sccs: 1
    #rules: 2
    #arcs: 8/16
    DPs:
     a#(c(x1)) -> a#(a(x1))
     a#(c(x1)) -> a#(x1)
    TRS:
     a(b(x1)) -> x1
     a(c(x1)) -> b(b(c(c(a(a(x1))))))
     b(c(x1)) -> x1
    Matrix Interpretation Processor: dim=1
     
     interpretation:
      [a#](x0) = 4x0 + 2,
      
      [c](x0) = 2x0 + 5,
      
      [a](x0) = 2x0 + 1/2,
      
      [b](x0) = 1/2x0 + 7/2
     orientation:
      a#(c(x1)) = 8x1 + 22 >= 8x1 + 4 = a#(a(x1))
      
      a#(c(x1)) = 8x1 + 22 >= 4x1 + 2 = a#(x1)
      
      a(b(x1)) = x1 + 15/2 >= x1 = x1
      
      a(c(x1)) = 4x1 + 21/2 >= 4x1 + 21/2 = b(b(c(c(a(a(x1))))))
      
      b(c(x1)) = x1 + 6 >= x1 = x1
     problem:
      DPs:
       
      TRS:
       a(b(x1)) -> x1
       a(c(x1)) -> b(b(c(c(a(a(x1))))))
       b(c(x1)) -> x1
     Qed