YES

Problem:
 a(b(x1)) -> x1
 a(c(x1)) -> b(c(c(b(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#(a(a(x1)))
   a#(c(x1)) -> b#(c(c(b(a(a(x1))))))
  TRS:
   a(b(x1)) -> x1
   a(c(x1)) -> b(c(c(b(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#(a(a(x1)))
    a#(c(x1)) -> b#(c(c(b(a(a(x1))))))
   TRS:
    a(b(x1)) -> x1
    a(c(x1)) -> b(c(c(b(a(a(x1))))))
    b(c(x1)) -> x1
   graph:
    a#(c(x1)) -> a#(a(x1)) -> a#(c(x1)) -> b#(c(c(b(a(a(x1))))))
    a#(c(x1)) -> a#(a(x1)) -> a#(c(x1)) -> b#(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#(c(c(b(a(a(x1))))))
    a#(c(x1)) -> a#(x1) -> a#(c(x1)) -> b#(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(c(c(b(a(a(x1))))))
     b(c(x1)) -> x1
    Arctic Interpretation Processor:
     dimension: 1
     interpretation:
      [a#](x0) = x0,
      
      [c](x0) = 1x0 + 15,
      
      [a](x0) = 1x0 + 0,
      
      [b](x0) = -1x0 + 15
     orientation:
      a#(c(x1)) = 1x1 + 15 >= 1x1 + 0 = a#(a(x1))
      
      a#(c(x1)) = 1x1 + 15 >= x1 = a#(x1)
      
      a(b(x1)) = x1 + 16 >= x1 = x1
      
      a(c(x1)) = 2x1 + 16 >= 2x1 + 16 = b(c(c(b(a(a(x1))))))
      
      b(c(x1)) = x1 + 15 >= x1 = x1
     problem:
      DPs:
       a#(c(x1)) -> a#(a(x1))
      TRS:
       a(b(x1)) -> x1
       a(c(x1)) -> b(c(c(b(a(a(x1))))))
       b(c(x1)) -> x1
     EDG Processor:
      DPs:
       a#(c(x1)) -> a#(a(x1))
      TRS:
       a(b(x1)) -> x1
       a(c(x1)) -> b(c(c(b(a(a(x1))))))
       b(c(x1)) -> x1
      graph:
       a#(c(x1)) -> a#(a(x1)) -> a#(c(x1)) -> a#(a(x1))
      Matrix Interpretation Processor: dim=1
       
       interpretation:
        [a#](x0) = 9/2x0 + 2,
        
        [c](x0) = 2x0 + 9/2,
        
        [a](x0) = 2x0,
        
        [b](x0) = 1/2x0
       orientation:
        a#(c(x1)) = 9x1 + 89/4 >= 9x1 + 2 = a#(a(x1))
        
        a(b(x1)) = x1 >= x1 = x1
        
        a(c(x1)) = 4x1 + 9 >= 4x1 + 27/4 = b(c(c(b(a(a(x1))))))
        
        b(c(x1)) = x1 + 9/4 >= x1 = x1
       problem:
        DPs:
         
        TRS:
         a(b(x1)) -> x1
         a(c(x1)) -> b(c(c(b(a(a(x1))))))
         b(c(x1)) -> x1
       Qed