YES

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

Proof:
 DP Processor:
  DPs:
   a#(b(x1)) -> a#(x1)
   a#(b(x1)) -> a#(a(x1))
   a#(b(x1)) -> b#(a(a(x1)))
   b#(x1) -> a#(c(x1))
   a#(a(x1)) -> a#(c(a(x1)))
  TRS:
   a(b(x1)) -> b(a(a(x1)))
   b(x1) -> c(a(c(x1)))
   a(a(x1)) -> a(c(a(x1)))
  TDG Processor:
   DPs:
    a#(b(x1)) -> a#(x1)
    a#(b(x1)) -> a#(a(x1))
    a#(b(x1)) -> b#(a(a(x1)))
    b#(x1) -> a#(c(x1))
    a#(a(x1)) -> a#(c(a(x1)))
   TRS:
    a(b(x1)) -> b(a(a(x1)))
    b(x1) -> c(a(c(x1)))
    a(a(x1)) -> a(c(a(x1)))
   graph:
    b#(x1) -> a#(c(x1)) -> a#(a(x1)) -> a#(c(a(x1)))
    b#(x1) -> a#(c(x1)) -> a#(b(x1)) -> b#(a(a(x1)))
    b#(x1) -> a#(c(x1)) -> a#(b(x1)) -> a#(a(x1))
    b#(x1) -> a#(c(x1)) -> a#(b(x1)) -> a#(x1)
    a#(a(x1)) -> a#(c(a(x1))) -> a#(a(x1)) -> a#(c(a(x1)))
    a#(a(x1)) -> a#(c(a(x1))) -> a#(b(x1)) -> b#(a(a(x1)))
    a#(a(x1)) -> a#(c(a(x1))) -> a#(b(x1)) -> a#(a(x1))
    a#(a(x1)) -> a#(c(a(x1))) -> a#(b(x1)) -> a#(x1)
    a#(b(x1)) -> b#(a(a(x1))) -> b#(x1) -> a#(c(x1))
    a#(b(x1)) -> a#(a(x1)) -> a#(a(x1)) -> a#(c(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#(a(x1)) -> a#(c(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)
   EDG Processor:
    DPs:
     a#(b(x1)) -> a#(x1)
     a#(b(x1)) -> a#(a(x1))
     a#(b(x1)) -> b#(a(a(x1)))
     b#(x1) -> a#(c(x1))
     a#(a(x1)) -> a#(c(a(x1)))
    TRS:
     a(b(x1)) -> b(a(a(x1)))
     b(x1) -> c(a(c(x1)))
     a(a(x1)) -> a(c(a(x1)))
    graph:
     a#(b(x1)) -> b#(a(a(x1))) -> b#(x1) -> a#(c(x1))
     a#(b(x1)) -> a#(a(x1)) -> a#(b(x1)) -> a#(x1)
     a#(b(x1)) -> a#(a(x1)) -> a#(b(x1)) -> a#(a(x1))
     a#(b(x1)) -> a#(a(x1)) -> a#(b(x1)) -> b#(a(a(x1)))
     a#(b(x1)) -> a#(a(x1)) -> a#(a(x1)) -> a#(c(a(x1)))
     a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> a#(x1)
     a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> a#(a(x1))
     a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> b#(a(a(x1)))
     a#(b(x1)) -> a#(x1) -> a#(a(x1)) -> a#(c(a(x1)))
    SCC Processor:
     #sccs: 1
     #rules: 2
     #arcs: 9/25
     DPs:
      a#(b(x1)) -> a#(a(x1))
      a#(b(x1)) -> a#(x1)
     TRS:
      a(b(x1)) -> b(a(a(x1)))
      b(x1) -> c(a(c(x1)))
      a(a(x1)) -> a(c(a(x1)))
     Arctic Interpretation Processor:
      dimension: 1
      interpretation:
       [a#](x0) = 8x0,
       
       [c](x0) = x0,
       
       [a](x0) = x0,
       
       [b](x0) = 1x0
      orientation:
       a#(b(x1)) = 9x1 >= 8x1 = a#(a(x1))
       
       a#(b(x1)) = 9x1 >= 8x1 = a#(x1)
       
       a(b(x1)) = 1x1 >= 1x1 = b(a(a(x1)))
       
       b(x1) = 1x1 >= x1 = c(a(c(x1)))
       
       a(a(x1)) = x1 >= x1 = a(c(a(x1)))
      problem:
       DPs:
        
       TRS:
        a(b(x1)) -> b(a(a(x1)))
        b(x1) -> c(a(c(x1)))
        a(a(x1)) -> a(c(a(x1)))
      Qed