YES

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

Proof:
 DP Processor:
  DPs:
   a#(a(x1)) -> b#(a(x1))
   a#(a(x1)) -> a#(b(a(x1)))
   b#(a(b(x1))) -> a#(x1)
   b#(a(b(x1))) -> a#(c(a(x1)))
  TRS:
   a(a(x1)) -> a(b(a(x1)))
   b(a(b(x1))) -> a(c(a(x1)))
  TDG Processor:
   DPs:
    a#(a(x1)) -> b#(a(x1))
    a#(a(x1)) -> a#(b(a(x1)))
    b#(a(b(x1))) -> a#(x1)
    b#(a(b(x1))) -> a#(c(a(x1)))
   TRS:
    a(a(x1)) -> a(b(a(x1)))
    b(a(b(x1))) -> a(c(a(x1)))
   graph:
    b#(a(b(x1))) -> a#(c(a(x1))) -> a#(a(x1)) -> a#(b(a(x1)))
    b#(a(b(x1))) -> a#(c(a(x1))) -> a#(a(x1)) -> b#(a(x1))
    b#(a(b(x1))) -> a#(x1) -> a#(a(x1)) -> a#(b(a(x1)))
    b#(a(b(x1))) -> a#(x1) -> a#(a(x1)) -> b#(a(x1))
    a#(a(x1)) -> b#(a(x1)) -> b#(a(b(x1))) -> a#(c(a(x1)))
    a#(a(x1)) -> b#(a(x1)) -> b#(a(b(x1))) -> a#(x1)
    a#(a(x1)) -> a#(b(a(x1))) -> a#(a(x1)) -> a#(b(a(x1)))
    a#(a(x1)) -> a#(b(a(x1))) -> a#(a(x1)) -> b#(a(x1))
   EDG Processor:
    DPs:
     a#(a(x1)) -> b#(a(x1))
     a#(a(x1)) -> a#(b(a(x1)))
     b#(a(b(x1))) -> a#(x1)
     b#(a(b(x1))) -> a#(c(a(x1)))
    TRS:
     a(a(x1)) -> a(b(a(x1)))
     b(a(b(x1))) -> a(c(a(x1)))
    graph:
     b#(a(b(x1))) -> a#(x1) -> a#(a(x1)) -> b#(a(x1))
     b#(a(b(x1))) -> a#(x1) -> a#(a(x1)) -> a#(b(a(x1)))
     a#(a(x1)) -> b#(a(x1)) -> b#(a(b(x1))) -> a#(x1)
     a#(a(x1)) -> b#(a(x1)) -> b#(a(b(x1))) -> a#(c(a(x1)))
     a#(a(x1)) -> a#(b(a(x1))) -> a#(a(x1)) -> b#(a(x1))
     a#(a(x1)) -> a#(b(a(x1))) -> a#(a(x1)) -> a#(b(a(x1)))
    SCC Processor:
     #sccs: 1
     #rules: 3
     #arcs: 6/16
     DPs:
      b#(a(b(x1))) -> a#(x1)
      a#(a(x1)) -> a#(b(a(x1)))
      a#(a(x1)) -> b#(a(x1))
     TRS:
      a(a(x1)) -> a(b(a(x1)))
      b(a(b(x1))) -> a(c(a(x1)))
     Arctic Interpretation Processor:
      dimension: 1
      interpretation:
       [b#](x0) = x0 + 0,
       
       [a#](x0) = 1x0,
       
       [c](x0) = -13x0 + 1,
       
       [b](x0) = -1x0 + 3,
       
       [a](x0) = 4x0 + 5
      orientation:
       b#(a(b(x1))) = 3x1 + 7 >= 1x1 = a#(x1)
       
       a#(a(x1)) = 5x1 + 6 >= 4x1 + 5 = a#(b(a(x1)))
       
       a#(a(x1)) = 5x1 + 6 >= 4x1 + 5 = b#(a(x1))
       
       a(a(x1)) = 8x1 + 9 >= 7x1 + 8 = a(b(a(x1)))
       
       b(a(b(x1))) = 2x1 + 6 >= -5x1 + 5 = a(c(a(x1)))
      problem:
       DPs:
        
       TRS:
        a(a(x1)) -> a(b(a(x1)))
        b(a(b(x1))) -> a(c(a(x1)))
      Qed