YES

Problem:
 0(x1) -> 1(x1)
 4(5(4(5(x1)))) -> 4(4(5(5(x1))))
 5(5(5(5(5(5(4(4(4(4(4(4(x1)))))))))))) -> 2(x1)

Proof:
 DP Processor:
  DPs:
   4#(5(4(5(x1)))) -> 5#(5(x1))
   4#(5(4(5(x1)))) -> 4#(5(5(x1)))
   4#(5(4(5(x1)))) -> 4#(4(5(5(x1))))
  TRS:
   0(x1) -> 1(x1)
   4(5(4(5(x1)))) -> 4(4(5(5(x1))))
   5(5(5(5(5(5(4(4(4(4(4(4(x1)))))))))))) -> 2(x1)
  TDG Processor:
   DPs:
    4#(5(4(5(x1)))) -> 5#(5(x1))
    4#(5(4(5(x1)))) -> 4#(5(5(x1)))
    4#(5(4(5(x1)))) -> 4#(4(5(5(x1))))
   TRS:
    0(x1) -> 1(x1)
    4(5(4(5(x1)))) -> 4(4(5(5(x1))))
    5(5(5(5(5(5(4(4(4(4(4(4(x1)))))))))))) -> 2(x1)
   graph:
    4#(5(4(5(x1)))) -> 4#(4(5(5(x1)))) ->
    4#(5(4(5(x1)))) -> 4#(4(5(5(x1))))
    4#(5(4(5(x1)))) -> 4#(4(5(5(x1)))) ->
    4#(5(4(5(x1)))) -> 4#(5(5(x1)))
    4#(5(4(5(x1)))) -> 4#(4(5(5(x1)))) ->
    4#(5(4(5(x1)))) -> 5#(5(x1))
    4#(5(4(5(x1)))) -> 4#(5(5(x1))) ->
    4#(5(4(5(x1)))) -> 4#(4(5(5(x1))))
    4#(5(4(5(x1)))) -> 4#(5(5(x1))) ->
    4#(5(4(5(x1)))) -> 4#(5(5(x1)))
    4#(5(4(5(x1)))) -> 4#(5(5(x1))) -> 4#(5(4(5(x1)))) -> 5#(5(x1))
   EDG Processor:
    DPs:
     4#(5(4(5(x1)))) -> 5#(5(x1))
     4#(5(4(5(x1)))) -> 4#(5(5(x1)))
     4#(5(4(5(x1)))) -> 4#(4(5(5(x1))))
    TRS:
     0(x1) -> 1(x1)
     4(5(4(5(x1)))) -> 4(4(5(5(x1))))
     5(5(5(5(5(5(4(4(4(4(4(4(x1)))))))))))) -> 2(x1)
    graph:
     
    Qed