YES

Problem:
 a__f(g(X),Y) -> a__f(mark(X),f(g(X),Y))
 mark(f(X1,X2)) -> a__f(mark(X1),X2)
 mark(g(X)) -> g(mark(X))
 a__f(X1,X2) -> f(X1,X2)

Proof:
 DP Processor:
  DPs:
   a__f#(g(X),Y) -> mark#(X)
   a__f#(g(X),Y) -> a__f#(mark(X),f(g(X),Y))
   mark#(f(X1,X2)) -> mark#(X1)
   mark#(f(X1,X2)) -> a__f#(mark(X1),X2)
   mark#(g(X)) -> mark#(X)
  TRS:
   a__f(g(X),Y) -> a__f(mark(X),f(g(X),Y))
   mark(f(X1,X2)) -> a__f(mark(X1),X2)
   mark(g(X)) -> g(mark(X))
   a__f(X1,X2) -> f(X1,X2)
  KBO Processor:
   argument filtering:
    pi(g) = [0]
    pi(a__f) = [0]
    pi(mark) = [0]
    pi(f) = [0]
    pi(a__f#) = [0]
    pi(mark#) = 0
   weight function:
    w0 = 1
    w(mark#) = w(a__f#) = w(f) = w(a__f) = w(g) = 1
    w(mark) = 0
   precedence:
    mark > a__f > f > mark# ~ a__f# ~ g
   problem:
    DPs:
     
    TRS:
     a__f(g(X),Y) -> a__f(mark(X),f(g(X),Y))
     mark(f(X1,X2)) -> a__f(mark(X1),X2)
     mark(g(X)) -> g(mark(X))
     a__f(X1,X2) -> f(X1,X2)
   Qed