YES

Problem:
 active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
 active(f(X1,X2)) -> f(active(X1),X2)
 active(g(X)) -> g(active(X))
 f(mark(X1),X2) -> mark(f(X1,X2))
 g(mark(X)) -> mark(g(X))
 proper(f(X1,X2)) -> f(proper(X1),proper(X2))
 proper(g(X)) -> g(proper(X))
 f(ok(X1),ok(X2)) -> ok(f(X1,X2))
 g(ok(X)) -> ok(g(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))

Proof:
 DP Processor:
  DPs:
   active#(f(g(X),Y)) -> f#(X,f(g(X),Y))
   active#(f(X1,X2)) -> active#(X1)
   active#(f(X1,X2)) -> f#(active(X1),X2)
   active#(g(X)) -> active#(X)
   active#(g(X)) -> g#(active(X))
   f#(mark(X1),X2) -> f#(X1,X2)
   g#(mark(X)) -> g#(X)
   proper#(f(X1,X2)) -> proper#(X2)
   proper#(f(X1,X2)) -> proper#(X1)
   proper#(f(X1,X2)) -> f#(proper(X1),proper(X2))
   proper#(g(X)) -> proper#(X)
   proper#(g(X)) -> g#(proper(X))
   f#(ok(X1),ok(X2)) -> f#(X1,X2)
   g#(ok(X)) -> g#(X)
   top#(mark(X)) -> proper#(X)
   top#(mark(X)) -> top#(proper(X))
   top#(ok(X)) -> active#(X)
   top#(ok(X)) -> top#(active(X))
  TRS:
   active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
   active(f(X1,X2)) -> f(active(X1),X2)
   active(g(X)) -> g(active(X))
   f(mark(X1),X2) -> mark(f(X1,X2))
   g(mark(X)) -> mark(g(X))
   proper(f(X1,X2)) -> f(proper(X1),proper(X2))
   proper(g(X)) -> g(proper(X))
   f(ok(X1),ok(X2)) -> ok(f(X1,X2))
   g(ok(X)) -> ok(g(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))
  TDG Processor:
   DPs:
    active#(f(g(X),Y)) -> f#(X,f(g(X),Y))
    active#(f(X1,X2)) -> active#(X1)
    active#(f(X1,X2)) -> f#(active(X1),X2)
    active#(g(X)) -> active#(X)
    active#(g(X)) -> g#(active(X))
    f#(mark(X1),X2) -> f#(X1,X2)
    g#(mark(X)) -> g#(X)
    proper#(f(X1,X2)) -> proper#(X2)
    proper#(f(X1,X2)) -> proper#(X1)
    proper#(f(X1,X2)) -> f#(proper(X1),proper(X2))
    proper#(g(X)) -> proper#(X)
    proper#(g(X)) -> g#(proper(X))
    f#(ok(X1),ok(X2)) -> f#(X1,X2)
    g#(ok(X)) -> g#(X)
    top#(mark(X)) -> proper#(X)
    top#(mark(X)) -> top#(proper(X))
    top#(ok(X)) -> active#(X)
    top#(ok(X)) -> top#(active(X))
   TRS:
    active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
    active(f(X1,X2)) -> f(active(X1),X2)
    active(g(X)) -> g(active(X))
    f(mark(X1),X2) -> mark(f(X1,X2))
    g(mark(X)) -> mark(g(X))
    proper(f(X1,X2)) -> f(proper(X1),proper(X2))
    proper(g(X)) -> g(proper(X))
    f(ok(X1),ok(X2)) -> ok(f(X1,X2))
    g(ok(X)) -> ok(g(X))
    top(mark(X)) -> top(proper(X))
    top(ok(X)) -> top(active(X))
   graph:
    top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> top#(active(X))
    top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> active#(X)
    top#(ok(X)) -> top#(active(X)) ->
    top#(mark(X)) -> top#(proper(X))
    top#(ok(X)) -> top#(active(X)) -> top#(mark(X)) -> proper#(X)
    top#(ok(X)) -> active#(X) -> active#(g(X)) -> g#(active(X))
    top#(ok(X)) -> active#(X) -> active#(g(X)) -> active#(X)
    top#(ok(X)) -> active#(X) -> active#(f(X1,X2)) -> f#(active(X1),X2)
    top#(ok(X)) -> active#(X) -> active#(f(X1,X2)) -> active#(X1)
    top#(ok(X)) -> active#(X) ->
    active#(f(g(X),Y)) -> f#(X,f(g(X),Y))
    top#(mark(X)) -> top#(proper(X)) ->
    top#(ok(X)) -> top#(active(X))
    top#(mark(X)) -> top#(proper(X)) -> top#(ok(X)) -> active#(X)
    top#(mark(X)) -> top#(proper(X)) ->
    top#(mark(X)) -> top#(proper(X))
    top#(mark(X)) -> top#(proper(X)) -> top#(mark(X)) -> proper#(X)
    top#(mark(X)) -> proper#(X) -> proper#(g(X)) -> g#(proper(X))
    top#(mark(X)) -> proper#(X) -> proper#(g(X)) -> proper#(X)
    top#(mark(X)) -> proper#(X) ->
    proper#(f(X1,X2)) -> f#(proper(X1),proper(X2))
    top#(mark(X)) -> proper#(X) -> proper#(f(X1,X2)) -> proper#(X1)
    top#(mark(X)) -> proper#(X) -> proper#(f(X1,X2)) -> proper#(X2)
    proper#(f(X1,X2)) -> proper#(X2) ->
    proper#(g(X)) -> g#(proper(X))
    proper#(f(X1,X2)) -> proper#(X2) -> proper#(g(X)) -> proper#(X)
    proper#(f(X1,X2)) -> proper#(X2) ->
    proper#(f(X1,X2)) -> f#(proper(X1),proper(X2))
    proper#(f(X1,X2)) -> proper#(X2) ->
    proper#(f(X1,X2)) -> proper#(X1)
    proper#(f(X1,X2)) -> proper#(X2) ->
    proper#(f(X1,X2)) -> proper#(X2)
    proper#(f(X1,X2)) -> proper#(X1) ->
    proper#(g(X)) -> g#(proper(X))
    proper#(f(X1,X2)) -> proper#(X1) -> proper#(g(X)) -> proper#(X)
    proper#(f(X1,X2)) -> proper#(X1) ->
    proper#(f(X1,X2)) -> f#(proper(X1),proper(X2))
    proper#(f(X1,X2)) -> proper#(X1) ->
    proper#(f(X1,X2)) -> proper#(X1)
    proper#(f(X1,X2)) -> proper#(X1) ->
    proper#(f(X1,X2)) -> proper#(X2)
    proper#(f(X1,X2)) -> f#(proper(X1),proper(X2)) ->
    f#(ok(X1),ok(X2)) -> f#(X1,X2)
    proper#(f(X1,X2)) -> f#(proper(X1),proper(X2)) ->
    f#(mark(X1),X2) -> f#(X1,X2)
    proper#(g(X)) -> proper#(X) -> proper#(g(X)) -> g#(proper(X))
    proper#(g(X)) -> proper#(X) -> proper#(g(X)) -> proper#(X)
    proper#(g(X)) -> proper#(X) ->
    proper#(f(X1,X2)) -> f#(proper(X1),proper(X2))
    proper#(g(X)) -> proper#(X) -> proper#(f(X1,X2)) -> proper#(X1)
    proper#(g(X)) -> proper#(X) -> proper#(f(X1,X2)) -> proper#(X2)
    proper#(g(X)) -> g#(proper(X)) -> g#(ok(X)) -> g#(X)
    proper#(g(X)) -> g#(proper(X)) -> g#(mark(X)) -> g#(X)
    g#(ok(X)) -> g#(X) -> g#(ok(X)) -> g#(X)
    g#(ok(X)) -> g#(X) -> g#(mark(X)) -> g#(X)
    g#(mark(X)) -> g#(X) -> g#(ok(X)) -> g#(X)
    g#(mark(X)) -> g#(X) -> g#(mark(X)) -> g#(X)
    f#(ok(X1),ok(X2)) -> f#(X1,X2) -> f#(ok(X1),ok(X2)) -> f#(X1,X2)
    f#(ok(X1),ok(X2)) -> f#(X1,X2) -> f#(mark(X1),X2) -> f#(X1,X2)
    f#(mark(X1),X2) -> f#(X1,X2) -> f#(ok(X1),ok(X2)) -> f#(X1,X2)
    f#(mark(X1),X2) -> f#(X1,X2) ->
    f#(mark(X1),X2) -> f#(X1,X2)
    active#(f(g(X),Y)) -> f#(X,f(g(X),Y)) ->
    f#(ok(X1),ok(X2)) -> f#(X1,X2)
    active#(f(g(X),Y)) -> f#(X,f(g(X),Y)) ->
    f#(mark(X1),X2) -> f#(X1,X2)
    active#(f(X1,X2)) -> f#(active(X1),X2) ->
    f#(ok(X1),ok(X2)) -> f#(X1,X2)
    active#(f(X1,X2)) -> f#(active(X1),X2) ->
    f#(mark(X1),X2) -> f#(X1,X2)
    active#(f(X1,X2)) -> active#(X1) ->
    active#(g(X)) -> g#(active(X))
    active#(f(X1,X2)) -> active#(X1) -> active#(g(X)) -> active#(X)
    active#(f(X1,X2)) -> active#(X1) ->
    active#(f(X1,X2)) -> f#(active(X1),X2)
    active#(f(X1,X2)) -> active#(X1) ->
    active#(f(X1,X2)) -> active#(X1)
    active#(f(X1,X2)) -> active#(X1) ->
    active#(f(g(X),Y)) -> f#(X,f(g(X),Y))
    active#(g(X)) -> g#(active(X)) -> g#(ok(X)) -> g#(X)
    active#(g(X)) -> g#(active(X)) -> g#(mark(X)) -> g#(X)
    active#(g(X)) -> active#(X) -> active#(g(X)) -> g#(active(X))
    active#(g(X)) -> active#(X) -> active#(g(X)) -> active#(X)
    active#(g(X)) -> active#(X) ->
    active#(f(X1,X2)) -> f#(active(X1),X2)
    active#(g(X)) -> active#(X) -> active#(f(X1,X2)) -> active#(X1)
    active#(g(X)) -> active#(X) -> active#(f(g(X),Y)) -> f#(X,f(g(X),Y))
   SCC Processor:
    #sccs: 5
    #rules: 11
    #arcs: 61/324
    DPs:
     top#(ok(X)) -> top#(active(X))
     top#(mark(X)) -> top#(proper(X))
    TRS:
     active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
     active(f(X1,X2)) -> f(active(X1),X2)
     active(g(X)) -> g(active(X))
     f(mark(X1),X2) -> mark(f(X1,X2))
     g(mark(X)) -> mark(g(X))
     proper(f(X1,X2)) -> f(proper(X1),proper(X2))
     proper(g(X)) -> g(proper(X))
     f(ok(X1),ok(X2)) -> ok(f(X1,X2))
     g(ok(X)) -> ok(g(X))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    KBO Processor:
     argument filtering:
      pi(g) = [0]
      pi(f) = 0
      pi(active) = 0
      pi(mark) = [0]
      pi(proper) = 0
      pi(ok) = 0
      pi(top) = 0
      pi(top#) = 0
     weight function:
      w0 = 1
      w(top#) = w(top) = w(ok) = w(mark) = w(active) = w(g) = 1
      w(proper) = w(f) = 0
     precedence:
      top# ~ ok ~ active ~ g > top ~ proper ~ mark ~ f
     problem:
      DPs:
       top#(ok(X)) -> top#(active(X))
      TRS:
       active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
       active(f(X1,X2)) -> f(active(X1),X2)
       active(g(X)) -> g(active(X))
       f(mark(X1),X2) -> mark(f(X1,X2))
       g(mark(X)) -> mark(g(X))
       proper(f(X1,X2)) -> f(proper(X1),proper(X2))
       proper(g(X)) -> g(proper(X))
       f(ok(X1),ok(X2)) -> ok(f(X1,X2))
       g(ok(X)) -> ok(g(X))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     KBO Processor:
      argument filtering:
       pi(g) = 0
       pi(f) = 0
       pi(active) = []
       pi(mark) = []
       pi(proper) = 0
       pi(ok) = []
       pi(top) = []
       pi(top#) = 0
      weight function:
       w0 = 1
       w(top#) = w(top) = w(ok) = w(mark) = w(active) = w(g) = 1
       w(proper) = w(f) = 0
      precedence:
       top# ~ top ~ proper > ok > active > mark ~ f ~ g
      problem:
       DPs:
        
       TRS:
        active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
        active(f(X1,X2)) -> f(active(X1),X2)
        active(g(X)) -> g(active(X))
        f(mark(X1),X2) -> mark(f(X1,X2))
        g(mark(X)) -> mark(g(X))
        proper(f(X1,X2)) -> f(proper(X1),proper(X2))
        proper(g(X)) -> g(proper(X))
        f(ok(X1),ok(X2)) -> ok(f(X1,X2))
        g(ok(X)) -> ok(g(X))
        top(mark(X)) -> top(proper(X))
        top(ok(X)) -> top(active(X))
      Qed
    
    DPs:
     active#(f(X1,X2)) -> active#(X1)
     active#(g(X)) -> active#(X)
    TRS:
     active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
     active(f(X1,X2)) -> f(active(X1),X2)
     active(g(X)) -> g(active(X))
     f(mark(X1),X2) -> mark(f(X1,X2))
     g(mark(X)) -> mark(g(X))
     proper(f(X1,X2)) -> f(proper(X1),proper(X2))
     proper(g(X)) -> g(proper(X))
     f(ok(X1),ok(X2)) -> ok(f(X1,X2))
     g(ok(X)) -> ok(g(X))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Subterm Criterion Processor:
     simple projection:
      pi(active#) = 0
     problem:
      DPs:
       
      TRS:
       active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
       active(f(X1,X2)) -> f(active(X1),X2)
       active(g(X)) -> g(active(X))
       f(mark(X1),X2) -> mark(f(X1,X2))
       g(mark(X)) -> mark(g(X))
       proper(f(X1,X2)) -> f(proper(X1),proper(X2))
       proper(g(X)) -> g(proper(X))
       f(ok(X1),ok(X2)) -> ok(f(X1,X2))
       g(ok(X)) -> ok(g(X))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Qed
    
    DPs:
     proper#(f(X1,X2)) -> proper#(X2)
     proper#(f(X1,X2)) -> proper#(X1)
     proper#(g(X)) -> proper#(X)
    TRS:
     active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
     active(f(X1,X2)) -> f(active(X1),X2)
     active(g(X)) -> g(active(X))
     f(mark(X1),X2) -> mark(f(X1,X2))
     g(mark(X)) -> mark(g(X))
     proper(f(X1,X2)) -> f(proper(X1),proper(X2))
     proper(g(X)) -> g(proper(X))
     f(ok(X1),ok(X2)) -> ok(f(X1,X2))
     g(ok(X)) -> ok(g(X))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Subterm Criterion Processor:
     simple projection:
      pi(proper#) = 0
     problem:
      DPs:
       
      TRS:
       active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
       active(f(X1,X2)) -> f(active(X1),X2)
       active(g(X)) -> g(active(X))
       f(mark(X1),X2) -> mark(f(X1,X2))
       g(mark(X)) -> mark(g(X))
       proper(f(X1,X2)) -> f(proper(X1),proper(X2))
       proper(g(X)) -> g(proper(X))
       f(ok(X1),ok(X2)) -> ok(f(X1,X2))
       g(ok(X)) -> ok(g(X))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Qed
    
    DPs:
     g#(mark(X)) -> g#(X)
     g#(ok(X)) -> g#(X)
    TRS:
     active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
     active(f(X1,X2)) -> f(active(X1),X2)
     active(g(X)) -> g(active(X))
     f(mark(X1),X2) -> mark(f(X1,X2))
     g(mark(X)) -> mark(g(X))
     proper(f(X1,X2)) -> f(proper(X1),proper(X2))
     proper(g(X)) -> g(proper(X))
     f(ok(X1),ok(X2)) -> ok(f(X1,X2))
     g(ok(X)) -> ok(g(X))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Subterm Criterion Processor:
     simple projection:
      pi(g#) = 0
     problem:
      DPs:
       
      TRS:
       active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
       active(f(X1,X2)) -> f(active(X1),X2)
       active(g(X)) -> g(active(X))
       f(mark(X1),X2) -> mark(f(X1,X2))
       g(mark(X)) -> mark(g(X))
       proper(f(X1,X2)) -> f(proper(X1),proper(X2))
       proper(g(X)) -> g(proper(X))
       f(ok(X1),ok(X2)) -> ok(f(X1,X2))
       g(ok(X)) -> ok(g(X))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Qed
    
    DPs:
     f#(mark(X1),X2) -> f#(X1,X2)
     f#(ok(X1),ok(X2)) -> f#(X1,X2)
    TRS:
     active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
     active(f(X1,X2)) -> f(active(X1),X2)
     active(g(X)) -> g(active(X))
     f(mark(X1),X2) -> mark(f(X1,X2))
     g(mark(X)) -> mark(g(X))
     proper(f(X1,X2)) -> f(proper(X1),proper(X2))
     proper(g(X)) -> g(proper(X))
     f(ok(X1),ok(X2)) -> ok(f(X1,X2))
     g(ok(X)) -> ok(g(X))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Subterm Criterion Processor:
     simple projection:
      pi(f#) = 1
     problem:
      DPs:
       f#(mark(X1),X2) -> f#(X1,X2)
      TRS:
       active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
       active(f(X1,X2)) -> f(active(X1),X2)
       active(g(X)) -> g(active(X))
       f(mark(X1),X2) -> mark(f(X1,X2))
       g(mark(X)) -> mark(g(X))
       proper(f(X1,X2)) -> f(proper(X1),proper(X2))
       proper(g(X)) -> g(proper(X))
       f(ok(X1),ok(X2)) -> ok(f(X1,X2))
       g(ok(X)) -> ok(g(X))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Subterm Criterion Processor:
      simple projection:
       pi(f#) = 0
      problem:
       DPs:
        
       TRS:
        active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
        active(f(X1,X2)) -> f(active(X1),X2)
        active(g(X)) -> g(active(X))
        f(mark(X1),X2) -> mark(f(X1,X2))
        g(mark(X)) -> mark(g(X))
        proper(f(X1,X2)) -> f(proper(X1),proper(X2))
        proper(g(X)) -> g(proper(X))
        f(ok(X1),ok(X2)) -> ok(f(X1,X2))
        g(ok(X)) -> ok(g(X))
        top(mark(X)) -> top(proper(X))
        top(ok(X)) -> top(active(X))
      Qed