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))
  CDG 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#(mark(X)) -> proper#(X)
    top#(ok(X)) -> top#(active(X)) -> top#(mark(X)) -> top#(proper(X))
    top#(ok(X)) -> active#(X) -> active#(f(g(X),Y)) -> f#(X,f(g(X),Y))
    top#(ok(X)) -> active#(X) -> active#(f(X1,X2)) -> active#(X1)
    top#(ok(X)) -> active#(X) -> active#(f(X1,X2)) -> f#(active(X1),X2)
    top#(ok(X)) -> active#(X) -> active#(g(X)) -> active#(X)
    top#(ok(X)) -> active#(X) -> active#(g(X)) -> g#(active(X))
    top#(mark(X)) -> proper#(X) -> proper#(f(X1,X2)) -> proper#(X2)
    top#(mark(X)) -> proper#(X) -> proper#(f(X1,X2)) -> proper#(X1)
    top#(mark(X)) -> proper#(X) ->
    proper#(f(X1,X2)) -> f#(proper(X1),proper(X2))
    top#(mark(X)) -> proper#(X) -> proper#(g(X)) -> proper#(X)
    top#(mark(X)) -> proper#(X) -> proper#(g(X)) -> g#(proper(X))
    proper#(f(X1,X2)) -> proper#(X2) ->
    proper#(f(X1,X2)) -> proper#(X2)
    proper#(f(X1,X2)) -> proper#(X2) ->
    proper#(f(X1,X2)) -> proper#(X1)
    proper#(f(X1,X2)) -> proper#(X2) ->
    proper#(f(X1,X2)) -> f#(proper(X1),proper(X2))
    proper#(f(X1,X2)) -> proper#(X2) -> proper#(g(X)) -> proper#(X)
    proper#(f(X1,X2)) -> proper#(X2) ->
    proper#(g(X)) -> g#(proper(X))
    proper#(f(X1,X2)) -> proper#(X1) ->
    proper#(f(X1,X2)) -> proper#(X2)
    proper#(f(X1,X2)) -> proper#(X1) ->
    proper#(f(X1,X2)) -> proper#(X1)
    proper#(f(X1,X2)) -> proper#(X1) ->
    proper#(f(X1,X2)) -> f#(proper(X1),proper(X2))
    proper#(f(X1,X2)) -> proper#(X1) -> proper#(g(X)) -> proper#(X)
    proper#(f(X1,X2)) -> proper#(X1) -> proper#(g(X)) -> g#(proper(X))
    proper#(g(X)) -> proper#(X) -> proper#(f(X1,X2)) -> proper#(X2)
    proper#(g(X)) -> proper#(X) -> proper#(f(X1,X2)) -> proper#(X1)
    proper#(g(X)) -> proper#(X) ->
    proper#(f(X1,X2)) -> f#(proper(X1),proper(X2))
    proper#(g(X)) -> proper#(X) -> proper#(g(X)) -> proper#(X)
    proper#(g(X)) -> proper#(X) -> proper#(g(X)) -> g#(proper(X))
    g#(ok(X)) -> g#(X) -> g#(mark(X)) -> g#(X)
    g#(ok(X)) -> g#(X) -> g#(ok(X)) -> g#(X)
    g#(mark(X)) -> g#(X) -> g#(mark(X)) -> g#(X)
    g#(mark(X)) -> g#(X) -> g#(ok(X)) -> g#(X)
    f#(ok(X1),ok(X2)) -> f#(X1,X2) -> f#(mark(X1),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#(mark(X1),X2) -> f#(X1,X2) ->
    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(g(X),Y)) -> f#(X,f(g(X),Y)) ->
    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#(f(g(X),Y)) -> f#(X,f(g(X),Y))
    active#(f(X1,X2)) -> active#(X1) ->
    active#(f(X1,X2)) -> active#(X1)
    active#(f(X1,X2)) -> active#(X1) ->
    active#(f(X1,X2)) -> f#(active(X1),X2)
    active#(f(X1,X2)) -> active#(X1) -> active#(g(X)) -> active#(X)
    active#(f(X1,X2)) -> active#(X1) ->
    active#(g(X)) -> g#(active(X))
    active#(g(X)) -> g#(active(X)) -> g#(mark(X)) -> g#(X)
    active#(g(X)) -> active#(X) -> active#(f(g(X),Y)) -> f#(X,f(g(X),Y))
    active#(g(X)) -> active#(X) -> active#(f(X1,X2)) -> active#(X1)
    active#(g(X)) -> active#(X) ->
    active#(f(X1,X2)) -> f#(active(X1),X2)
    active#(g(X)) -> active#(X) -> active#(g(X)) -> active#(X)
    active#(g(X)) -> active#(X) -> active#(g(X)) -> g#(active(X))
   SCC Processor:
    #sccs: 4
    #rules: 9
    #arcs: 49/324
    DPs:
     active#(g(X)) -> active#(X)
     active#(f(X1,X2)) -> active#(X1)
    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))
    Matrix Interpretation Processor:
     dimension: 1
     interpretation:
      [active#](x0) = x0 + 1,
      
      [top](x0) = 0,
      
      [ok](x0) = x0,
      
      [proper](x0) = x0 + 1,
      
      [mark](x0) = x0 + 1,
      
      [active](x0) = x0 + 1,
      
      [f](x0, x1) = x0 + 1,
      
      [g](x0) = x0 + 1
     orientation:
      active#(g(X)) = X + 2 >= X + 1 = active#(X)
      
      active#(f(X1,X2)) = X1 + 2 >= X1 + 1 = active#(X1)
      
      active(f(g(X),Y)) = X + 3 >= X + 2 = mark(f(X,f(g(X),Y)))
      
      active(f(X1,X2)) = X1 + 2 >= X1 + 2 = f(active(X1),X2)
      
      active(g(X)) = X + 2 >= X + 2 = g(active(X))
      
      f(mark(X1),X2) = X1 + 2 >= X1 + 2 = mark(f(X1,X2))
      
      g(mark(X)) = X + 2 >= X + 2 = mark(g(X))
      
      proper(f(X1,X2)) = X1 + 2 >= X1 + 2 = f(proper(X1),proper(X2))
      
      proper(g(X)) = X + 2 >= X + 2 = g(proper(X))
      
      f(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(f(X1,X2))
      
      g(ok(X)) = X + 1 >= X + 1 = ok(g(X))
      
      top(mark(X)) = 0 >= 0 = top(proper(X))
      
      top(ok(X)) = 0 >= 0 = top(active(X))
     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))
    Matrix Interpretation Processor:
     dimension: 1
     interpretation:
      [f#](x0, x1) = x0 + 1,
      
      [top](x0) = 0,
      
      [ok](x0) = x0 + 1,
      
      [proper](x0) = x0,
      
      [mark](x0) = x0 + 1,
      
      [active](x0) = x0 + 1,
      
      [f](x0, x1) = x0 + 1,
      
      [g](x0) = x0
     orientation:
      f#(mark(X1),X2) = X1 + 2 >= X1 + 1 = f#(X1,X2)
      
      f#(ok(X1),ok(X2)) = X1 + 2 >= X1 + 1 = f#(X1,X2)
      
      active(f(g(X),Y)) = X + 2 >= X + 2 = mark(f(X,f(g(X),Y)))
      
      active(f(X1,X2)) = X1 + 2 >= X1 + 2 = f(active(X1),X2)
      
      active(g(X)) = X + 1 >= X + 1 = g(active(X))
      
      f(mark(X1),X2) = X1 + 2 >= X1 + 2 = mark(f(X1,X2))
      
      g(mark(X)) = X + 1 >= X + 1 = mark(g(X))
      
      proper(f(X1,X2)) = X1 + 1 >= X1 + 1 = f(proper(X1),proper(X2))
      
      proper(g(X)) = X >= X = g(proper(X))
      
      f(ok(X1),ok(X2)) = X1 + 2 >= X1 + 2 = ok(f(X1,X2))
      
      g(ok(X)) = X + 1 >= X + 1 = ok(g(X))
      
      top(mark(X)) = 0 >= 0 = top(proper(X))
      
      top(ok(X)) = 0 >= 0 = top(active(X))
     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))
    Matrix Interpretation Processor:
     dimension: 1
     interpretation:
      [g#](x0) = x0 + 1,
      
      [top](x0) = 0,
      
      [ok](x0) = x0 + 1,
      
      [proper](x0) = 0,
      
      [mark](x0) = x0 + 1,
      
      [active](x0) = x0 + 1,
      
      [f](x0, x1) = x0,
      
      [g](x0) = x0
     orientation:
      g#(mark(X)) = X + 2 >= X + 1 = g#(X)
      
      g#(ok(X)) = X + 2 >= X + 1 = g#(X)
      
      active(f(g(X),Y)) = X + 1 >= X + 1 = mark(f(X,f(g(X),Y)))
      
      active(f(X1,X2)) = X1 + 1 >= X1 + 1 = f(active(X1),X2)
      
      active(g(X)) = X + 1 >= X + 1 = g(active(X))
      
      f(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(f(X1,X2))
      
      g(mark(X)) = X + 1 >= X + 1 = mark(g(X))
      
      proper(f(X1,X2)) = 0 >= 0 = f(proper(X1),proper(X2))
      
      proper(g(X)) = 0 >= 0 = g(proper(X))
      
      f(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(f(X1,X2))
      
      g(ok(X)) = X + 1 >= X + 1 = ok(g(X))
      
      top(mark(X)) = 0 >= 0 = top(proper(X))
      
      top(ok(X)) = 0 >= 0 = top(active(X))
     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#(g(X)) -> proper#(X)
     proper#(f(X1,X2)) -> proper#(X1)
     proper#(f(X1,X2)) -> proper#(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))
    Matrix Interpretation Processor:
     dimension: 1
     interpretation:
      [proper#](x0) = x0,
      
      [top](x0) = 0,
      
      [ok](x0) = 1,
      
      [proper](x0) = x0,
      
      [mark](x0) = 0,
      
      [active](x0) = x0,
      
      [f](x0, x1) = x0 + x1 + 1,
      
      [g](x0) = x0
     orientation:
      proper#(g(X)) = X >= X = proper#(X)
      
      proper#(f(X1,X2)) = X1 + X2 + 1 >= X1 = proper#(X1)
      
      proper#(f(X1,X2)) = X1 + X2 + 1 >= X2 = proper#(X2)
      
      active(f(g(X),Y)) = X + Y + 1 >= 0 = mark(f(X,f(g(X),Y)))
      
      active(f(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = f(active(X1),X2)
      
      active(g(X)) = X >= X = g(active(X))
      
      f(mark(X1),X2) = X2 + 1 >= 0 = mark(f(X1,X2))
      
      g(mark(X)) = 0 >= 0 = mark(g(X))
      
      proper(f(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = f(proper(X1),proper(X2))
      
      proper(g(X)) = X >= X = g(proper(X))
      
      f(ok(X1),ok(X2)) = 3 >= 1 = ok(f(X1,X2))
      
      g(ok(X)) = 1 >= 1 = ok(g(X))
      
      top(mark(X)) = 0 >= 0 = top(proper(X))
      
      top(ok(X)) = 0 >= 0 = top(active(X))
     problem:
      DPs:
       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))
     Matrix Interpretation Processor:
      dimension: 1
      interpretation:
       [proper#](x0) = x0 + 1,
       
       [top](x0) = 0,
       
       [ok](x0) = x0,
       
       [proper](x0) = x0,
       
       [mark](x0) = 0,
       
       [active](x0) = x0,
       
       [f](x0, x1) = 0,
       
       [g](x0) = x0 + 1
      orientation:
       proper#(g(X)) = X + 2 >= X + 1 = proper#(X)
       
       active(f(g(X),Y)) = 0 >= 0 = mark(f(X,f(g(X),Y)))
       
       active(f(X1,X2)) = 0 >= 0 = f(active(X1),X2)
       
       active(g(X)) = X + 1 >= X + 1 = g(active(X))
       
       f(mark(X1),X2) = 0 >= 0 = mark(f(X1,X2))
       
       g(mark(X)) = 1 >= 0 = mark(g(X))
       
       proper(f(X1,X2)) = 0 >= 0 = f(proper(X1),proper(X2))
       
       proper(g(X)) = X + 1 >= X + 1 = g(proper(X))
       
       f(ok(X1),ok(X2)) = 0 >= 0 = ok(f(X1,X2))
       
       g(ok(X)) = X + 1 >= X + 1 = ok(g(X))
       
       top(mark(X)) = 0 >= 0 = top(proper(X))
       
       top(ok(X)) = 0 >= 0 = top(active(X))
      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