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))
  Matrix Interpretation Processor: dim=1
   
   interpretation:
    [top#](x0) = 6x0 + 2,
    
    [proper#](x0) = 5x0,
    
    [g#](x0) = x0 + 2,
    
    [f#](x0, x1) = x0 + 4,
    
    [active#](x0) = 2x0 + 1,
    
    [top](x0) = x0 + 3,
    
    [ok](x0) = 3x0 + 4,
    
    [proper](x0) = x0,
    
    [mark](x0) = x0 + 1,
    
    [active](x0) = 3x0,
    
    [f](x0, x1) = 2x0 + x1 + 2,
    
    [g](x0) = 4x0 + 2
   orientation:
    active#(f(g(X),Y)) = 16X + 2Y + 13 >= X + 4 = f#(X,f(g(X),Y))
    
    active#(f(X1,X2)) = 4X1 + 2X2 + 5 >= 2X1 + 1 = active#(X1)
    
    active#(f(X1,X2)) = 4X1 + 2X2 + 5 >= 3X1 + 4 = f#(active(X1),X2)
    
    active#(g(X)) = 8X + 5 >= 2X + 1 = active#(X)
    
    active#(g(X)) = 8X + 5 >= 3X + 2 = g#(active(X))
    
    f#(mark(X1),X2) = X1 + 5 >= X1 + 4 = f#(X1,X2)
    
    g#(mark(X)) = X + 3 >= X + 2 = g#(X)
    
    proper#(f(X1,X2)) = 10X1 + 5X2 + 10 >= 5X2 = proper#(X2)
    
    proper#(f(X1,X2)) = 10X1 + 5X2 + 10 >= 5X1 = proper#(X1)
    
    proper#(f(X1,X2)) = 10X1 + 5X2 + 10 >= X1 + 4 = f#(proper(X1),proper(X2))
    
    proper#(g(X)) = 20X + 10 >= 5X = proper#(X)
    
    proper#(g(X)) = 20X + 10 >= X + 2 = g#(proper(X))
    
    f#(ok(X1),ok(X2)) = 3X1 + 8 >= X1 + 4 = f#(X1,X2)
    
    g#(ok(X)) = 3X + 6 >= X + 2 = g#(X)
    
    top#(mark(X)) = 6X + 8 >= 5X = proper#(X)
    
    top#(mark(X)) = 6X + 8 >= 6X + 2 = top#(proper(X))
    
    top#(ok(X)) = 18X + 26 >= 2X + 1 = active#(X)
    
    top#(ok(X)) = 18X + 26 >= 18X + 2 = top#(active(X))
    
    active(f(g(X),Y)) = 24X + 3Y + 18 >= 10X + Y + 9 = mark(f(X,f(g(X),Y)))
    
    active(f(X1,X2)) = 6X1 + 3X2 + 6 >= 6X1 + X2 + 2 = f(active(X1),X2)
    
    active(g(X)) = 12X + 6 >= 12X + 2 = g(active(X))
    
    f(mark(X1),X2) = 2X1 + X2 + 4 >= 2X1 + X2 + 3 = mark(f(X1,X2))
    
    g(mark(X)) = 4X + 6 >= 4X + 3 = mark(g(X))
    
    proper(f(X1,X2)) = 2X1 + X2 + 2 >= 2X1 + X2 + 2 = f(proper(X1),proper(X2))
    
    proper(g(X)) = 4X + 2 >= 4X + 2 = g(proper(X))
    
    f(ok(X1),ok(X2)) = 6X1 + 3X2 + 14 >= 6X1 + 3X2 + 10 = ok(f(X1,X2))
    
    g(ok(X)) = 12X + 18 >= 12X + 10 = ok(g(X))
    
    top(mark(X)) = X + 4 >= X + 3 = top(proper(X))
    
    top(ok(X)) = 3X + 7 >= 3X + 3 = 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