YES

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

Proof:
 DP Processor:
  DPs:
   active#(f(g(X),Y)) -> f#(X,f(g(X),Y))
   active#(f(g(X),Y)) -> mark#(f(X,f(g(X),Y)))
   mark#(f(X1,X2)) -> mark#(X1)
   mark#(f(X1,X2)) -> f#(mark(X1),X2)
   mark#(f(X1,X2)) -> active#(f(mark(X1),X2))
   mark#(g(X)) -> mark#(X)
   mark#(g(X)) -> g#(mark(X))
   mark#(g(X)) -> active#(g(mark(X)))
   f#(mark(X1),X2) -> f#(X1,X2)
   f#(X1,mark(X2)) -> f#(X1,X2)
   f#(active(X1),X2) -> f#(X1,X2)
   f#(X1,active(X2)) -> f#(X1,X2)
   g#(mark(X)) -> g#(X)
   g#(active(X)) -> g#(X)
  TRS:
   active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
   mark(f(X1,X2)) -> active(f(mark(X1),X2))
   mark(g(X)) -> active(g(mark(X)))
   f(mark(X1),X2) -> f(X1,X2)
   f(X1,mark(X2)) -> f(X1,X2)
   f(active(X1),X2) -> f(X1,X2)
   f(X1,active(X2)) -> f(X1,X2)
   g(mark(X)) -> g(X)
   g(active(X)) -> g(X)
  Matrix Interpretation Processor: dim=2
   
   interpretation:
    [g#](x0) = [1 0]x0 + [3],
    
    [mark#](x0) = [0 2]x0 + [1],
    
    [f#](x0, x1) = [1 0]x0 + [1 0]x1,
    
    [active#](x0) = [0 1]x0,
    
                 [2 2]     [1]
    [mark](x0) = [0 0]x0 + [0],
    
                   [1 1]     [1]
    [active](x0) = [0 0]x0 + [0],
    
                  [0 0]     [0 0]     [0]
    [f](x0, x1) = [1 1]x0 + [1 0]x1 + [1],
    
              [0 0]     [1]
    [g](x0) = [2 2]x0 + [2]
   orientation:
    active#(f(g(X),Y)) = [2 2]X + [1 0]Y + [4] >= [1 0]X = f#(X,f(g(X),Y))
    
    active#(f(g(X),Y)) = [2 2]X + [1 0]Y + [4] >= [2 2]X + [3] = mark#(f(X,f(g(X),Y)))
    
    mark#(f(X1,X2)) = [2 2]X1 + [2 0]X2 + [3] >= [0 2]X1 + [1] = mark#(X1)
    
    mark#(f(X1,X2)) = [2 2]X1 + [2 0]X2 + [3] >= [2 2]X1 + [1 0]X2 + [1] = f#(mark(X1),X2)
    
    mark#(f(X1,X2)) = [2 2]X1 + [2 0]X2 + [3] >= [2 2]X1 + [1 0]X2 + [2] = active#(f(mark(X1),X2))
    
    mark#(g(X)) = [4 4]X + [5] >= [0 2]X + [1] = mark#(X)
    
    mark#(g(X)) = [4 4]X + [5] >= [2 2]X + [4] = g#(mark(X))
    
    mark#(g(X)) = [4 4]X + [5] >= [4 4]X + [4] = active#(g(mark(X)))
    
    f#(mark(X1),X2) = [2 2]X1 + [1 0]X2 + [1] >= [1 0]X1 + [1 0]X2 = f#(X1,X2)
    
    f#(X1,mark(X2)) = [1 0]X1 + [2 2]X2 + [1] >= [1 0]X1 + [1 0]X2 = f#(X1,X2)
    
    f#(active(X1),X2) = [1 1]X1 + [1 0]X2 + [1] >= [1 0]X1 + [1 0]X2 = f#(X1,X2)
    
    f#(X1,active(X2)) = [1 0]X1 + [1 1]X2 + [1] >= [1 0]X1 + [1 0]X2 = f#(X1,X2)
    
    g#(mark(X)) = [2 2]X + [4] >= [1 0]X + [3] = g#(X)
    
    g#(active(X)) = [1 1]X + [4] >= [1 0]X + [3] = g#(X)
    
                        [2 2]    [1 0]    [5]    [2 2]    [3]                       
    active(f(g(X),Y)) = [0 0]X + [0 0]Y + [0] >= [0 0]X + [0] = mark(f(X,f(g(X),Y)))
    
                     [2 2]     [2 0]     [3]    [2 2]     [1 0]     [3]                         
    mark(f(X1,X2)) = [0 0]X1 + [0 0]X2 + [0] >= [0 0]X1 + [0 0]X2 + [0] = active(f(mark(X1),X2))
    
                 [4 4]    [7]    [4 4]    [6]                     
    mark(g(X)) = [0 0]X + [0] >= [0 0]X + [0] = active(g(mark(X)))
    
                     [0 0]     [0 0]     [0]    [0 0]     [0 0]     [0]           
    f(mark(X1),X2) = [2 2]X1 + [1 0]X2 + [2] >= [1 1]X1 + [1 0]X2 + [1] = f(X1,X2)
    
                     [0 0]     [0 0]     [0]    [0 0]     [0 0]     [0]           
    f(X1,mark(X2)) = [1 1]X1 + [2 2]X2 + [2] >= [1 1]X1 + [1 0]X2 + [1] = f(X1,X2)
    
                       [0 0]     [0 0]     [0]    [0 0]     [0 0]     [0]           
    f(active(X1),X2) = [1 1]X1 + [1 0]X2 + [2] >= [1 1]X1 + [1 0]X2 + [1] = f(X1,X2)
    
                       [0 0]     [0 0]     [0]    [0 0]     [0 0]     [0]           
    f(X1,active(X2)) = [1 1]X1 + [1 1]X2 + [2] >= [1 1]X1 + [1 0]X2 + [1] = f(X1,X2)
    
                 [0 0]    [1]    [0 0]    [1]       
    g(mark(X)) = [4 4]X + [4] >= [2 2]X + [2] = g(X)
    
                   [0 0]    [1]    [0 0]    [1]       
    g(active(X)) = [2 2]X + [4] >= [2 2]X + [2] = g(X)
   problem:
    DPs:
     
    TRS:
     active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
     mark(f(X1,X2)) -> active(f(mark(X1),X2))
     mark(g(X)) -> active(g(mark(X)))
     f(mark(X1),X2) -> f(X1,X2)
     f(X1,mark(X2)) -> f(X1,X2)
     f(active(X1),X2) -> f(X1,X2)
     f(X1,active(X2)) -> f(X1,X2)
     g(mark(X)) -> g(X)
     g(active(X)) -> g(X)
   Qed