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)
  TDG 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)
   graph:
    mark#(f(X1,X2)) -> mark#(X1) -> mark#(g(X)) -> mark#(X)
    mark#(f(X1,X2)) -> mark#(X1) ->
    mark#(f(X1,X2)) -> a__f#(mark(X1),X2)
    mark#(f(X1,X2)) -> mark#(X1) ->
    mark#(f(X1,X2)) -> mark#(X1)
    mark#(f(X1,X2)) -> a__f#(mark(X1),X2) ->
    a__f#(g(X),Y) -> a__f#(mark(X),f(g(X),Y))
    mark#(f(X1,X2)) -> a__f#(mark(X1),X2) -> a__f#(g(X),Y) -> mark#(X)
    mark#(g(X)) -> mark#(X) -> mark#(g(X)) -> mark#(X)
    mark#(g(X)) -> mark#(X) -> mark#(f(X1,X2)) -> a__f#(mark(X1),X2)
    mark#(g(X)) -> mark#(X) -> mark#(f(X1,X2)) -> mark#(X1)
    a__f#(g(X),Y) -> mark#(X) -> mark#(g(X)) -> mark#(X)
    a__f#(g(X),Y) -> mark#(X) -> mark#(f(X1,X2)) -> a__f#(mark(X1),X2)
    a__f#(g(X),Y) -> mark#(X) ->
    mark#(f(X1,X2)) -> mark#(X1)
    a__f#(g(X),Y) -> a__f#(mark(X),f(g(X),Y)) ->
    a__f#(g(X),Y) -> a__f#(mark(X),f(g(X),Y))
    a__f#(g(X),Y) -> a__f#(mark(X),f(g(X),Y)) -> a__f#(g(X),Y) -> mark#(X)
   SCC Processor:
    #sccs: 1
    #rules: 5
    #arcs: 13/25
    DPs:
     mark#(f(X1,X2)) -> mark#(X1)
     mark#(f(X1,X2)) -> a__f#(mark(X1),X2)
     a__f#(g(X),Y) -> mark#(X)
     mark#(g(X)) -> mark#(X)
     a__f#(g(X),Y) -> a__f#(mark(X),f(g(X),Y))
    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)
    Matrix Interpretation Processor:
     dimension: 1
     interpretation:
      [mark#](x0) = x0,
      
      [a__f#](x0, x1) = x0,
      
      [f](x0, x1) = x0,
      
      [mark](x0) = x0,
      
      [a__f](x0, x1) = x0,
      
      [g](x0) = x0 + 1
     orientation:
      mark#(f(X1,X2)) = X1 >= X1 = mark#(X1)
      
      mark#(f(X1,X2)) = X1 >= X1 = a__f#(mark(X1),X2)
      
      a__f#(g(X),Y) = X + 1 >= X = mark#(X)
      
      mark#(g(X)) = X + 1 >= X = mark#(X)
      
      a__f#(g(X),Y) = X + 1 >= X = a__f#(mark(X),f(g(X),Y))
      
      a__f(g(X),Y) = X + 1 >= X = a__f(mark(X),f(g(X),Y))
      
      mark(f(X1,X2)) = X1 >= X1 = a__f(mark(X1),X2)
      
      mark(g(X)) = X + 1 >= X + 1 = g(mark(X))
      
      a__f(X1,X2) = X1 >= X1 = f(X1,X2)
     problem:
      DPs:
       mark#(f(X1,X2)) -> mark#(X1)
       mark#(f(X1,X2)) -> a__f#(mark(X1),X2)
      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)
     Matrix Interpretation Processor:
      dimension: 1
      interpretation:
       [mark#](x0) = x0,
       
       [a__f#](x0, x1) = x0,
       
       [f](x0, x1) = x0 + 1,
       
       [mark](x0) = x0 + 1,
       
       [a__f](x0, x1) = x0 + 1,
       
       [g](x0) = x0 + 1
      orientation:
       mark#(f(X1,X2)) = X1 + 1 >= X1 = mark#(X1)
       
       mark#(f(X1,X2)) = X1 + 1 >= X1 + 1 = a__f#(mark(X1),X2)
       
       a__f(g(X),Y) = X + 2 >= X + 2 = a__f(mark(X),f(g(X),Y))
       
       mark(f(X1,X2)) = X1 + 2 >= X1 + 2 = a__f(mark(X1),X2)
       
       mark(g(X)) = X + 2 >= X + 2 = g(mark(X))
       
       a__f(X1,X2) = X1 + 1 >= X1 + 1 = f(X1,X2)
      problem:
       DPs:
        mark#(f(X1,X2)) -> a__f#(mark(X1),X2)
       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)
      Matrix Interpretation Processor:
       dimension: 1
       interpretation:
        [mark#](x0) = x0,
        
        [a__f#](x0, x1) = 0,
        
        [f](x0, x1) = 1,
        
        [mark](x0) = x0,
        
        [a__f](x0, x1) = 1,
        
        [g](x0) = 0
       orientation:
        mark#(f(X1,X2)) = 1 >= 0 = a__f#(mark(X1),X2)
        
        a__f(g(X),Y) = 1 >= 1 = a__f(mark(X),f(g(X),Y))
        
        mark(f(X1,X2)) = 1 >= 1 = a__f(mark(X1),X2)
        
        mark(g(X)) = 0 >= 0 = g(mark(X))
        
        a__f(X1,X2) = 1 >= 1 = f(X1,X2)
       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