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)
   Arctic Interpretation Processor:
    dimension: 1
    usable rules:
     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)
    interpretation:
     [mark#](x0) = x0,
     
     [a__f#](x0, x1) = 7x0 + x1,
     
     [f](x0, x1) = 7x0 + x1,
     
     [mark](x0) = x0,
     
     [a__f](x0, x1) = 7x0 + x1,
     
     [g](x0) = x0 + -7
    orientation:
     a__f#(g(X),Y) = 7X + Y + 0 >= X = mark#(X)
     
     a__f#(g(X),Y) = 7X + Y + 0 >= 7X + Y + 0 = a__f#(mark(X),f(g(X),Y))
     
     mark#(f(X1,X2)) = 7X1 + X2 >= X1 = mark#(X1)
     
     mark#(f(X1,X2)) = 7X1 + X2 >= 7X1 + X2 = a__f#(mark(X1),X2)
     
     mark#(g(X)) = X + -7 >= X = mark#(X)
     
     a__f(g(X),Y) = 7X + Y + 0 >= 7X + Y + 0 = a__f(mark(X),f(g(X),Y))
     
     mark(f(X1,X2)) = 7X1 + X2 >= 7X1 + X2 = a__f(mark(X1),X2)
     
     mark(g(X)) = X + -7 >= X + -7 = g(mark(X))
     
     a__f(X1,X2) = 7X1 + X2 >= 7X1 + X2 = f(X1,X2)
    problem:
     DPs:
      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)) -> 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)
    Restore Modifier:
     DPs:
      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)) -> 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)
     SCC Processor:
      #sccs: 2
      #rules: 2
      #arcs: 13/9
      DPs:
       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)
      Subterm Criterion Processor:
       simple projection:
        pi(mark#) = 0
       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
      
      DPs:
       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)
      Arctic Interpretation Processor:
       dimension: 1
       usable rules:
        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)
       interpretation:
        [a__f#](x0, x1) = 3x0 + x1,
        
        [f](x0, x1) = x0 + -4x1,
        
        [mark](x0) = x0,
        
        [a__f](x0, x1) = x0 + -4x1,
        
        [g](x0) = 3x0
       orientation:
        a__f#(g(X),Y) = 6X + Y >= 3X + -4Y = a__f#(mark(X),f(g(X),Y))
        
        a__f(g(X),Y) = 3X + -4Y >= X + -8Y = a__f(mark(X),f(g(X),Y))
        
        mark(f(X1,X2)) = X1 + -4X2 >= X1 + -4X2 = a__f(mark(X1),X2)
        
        mark(g(X)) = 3X >= 3X = g(mark(X))
        
        a__f(X1,X2) = X1 + -4X2 >= X1 + -4X2 = 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