YES

Problem:
 active(f(a(),X,X)) -> mark(f(X,b(),b()))
 active(b()) -> mark(a())
 mark(f(X1,X2,X3)) -> active(f(X1,mark(X2),X3))
 mark(a()) -> active(a())
 mark(b()) -> active(b())
 f(mark(X1),X2,X3) -> f(X1,X2,X3)
 f(X1,mark(X2),X3) -> f(X1,X2,X3)
 f(X1,X2,mark(X3)) -> f(X1,X2,X3)
 f(active(X1),X2,X3) -> f(X1,X2,X3)
 f(X1,active(X2),X3) -> f(X1,X2,X3)
 f(X1,X2,active(X3)) -> f(X1,X2,X3)

Proof:
 Matrix Interpretation Processor: dim=3
  
  interpretation:
                [1 0 0]     [0]
   [mark](x0) = [0 1 0]x0 + [1]
                [0 0 0]     [0],
   
         [0]
   [b] = [0]
         [0],
   
                  [1 0 0]     [0]
   [active](x0) = [0 1 0]x0 + [1]
                  [0 0 0]     [0],
   
                     [1 0 0]     [1 0 0]     [1 1 0]  
   [f](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [1 1 0]x2
                     [0 0 0]     [0 0 0]     [0 0 0]  ,
   
         [0]
   [a] = [0]
         [0]
  orientation:
                        [2 1 0]    [0]    [1 0 0]    [0]                     
   active(f(a(),X,X)) = [1 1 0]X + [1] >= [0 0 0]X + [1] = mark(f(X,b(),b()))
                        [0 0 0]    [0]    [0 0 0]    [0]                     
   
                 [0]    [0]            
   active(b()) = [1] >= [1] = mark(a())
                 [0]    [0]            
   
                       [1 0 0]     [1 0 0]     [1 1 0]     [0]    [1 0 0]     [1 0 0]     [1 1 0]     [0]                            
   mark(f(X1,X2,X3)) = [0 0 0]X1 + [0 0 0]X2 + [1 1 0]X3 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1 1 0]X3 + [1] = active(f(X1,mark(X2),X3))
                       [0 0 0]     [0 0 0]     [0 0 0]     [0]    [0 0 0]     [0 0 0]     [0 0 0]     [0]                            
   
               [0]    [0]              
   mark(a()) = [1] >= [1] = active(a())
               [0]    [0]              
   
               [0]    [0]              
   mark(b()) = [1] >= [1] = active(b())
               [0]    [0]              
   
                       [1 0 0]     [1 0 0]     [1 1 0]      [1 0 0]     [1 0 0]     [1 1 0]                
   f(mark(X1),X2,X3) = [0 0 0]X1 + [0 0 0]X2 + [1 1 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [1 1 0]X3 = f(X1,X2,X3)
                       [0 0 0]     [0 0 0]     [0 0 0]      [0 0 0]     [0 0 0]     [0 0 0]                
   
                       [1 0 0]     [1 0 0]     [1 1 0]      [1 0 0]     [1 0 0]     [1 1 0]                
   f(X1,mark(X2),X3) = [0 0 0]X1 + [0 0 0]X2 + [1 1 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [1 1 0]X3 = f(X1,X2,X3)
                       [0 0 0]     [0 0 0]     [0 0 0]      [0 0 0]     [0 0 0]     [0 0 0]                
   
                       [1 0 0]     [1 0 0]     [1 1 0]     [1]    [1 0 0]     [1 0 0]     [1 1 0]                
   f(X1,X2,mark(X3)) = [0 0 0]X1 + [0 0 0]X2 + [1 1 0]X3 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1 1 0]X3 = f(X1,X2,X3)
                       [0 0 0]     [0 0 0]     [0 0 0]     [0]    [0 0 0]     [0 0 0]     [0 0 0]                
   
                         [1 0 0]     [1 0 0]     [1 1 0]      [1 0 0]     [1 0 0]     [1 1 0]                
   f(active(X1),X2,X3) = [0 0 0]X1 + [0 0 0]X2 + [1 1 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [1 1 0]X3 = f(X1,X2,X3)
                         [0 0 0]     [0 0 0]     [0 0 0]      [0 0 0]     [0 0 0]     [0 0 0]                
   
                         [1 0 0]     [1 0 0]     [1 1 0]      [1 0 0]     [1 0 0]     [1 1 0]                
   f(X1,active(X2),X3) = [0 0 0]X1 + [0 0 0]X2 + [1 1 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [1 1 0]X3 = f(X1,X2,X3)
                         [0 0 0]     [0 0 0]     [0 0 0]      [0 0 0]     [0 0 0]     [0 0 0]                
   
                         [1 0 0]     [1 0 0]     [1 1 0]     [1]    [1 0 0]     [1 0 0]     [1 1 0]                
   f(X1,X2,active(X3)) = [0 0 0]X1 + [0 0 0]X2 + [1 1 0]X3 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1 1 0]X3 = f(X1,X2,X3)
                         [0 0 0]     [0 0 0]     [0 0 0]     [0]    [0 0 0]     [0 0 0]     [0 0 0]                
  problem:
   active(f(a(),X,X)) -> mark(f(X,b(),b()))
   active(b()) -> mark(a())
   mark(f(X1,X2,X3)) -> active(f(X1,mark(X2),X3))
   mark(a()) -> active(a())
   mark(b()) -> active(b())
   f(mark(X1),X2,X3) -> f(X1,X2,X3)
   f(X1,mark(X2),X3) -> f(X1,X2,X3)
   f(active(X1),X2,X3) -> f(X1,X2,X3)
   f(X1,active(X2),X3) -> f(X1,X2,X3)
  Matrix Interpretation Processor: dim=3
   
   interpretation:
                 [1 0 1]     [0]
    [mark](x0) = [0 1 0]x0 + [1]
                 [1 0 0]     [0],
    
          [0]
    [b] = [0]
          [0],
    
                   [1 1 0]     [0]
    [active](x0) = [0 0 0]x0 + [1]
                   [0 0 1]     [0],
    
                      [1 1 0]     [1 0 1]     [1 1 0]  
    [f](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [0 0 0]x2
                      [0 0 0]     [1 0 0]     [0 1 0]  ,
    
          [0]
    [a] = [0]
          [0]
   orientation:
                         [2 1 1]    [0]    [1 1 0]    [0]                     
    active(f(a(),X,X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = mark(f(X,b(),b()))
                         [1 1 0]    [0]    [1 1 0]    [0]                     
    
                  [0]    [0]            
    active(b()) = [1] >= [1] = mark(a())
                  [0]    [0]            
    
                        [1 1 0]     [2 0 1]     [1 2 0]     [0]    [1 1 0]     [2 0 1]     [1 1 0]     [0]                            
    mark(f(X1,X2,X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [1] = active(f(X1,mark(X2),X3))
                        [1 1 0]     [1 0 1]     [1 1 0]     [0]    [0 0 0]     [1 0 1]     [0 1 0]     [0]                            
    
                [0]    [0]              
    mark(a()) = [1] >= [1] = active(a())
                [0]    [0]              
    
                [0]    [0]              
    mark(b()) = [1] >= [1] = active(b())
                [0]    [0]              
    
                        [1 1 1]     [1 0 1]     [1 1 0]     [1]    [1 1 0]     [1 0 1]     [1 1 0]                
    f(mark(X1),X2,X3) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = f(X1,X2,X3)
                        [0 0 0]     [1 0 0]     [0 1 0]     [0]    [0 0 0]     [1 0 0]     [0 1 0]                
    
                        [1 1 0]     [2 0 1]     [1 1 0]      [1 1 0]     [1 0 1]     [1 1 0]                
    f(X1,mark(X2),X3) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = f(X1,X2,X3)
                        [0 0 0]     [1 0 1]     [0 1 0]      [0 0 0]     [1 0 0]     [0 1 0]                
    
                          [1 1 0]     [1 0 1]     [1 1 0]     [1]    [1 1 0]     [1 0 1]     [1 1 0]                
    f(active(X1),X2,X3) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = f(X1,X2,X3)
                          [0 0 0]     [1 0 0]     [0 1 0]     [0]    [0 0 0]     [1 0 0]     [0 1 0]                
    
                          [1 1 0]     [1 1 1]     [1 1 0]      [1 1 0]     [1 0 1]     [1 1 0]                
    f(X1,active(X2),X3) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = f(X1,X2,X3)
                          [0 0 0]     [1 1 0]     [0 1 0]      [0 0 0]     [1 0 0]     [0 1 0]                
   problem:
    active(f(a(),X,X)) -> mark(f(X,b(),b()))
    active(b()) -> mark(a())
    mark(f(X1,X2,X3)) -> active(f(X1,mark(X2),X3))
    mark(a()) -> active(a())
    mark(b()) -> active(b())
    f(X1,mark(X2),X3) -> f(X1,X2,X3)
    f(X1,active(X2),X3) -> f(X1,X2,X3)
   DP Processor:
    DPs:
     active#(f(a(),X,X)) -> f#(X,b(),b())
     active#(f(a(),X,X)) -> mark#(f(X,b(),b()))
     active#(b()) -> mark#(a())
     mark#(f(X1,X2,X3)) -> mark#(X2)
     mark#(f(X1,X2,X3)) -> f#(X1,mark(X2),X3)
     mark#(f(X1,X2,X3)) -> active#(f(X1,mark(X2),X3))
     mark#(a()) -> active#(a())
     mark#(b()) -> active#(b())
     f#(X1,mark(X2),X3) -> f#(X1,X2,X3)
     f#(X1,active(X2),X3) -> f#(X1,X2,X3)
    TRS:
     active(f(a(),X,X)) -> mark(f(X,b(),b()))
     active(b()) -> mark(a())
     mark(f(X1,X2,X3)) -> active(f(X1,mark(X2),X3))
     mark(a()) -> active(a())
     mark(b()) -> active(b())
     f(X1,mark(X2),X3) -> f(X1,X2,X3)
     f(X1,active(X2),X3) -> f(X1,X2,X3)
    TDG Processor:
     DPs:
      active#(f(a(),X,X)) -> f#(X,b(),b())
      active#(f(a(),X,X)) -> mark#(f(X,b(),b()))
      active#(b()) -> mark#(a())
      mark#(f(X1,X2,X3)) -> mark#(X2)
      mark#(f(X1,X2,X3)) -> f#(X1,mark(X2),X3)
      mark#(f(X1,X2,X3)) -> active#(f(X1,mark(X2),X3))
      mark#(a()) -> active#(a())
      mark#(b()) -> active#(b())
      f#(X1,mark(X2),X3) -> f#(X1,X2,X3)
      f#(X1,active(X2),X3) -> f#(X1,X2,X3)
     TRS:
      active(f(a(),X,X)) -> mark(f(X,b(),b()))
      active(b()) -> mark(a())
      mark(f(X1,X2,X3)) -> active(f(X1,mark(X2),X3))
      mark(a()) -> active(a())
      mark(b()) -> active(b())
      f(X1,mark(X2),X3) -> f(X1,X2,X3)
      f(X1,active(X2),X3) -> f(X1,X2,X3)
     graph:
      mark#(b()) -> active#(b()) -> active#(b()) -> mark#(a())
      mark#(b()) -> active#(b()) ->
      active#(f(a(),X,X)) -> mark#(f(X,b(),b()))
      mark#(b()) -> active#(b()) ->
      active#(f(a(),X,X)) -> f#(X,b(),b())
      mark#(f(X1,X2,X3)) -> mark#(X2) -> mark#(b()) -> active#(b())
      mark#(f(X1,X2,X3)) -> mark#(X2) -> mark#(a()) -> active#(a())
      mark#(f(X1,X2,X3)) -> mark#(X2) ->
      mark#(f(X1,X2,X3)) -> active#(f(X1,mark(X2),X3))
      mark#(f(X1,X2,X3)) -> mark#(X2) ->
      mark#(f(X1,X2,X3)) -> f#(X1,mark(X2),X3)
      mark#(f(X1,X2,X3)) -> mark#(X2) ->
      mark#(f(X1,X2,X3)) -> mark#(X2)
      mark#(f(X1,X2,X3)) -> f#(X1,mark(X2),X3) ->
      f#(X1,active(X2),X3) -> f#(X1,X2,X3)
      mark#(f(X1,X2,X3)) -> f#(X1,mark(X2),X3) ->
      f#(X1,mark(X2),X3) -> f#(X1,X2,X3)
      mark#(f(X1,X2,X3)) -> active#(f(X1,mark(X2),X3)) ->
      active#(b()) -> mark#(a())
      mark#(f(X1,X2,X3)) -> active#(f(X1,mark(X2),X3)) ->
      active#(f(a(),X,X)) -> mark#(f(X,b(),b()))
      mark#(f(X1,X2,X3)) -> active#(f(X1,mark(X2),X3)) ->
      active#(f(a(),X,X)) -> f#(X,b(),b())
      mark#(a()) -> active#(a()) -> active#(b()) -> mark#(a())
      mark#(a()) -> active#(a()) ->
      active#(f(a(),X,X)) -> mark#(f(X,b(),b()))
      mark#(a()) -> active#(a()) ->
      active#(f(a(),X,X)) -> f#(X,b(),b())
      f#(X1,mark(X2),X3) -> f#(X1,X2,X3) ->
      f#(X1,active(X2),X3) -> f#(X1,X2,X3)
      f#(X1,mark(X2),X3) -> f#(X1,X2,X3) ->
      f#(X1,mark(X2),X3) -> f#(X1,X2,X3)
      f#(X1,active(X2),X3) -> f#(X1,X2,X3) ->
      f#(X1,active(X2),X3) -> f#(X1,X2,X3)
      f#(X1,active(X2),X3) -> f#(X1,X2,X3) ->
      f#(X1,mark(X2),X3) -> f#(X1,X2,X3)
      active#(b()) -> mark#(a()) -> mark#(b()) -> active#(b())
      active#(b()) -> mark#(a()) -> mark#(a()) -> active#(a())
      active#(b()) -> mark#(a()) ->
      mark#(f(X1,X2,X3)) -> active#(f(X1,mark(X2),X3))
      active#(b()) -> mark#(a()) ->
      mark#(f(X1,X2,X3)) -> f#(X1,mark(X2),X3)
      active#(b()) -> mark#(a()) ->
      mark#(f(X1,X2,X3)) -> mark#(X2)
      active#(f(a(),X,X)) -> mark#(f(X,b(),b())) ->
      mark#(b()) -> active#(b())
      active#(f(a(),X,X)) -> mark#(f(X,b(),b())) ->
      mark#(a()) -> active#(a())
      active#(f(a(),X,X)) -> mark#(f(X,b(),b())) ->
      mark#(f(X1,X2,X3)) -> active#(f(X1,mark(X2),X3))
      active#(f(a(),X,X)) -> mark#(f(X,b(),b())) ->
      mark#(f(X1,X2,X3)) -> f#(X1,mark(X2),X3)
      active#(f(a(),X,X)) -> mark#(f(X,b(),b())) ->
      mark#(f(X1,X2,X3)) -> mark#(X2)
      active#(f(a(),X,X)) -> f#(X,b(),b()) ->
      f#(X1,active(X2),X3) -> f#(X1,X2,X3)
      active#(f(a(),X,X)) -> f#(X,b(),b()) -> f#(X1,mark(X2),X3) -> f#(X1,X2,X3)
     SCC Processor:
      #sccs: 2
      #rules: 8
      #arcs: 32/100
      DPs:
       mark#(b()) -> active#(b())
       active#(f(a(),X,X)) -> mark#(f(X,b(),b()))
       mark#(f(X1,X2,X3)) -> mark#(X2)
       mark#(f(X1,X2,X3)) -> active#(f(X1,mark(X2),X3))
       active#(b()) -> mark#(a())
       mark#(a()) -> active#(a())
      TRS:
       active(f(a(),X,X)) -> mark(f(X,b(),b()))
       active(b()) -> mark(a())
       mark(f(X1,X2,X3)) -> active(f(X1,mark(X2),X3))
       mark(a()) -> active(a())
       mark(b()) -> active(b())
       f(X1,mark(X2),X3) -> f(X1,X2,X3)
       f(X1,active(X2),X3) -> f(X1,X2,X3)
      EDG Processor:
       DPs:
        mark#(b()) -> active#(b())
        active#(f(a(),X,X)) -> mark#(f(X,b(),b()))
        mark#(f(X1,X2,X3)) -> mark#(X2)
        mark#(f(X1,X2,X3)) -> active#(f(X1,mark(X2),X3))
        active#(b()) -> mark#(a())
        mark#(a()) -> active#(a())
       TRS:
        active(f(a(),X,X)) -> mark(f(X,b(),b()))
        active(b()) -> mark(a())
        mark(f(X1,X2,X3)) -> active(f(X1,mark(X2),X3))
        mark(a()) -> active(a())
        mark(b()) -> active(b())
        f(X1,mark(X2),X3) -> f(X1,X2,X3)
        f(X1,active(X2),X3) -> f(X1,X2,X3)
       graph:
        mark#(b()) -> active#(b()) -> active#(b()) -> mark#(a())
        mark#(f(X1,X2,X3)) -> mark#(X2) ->
        mark#(f(X1,X2,X3)) -> mark#(X2)
        mark#(f(X1,X2,X3)) -> mark#(X2) ->
        mark#(f(X1,X2,X3)) -> active#(f(X1,mark(X2),X3))
        mark#(f(X1,X2,X3)) -> mark#(X2) ->
        mark#(a()) -> active#(a())
        mark#(f(X1,X2,X3)) -> mark#(X2) ->
        mark#(b()) -> active#(b())
        mark#(f(X1,X2,X3)) -> active#(f(X1,mark(X2),X3)) ->
        active#(f(a(),X,X)) -> mark#(f(X,b(),b()))
        active#(b()) -> mark#(a()) ->
        mark#(a()) -> active#(a())
        active#(f(a(),X,X)) -> mark#(f(X,b(),b())) ->
        mark#(f(X1,X2,X3)) -> mark#(X2)
        active#(f(a(),X,X)) -> mark#(f(X,b(),b())) ->
        mark#(f(X1,X2,X3)) -> active#(f(X1,mark(X2),X3))
       SCC Processor:
        #sccs: 1
        #rules: 3
        #arcs: 9/36
        DPs:
         mark#(f(X1,X2,X3)) -> mark#(X2)
         mark#(f(X1,X2,X3)) -> active#(f(X1,mark(X2),X3))
         active#(f(a(),X,X)) -> mark#(f(X,b(),b()))
        TRS:
         active(f(a(),X,X)) -> mark(f(X,b(),b()))
         active(b()) -> mark(a())
         mark(f(X1,X2,X3)) -> active(f(X1,mark(X2),X3))
         mark(a()) -> active(a())
         mark(b()) -> active(b())
         f(X1,mark(X2),X3) -> f(X1,X2,X3)
         f(X1,active(X2),X3) -> f(X1,X2,X3)
        Semantic Labeling Processor:
         dimension: 1
         usable rules:
          f(X1,mark(X2),X3) -> f(X1,X2,X3)
          f(X1,active(X2),X3) -> f(X1,X2,X3)
         interpretation:
          [mark](x0) = 0,
          
          [b] = 0,
          
          [active](x0) = 0,
          
          [f](x0, x1, x2) = x0 + x2,
          
          [a] = 1
          labeled:
           f
           active#
           b
          usable (for model):
           mark#
           f
           active#
           mark
           a
           b
           active
          argument filtering:
           pi(a) = []
           pi(f) = [1]
           pi(active) = [0]
           pi(b) = []
           pi(mark) = 0
           pi(active#) = []
           pi(mark#) = 0
         precedence:
          active > f > mark# ~ active# ~ mark ~ b ~ a
         problem:
          DPs:
           
          TRS:
           active(f(a(),X,X)) -> mark(f(X,b(),b()))
           active(b()) -> mark(a())
           mark(f(X1,X2,X3)) -> active(f(X1,mark(X2),X3))
           mark(a()) -> active(a())
           mark(b()) -> active(b())
           f(X1,mark(X2),X3) -> f(X1,X2,X3)
           f(X1,active(X2),X3) -> f(X1,X2,X3)
         Qed
      
      DPs:
       f#(X1,mark(X2),X3) -> f#(X1,X2,X3)
       f#(X1,active(X2),X3) -> f#(X1,X2,X3)
      TRS:
       active(f(a(),X,X)) -> mark(f(X,b(),b()))
       active(b()) -> mark(a())
       mark(f(X1,X2,X3)) -> active(f(X1,mark(X2),X3))
       mark(a()) -> active(a())
       mark(b()) -> active(b())
       f(X1,mark(X2),X3) -> f(X1,X2,X3)
       f(X1,active(X2),X3) -> f(X1,X2,X3)
      Subterm Criterion Processor:
       simple projection:
        pi(f#) = 1
       problem:
        DPs:
         
        TRS:
         active(f(a(),X,X)) -> mark(f(X,b(),b()))
         active(b()) -> mark(a())
         mark(f(X1,X2,X3)) -> active(f(X1,mark(X2),X3))
         mark(a()) -> active(a())
         mark(b()) -> active(b())
         f(X1,mark(X2),X3) -> f(X1,X2,X3)
         f(X1,active(X2),X3) -> f(X1,X2,X3)
       Qed