MAYBE

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

Proof:
 Matrix Interpretation Processor: dim=3
  
  interpretation:
                [1 0 1]  
   [mark](x0) = [0 1 0]x0
                [0 0 0]  ,
   
         [0]
   [b] = [0]
         [1],
   
         [0]
   [a] = [0]
         [0],
   
                  [1 0 1]  
   [active](x0) = [0 1 0]x0
                  [0 0 0]  ,
   
                 [1 0 1]     [1 0 0]     [0]
   [f](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0]
                 [0 0 0]     [0 1 0]     [1]
  orientation:
                    [2 1 1]    [1]    [1]                   
   active(f(X,X)) = [0 0 0]X + [0] >= [0] = mark(f(a(),b()))
                    [0 0 0]    [0]    [0]                   
   
                 [1]    [0]            
   active(b()) = [0] >= [0] = mark(a())
                 [0]    [0]            
   
                    [1 0 1]     [1 1 0]     [1]    [1 0 1]     [1 1 0]     [1]                         
   mark(f(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = active(f(mark(X1),X2))
                    [0 0 0]     [0 0 0]     [0]    [0 0 0]     [0 0 0]     [0]                         
   
               [0]    [0]              
   mark(a()) = [0] >= [0] = active(a())
               [0]    [0]              
   
               [1]    [1]              
   mark(b()) = [0] >= [0] = active(b())
               [0]    [0]              
   
                    [1 0 1]     [1 0 0]     [0]    [1 0 1]     [1 0 0]     [0]           
   f(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = f(X1,X2)
                    [0 0 0]     [0 1 0]     [1]    [0 0 0]     [0 1 0]     [1]           
   
                    [1 0 1]     [1 0 1]     [0]    [1 0 1]     [1 0 0]     [0]           
   f(X1,mark(X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = f(X1,X2)
                    [0 0 0]     [0 1 0]     [1]    [0 0 0]     [0 1 0]     [1]           
   
                      [1 0 1]     [1 0 0]     [0]    [1 0 1]     [1 0 0]     [0]           
   f(active(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = f(X1,X2)
                      [0 0 0]     [0 1 0]     [1]    [0 0 0]     [0 1 0]     [1]           
   
                      [1 0 1]     [1 0 1]     [0]    [1 0 1]     [1 0 0]     [0]           
   f(X1,active(X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = f(X1,X2)
                      [0 0 0]     [0 1 0]     [1]    [0 0 0]     [0 1 0]     [1]           
  problem:
   active(f(X,X)) -> mark(f(a(),b()))
   mark(f(X1,X2)) -> active(f(mark(X1),X2))
   mark(a()) -> active(a())
   mark(b()) -> active(b())
   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)
  Matrix Interpretation Processor: dim=3
   
   interpretation:
                 [1 0 1]     [0]
    [mark](x0) = [0 0 0]x0 + [1]
                 [0 0 1]     [0],
    
          [0]
    [b] = [0]
          [1],
    
          [0]
    [a] = [0]
          [0],
    
                   [1 1 0]  
    [active](x0) = [0 1 0]x0
                   [0 0 1]  ,
    
                  [1 0 0]     [1 0 0]     [0]
    [f](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [1]
                  [0 0 1]     [0 0 0]     [1]
   orientation:
                     [2 0 0]    [1]    [1]                   
    active(f(X,X)) = [0 0 0]X + [1] >= [1] = mark(f(a(),b()))
                     [0 0 1]    [1]    [1]                   
    
                     [1 0 1]     [1 0 0]     [1]    [1 0 1]     [1 0 0]     [1]                         
    mark(f(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = active(f(mark(X1),X2))
                     [0 0 1]     [0 0 0]     [1]    [0 0 1]     [0 0 0]     [1]                         
    
                [0]    [0]              
    mark(a()) = [1] >= [0] = active(a())
                [0]    [0]              
    
                [1]    [0]              
    mark(b()) = [1] >= [0] = active(b())
                [1]    [1]              
    
                     [1 0 1]     [1 0 0]     [0]    [1 0 0]     [1 0 0]     [0]           
    f(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = f(X1,X2)
                     [0 0 1]     [0 0 0]     [1]    [0 0 1]     [0 0 0]     [1]           
    
                     [1 0 0]     [1 0 1]     [0]    [1 0 0]     [1 0 0]     [0]           
    f(X1,mark(X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = f(X1,X2)
                     [0 0 1]     [0 0 0]     [1]    [0 0 1]     [0 0 0]     [1]           
    
                       [1 1 0]     [1 0 0]     [0]    [1 0 0]     [1 0 0]     [0]           
    f(active(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = f(X1,X2)
                       [0 0 1]     [0 0 0]     [1]    [0 0 1]     [0 0 0]     [1]           
    
                       [1 0 0]     [1 1 0]     [0]    [1 0 0]     [1 0 0]     [0]           
    f(X1,active(X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = f(X1,X2)
                       [0 0 1]     [0 0 0]     [1]    [0 0 1]     [0 0 0]     [1]           
   problem:
    active(f(X,X)) -> mark(f(a(),b()))
    mark(f(X1,X2)) -> active(f(mark(X1),X2))
    mark(a()) -> active(a())
    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)
   Matrix Interpretation Processor: dim=3
    
    interpretation:
                       [0]
     [mark](x0) = x0 + [0]
                       [1],
     
           [0]
     [b] = [0]
           [0],
     
           [0]
     [a] = [0]
           [0],
     
                         [0]
     [active](x0) = x0 + [0]
                         [1],
     
                   [1 0 0]     [1 0 1]     [0]
     [f](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [1]
                   [0 0 0]     [0 0 0]     [0]
    orientation:
                      [2 0 1]    [0]    [0]                   
     active(f(X,X)) = [0 0 0]X + [1] >= [1] = mark(f(a(),b()))
                      [0 0 0]    [1]    [1]                   
     
                      [1 0 0]     [1 0 1]     [0]    [1 0 0]     [1 0 1]     [0]                         
     mark(f(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = active(f(mark(X1),X2))
                      [0 0 0]     [0 0 0]     [1]    [0 0 0]     [0 0 0]     [1]                         
     
                 [0]    [0]              
     mark(a()) = [0] >= [0] = active(a())
                 [1]    [1]              
     
                      [1 0 0]     [1 0 1]     [0]    [1 0 0]     [1 0 1]     [0]           
     f(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = f(X1,X2)
                      [0 0 0]     [0 0 0]     [0]    [0 0 0]     [0 0 0]     [0]           
     
                      [1 0 0]     [1 0 1]     [1]    [1 0 0]     [1 0 1]     [0]           
     f(X1,mark(X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = f(X1,X2)
                      [0 0 0]     [0 0 0]     [0]    [0 0 0]     [0 0 0]     [0]           
     
                        [1 0 0]     [1 0 1]     [0]    [1 0 0]     [1 0 1]     [0]           
     f(active(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = f(X1,X2)
                        [0 0 0]     [0 0 0]     [0]    [0 0 0]     [0 0 0]     [0]           
     
                        [1 0 0]     [1 0 1]     [1]    [1 0 0]     [1 0 1]     [0]           
     f(X1,active(X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = f(X1,X2)
                        [0 0 0]     [0 0 0]     [0]    [0 0 0]     [0 0 0]     [0]           
    problem:
     active(f(X,X)) -> mark(f(a(),b()))
     mark(f(X1,X2)) -> active(f(mark(X1),X2))
     mark(a()) -> active(a())
     f(mark(X1),X2) -> f(X1,X2)
     f(active(X1),X2) -> f(X1,X2)
    DP Processor:
     DPs:
      active#(f(X,X)) -> f#(a(),b())
      active#(f(X,X)) -> mark#(f(a(),b()))
      mark#(f(X1,X2)) -> mark#(X1)
      mark#(f(X1,X2)) -> f#(mark(X1),X2)
      mark#(f(X1,X2)) -> active#(f(mark(X1),X2))
      mark#(a()) -> active#(a())
      f#(mark(X1),X2) -> f#(X1,X2)
      f#(active(X1),X2) -> f#(X1,X2)
     TRS:
      active(f(X,X)) -> mark(f(a(),b()))
      mark(f(X1,X2)) -> active(f(mark(X1),X2))
      mark(a()) -> active(a())
      f(mark(X1),X2) -> f(X1,X2)
      f(active(X1),X2) -> f(X1,X2)
     TDG Processor:
      DPs:
       active#(f(X,X)) -> f#(a(),b())
       active#(f(X,X)) -> mark#(f(a(),b()))
       mark#(f(X1,X2)) -> mark#(X1)
       mark#(f(X1,X2)) -> f#(mark(X1),X2)
       mark#(f(X1,X2)) -> active#(f(mark(X1),X2))
       mark#(a()) -> active#(a())
       f#(mark(X1),X2) -> f#(X1,X2)
       f#(active(X1),X2) -> f#(X1,X2)
      TRS:
       active(f(X,X)) -> mark(f(a(),b()))
       mark(f(X1,X2)) -> active(f(mark(X1),X2))
       mark(a()) -> active(a())
       f(mark(X1),X2) -> f(X1,X2)
       f(active(X1),X2) -> f(X1,X2)
      graph:
       mark#(a()) -> active#(a()) -> active#(f(X,X)) -> mark#(f(a(),b()))
       mark#(a()) -> active#(a()) -> active#(f(X,X)) -> f#(a(),b())
       mark#(f(X1,X2)) -> mark#(X1) -> mark#(a()) -> active#(a())
       mark#(f(X1,X2)) -> mark#(X1) ->
       mark#(f(X1,X2)) -> active#(f(mark(X1),X2))
       mark#(f(X1,X2)) -> mark#(X1) ->
       mark#(f(X1,X2)) -> f#(mark(X1),X2)
       mark#(f(X1,X2)) -> mark#(X1) ->
       mark#(f(X1,X2)) -> mark#(X1)
       mark#(f(X1,X2)) -> f#(mark(X1),X2) ->
       f#(active(X1),X2) -> f#(X1,X2)
       mark#(f(X1,X2)) -> f#(mark(X1),X2) ->
       f#(mark(X1),X2) -> f#(X1,X2)
       mark#(f(X1,X2)) -> active#(f(mark(X1),X2)) ->
       active#(f(X,X)) -> mark#(f(a(),b()))
       mark#(f(X1,X2)) -> active#(f(mark(X1),X2)) ->
       active#(f(X,X)) -> f#(a(),b())
       f#(mark(X1),X2) -> f#(X1,X2) -> f#(active(X1),X2) -> f#(X1,X2)
       f#(mark(X1),X2) -> f#(X1,X2) -> f#(mark(X1),X2) -> f#(X1,X2)
       f#(active(X1),X2) -> f#(X1,X2) ->
       f#(active(X1),X2) -> f#(X1,X2)
       f#(active(X1),X2) -> f#(X1,X2) ->
       f#(mark(X1),X2) -> f#(X1,X2)
       active#(f(X,X)) -> mark#(f(a(),b())) ->
       mark#(a()) -> active#(a())
       active#(f(X,X)) -> mark#(f(a(),b())) ->
       mark#(f(X1,X2)) -> active#(f(mark(X1),X2))
       active#(f(X,X)) -> mark#(f(a(),b())) ->
       mark#(f(X1,X2)) -> f#(mark(X1),X2)
       active#(f(X,X)) -> mark#(f(a(),b())) ->
       mark#(f(X1,X2)) -> mark#(X1)
       active#(f(X,X)) -> f#(a(),b()) ->
       f#(active(X1),X2) -> f#(X1,X2)
       active#(f(X,X)) -> f#(a(),b()) -> f#(mark(X1),X2) -> f#(X1,X2)
      SCC Processor:
       #sccs: 2
       #rules: 6
       #arcs: 20/64
       DPs:
        mark#(a()) -> active#(a())
        active#(f(X,X)) -> mark#(f(a(),b()))
        mark#(f(X1,X2)) -> mark#(X1)
        mark#(f(X1,X2)) -> active#(f(mark(X1),X2))
       TRS:
        active(f(X,X)) -> mark(f(a(),b()))
        mark(f(X1,X2)) -> active(f(mark(X1),X2))
        mark(a()) -> active(a())
        f(mark(X1),X2) -> f(X1,X2)
        f(active(X1),X2) -> f(X1,X2)
       EDG Processor:
        DPs:
         mark#(a()) -> active#(a())
         active#(f(X,X)) -> mark#(f(a(),b()))
         mark#(f(X1,X2)) -> mark#(X1)
         mark#(f(X1,X2)) -> active#(f(mark(X1),X2))
        TRS:
         active(f(X,X)) -> mark(f(a(),b()))
         mark(f(X1,X2)) -> active(f(mark(X1),X2))
         mark(a()) -> active(a())
         f(mark(X1),X2) -> f(X1,X2)
         f(active(X1),X2) -> f(X1,X2)
        graph:
         mark#(f(X1,X2)) -> mark#(X1) -> mark#(f(X1,X2)) -> mark#(X1)
         mark#(f(X1,X2)) -> mark#(X1) ->
         mark#(f(X1,X2)) -> active#(f(mark(X1),X2))
         mark#(f(X1,X2)) -> mark#(X1) ->
         mark#(a()) -> active#(a())
         mark#(f(X1,X2)) -> active#(f(mark(X1),X2)) ->
         active#(f(X,X)) -> mark#(f(a(),b()))
         active#(f(X,X)) -> mark#(f(a(),b())) ->
         mark#(f(X1,X2)) -> mark#(X1)
         active#(f(X,X)) -> mark#(f(a(),b())) -> mark#(f(X1,X2)) -> active#(f(mark(X1),X2))
        SCC Processor:
         #sccs: 1
         #rules: 3
         #arcs: 6/16
         DPs:
          mark#(f(X1,X2)) -> mark#(X1)
          mark#(f(X1,X2)) -> active#(f(mark(X1),X2))
          active#(f(X,X)) -> mark#(f(a(),b()))
         TRS:
          active(f(X,X)) -> mark(f(a(),b()))
          mark(f(X1,X2)) -> active(f(mark(X1),X2))
          mark(a()) -> active(a())
          f(mark(X1),X2) -> f(X1,X2)
          f(active(X1),X2) -> f(X1,X2)
         Arctic Interpretation Processor:
          dimension: 1
          interpretation:
           [mark#](x0) = x0 + 0,
           
           [active#](x0) = x0 + 0,
           
           [mark](x0) = x0 + 0,
           
           [b] = 2,
           
           [a] = 0,
           
           [active](x0) = x0 + 0,
           
           [f](x0, x1) = 1x0 + x1 + 2
          orientation:
           mark#(f(X1,X2)) = 1X1 + X2 + 2 >= X1 + 0 = mark#(X1)
           
           mark#(f(X1,X2)) = 1X1 + X2 + 2 >= 1X1 + X2 + 2 = active#(f(mark(X1),X2))
           
           active#(f(X,X)) = 1X + 2 >= 2 = mark#(f(a(),b()))
           
           active(f(X,X)) = 1X + 2 >= 2 = mark(f(a(),b()))
           
           mark(f(X1,X2)) = 1X1 + X2 + 2 >= 1X1 + X2 + 2 = active(f(mark(X1),X2))
           
           mark(a()) = 0 >= 0 = active(a())
           
           f(mark(X1),X2) = 1X1 + X2 + 2 >= 1X1 + X2 + 2 = f(X1,X2)
           
           f(active(X1),X2) = 1X1 + X2 + 2 >= 1X1 + X2 + 2 = f(X1,X2)
          problem:
           DPs:
            mark#(f(X1,X2)) -> active#(f(mark(X1),X2))
            active#(f(X,X)) -> mark#(f(a(),b()))
           TRS:
            active(f(X,X)) -> mark(f(a(),b()))
            mark(f(X1,X2)) -> active(f(mark(X1),X2))
            mark(a()) -> active(a())
            f(mark(X1),X2) -> f(X1,X2)
            f(active(X1),X2) -> f(X1,X2)
          Open
       
       DPs:
        f#(mark(X1),X2) -> f#(X1,X2)
        f#(active(X1),X2) -> f#(X1,X2)
       TRS:
        active(f(X,X)) -> mark(f(a(),b()))
        mark(f(X1,X2)) -> active(f(mark(X1),X2))
        mark(a()) -> active(a())
        f(mark(X1),X2) -> f(X1,X2)
        f(active(X1),X2) -> f(X1,X2)
       Subterm Criterion Processor:
        simple projection:
         pi(f#) = 0
        problem:
         DPs:
          
         TRS:
          active(f(X,X)) -> mark(f(a(),b()))
          mark(f(X1,X2)) -> active(f(mark(X1),X2))
          mark(a()) -> active(a())
          f(mark(X1),X2) -> f(X1,X2)
          f(active(X1),X2) -> f(X1,X2)
        Qed