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:
 DP Processor:
  DPs:
   active#(f(X,X)) -> f#(a(),b())
   active#(f(X,X)) -> mark#(f(a(),b()))
   active#(b()) -> mark#(a())
   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())
   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)
  TRS:
   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)
  TDG Processor:
   DPs:
    active#(f(X,X)) -> f#(a(),b())
    active#(f(X,X)) -> mark#(f(a(),b()))
    active#(b()) -> mark#(a())
    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())
    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)
   TRS:
    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)
   graph:
    mark#(b()) -> active#(b()) -> active#(b()) -> mark#(a())
    mark#(b()) -> active#(b()) -> active#(f(X,X)) -> mark#(f(a(),b()))
    mark#(b()) -> active#(b()) -> active#(f(X,X)) -> f#(a(),b())
    mark#(a()) -> active#(a()) -> active#(b()) -> mark#(a())
    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#(b()) -> active#(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#(X1,active(X2)) -> f#(X1,X2)
    mark#(f(X1,X2)) -> f#(mark(X1),X2) ->
    f#(active(X1),X2) -> f#(X1,X2)
    mark#(f(X1,X2)) -> f#(mark(X1),X2) ->
    f#(X1,mark(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#(b()) -> mark#(a())
    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#(X1,active(X2)) -> f#(X1,X2)
    f#(mark(X1),X2) -> f#(X1,X2) -> f#(active(X1),X2) -> f#(X1,X2)
    f#(mark(X1),X2) -> f#(X1,X2) -> f#(X1,mark(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#(X1,active(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#(X1,mark(X2)) -> f#(X1,X2)
    f#(active(X1),X2) -> f#(X1,X2) -> f#(mark(X1),X2) -> f#(X1,X2)
    f#(X1,mark(X2)) -> f#(X1,X2) -> f#(X1,active(X2)) -> f#(X1,X2)
    f#(X1,mark(X2)) -> f#(X1,X2) -> f#(active(X1),X2) -> f#(X1,X2)
    f#(X1,mark(X2)) -> f#(X1,X2) -> f#(X1,mark(X2)) -> f#(X1,X2)
    f#(X1,mark(X2)) -> f#(X1,X2) -> f#(mark(X1),X2) -> f#(X1,X2)
    f#(X1,active(X2)) -> f#(X1,X2) -> f#(X1,active(X2)) -> f#(X1,X2)
    f#(X1,active(X2)) -> f#(X1,X2) -> f#(active(X1),X2) -> f#(X1,X2)
    f#(X1,active(X2)) -> f#(X1,X2) -> f#(X1,mark(X2)) -> f#(X1,X2)
    f#(X1,active(X2)) -> f#(X1,X2) -> f#(mark(X1),X2) -> f#(X1,X2)
    active#(b()) -> mark#(a()) -> mark#(b()) -> active#(b())
    active#(b()) -> mark#(a()) -> mark#(a()) -> active#(a())
    active#(b()) -> mark#(a()) ->
    mark#(f(X1,X2)) -> active#(f(mark(X1),X2))
    active#(b()) -> mark#(a()) -> mark#(f(X1,X2)) -> f#(mark(X1),X2)
    active#(b()) -> mark#(a()) -> mark#(f(X1,X2)) -> mark#(X1)
    active#(f(X,X)) -> mark#(f(a(),b())) ->
    mark#(b()) -> active#(b())
    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#(X1,active(X2)) -> f#(X1,X2)
    active#(f(X,X)) -> f#(a(),b()) -> f#(active(X1),X2) -> f#(X1,X2)
    active#(f(X,X)) -> f#(a(),b()) -> f#(X1,mark(X2)) -> f#(X1,X2)
    active#(f(X,X)) -> f#(a(),b()) -> f#(mark(X1),X2) -> f#(X1,X2)
   SCC Processor:
    #sccs: 2
    #rules: 10
    #arcs: 48/144
    DPs:
     mark#(b()) -> active#(b())
     active#(f(X,X)) -> mark#(f(a(),b()))
     mark#(f(X1,X2)) -> mark#(X1)
     mark#(f(X1,X2)) -> active#(f(mark(X1),X2))
     active#(b()) -> mark#(a())
     mark#(a()) -> active#(a())
    TRS:
     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)
    EDG Processor:
     DPs:
      mark#(b()) -> active#(b())
      active#(f(X,X)) -> mark#(f(a(),b()))
      mark#(f(X1,X2)) -> mark#(X1)
      mark#(f(X1,X2)) -> active#(f(mark(X1),X2))
      active#(b()) -> mark#(a())
      mark#(a()) -> active#(a())
     TRS:
      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)
     graph:
      mark#(b()) -> active#(b()) -> active#(b()) -> mark#(a())
      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)) -> mark#(X1) ->
      mark#(b()) -> active#(b())
      mark#(f(X1,X2)) -> active#(f(mark(X1),X2)) ->
      active#(f(X,X)) -> mark#(f(a(),b()))
      active#(b()) -> mark#(a()) -> mark#(a()) -> active#(a())
      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: 9/36
      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()))
       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)
      Arctic Interpretation Processor:
       dimension: 1
       interpretation:
        [mark#](x0) = x0 + 0,
        
        [active#](x0) = x0 + 0,
        
        [mark](x0) = x0 + 1,
        
        [b] = 4,
        
        [a] = 0,
        
        [active](x0) = x0 + 1,
        
        [f](x0, x1) = 1x0 + 2
       orientation:
        mark#(f(X1,X2)) = 1X1 + 2 >= X1 + 0 = mark#(X1)
        
        mark#(f(X1,X2)) = 1X1 + 2 >= 1X1 + 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()))
        
        active(b()) = 4 >= 1 = mark(a())
        
        mark(f(X1,X2)) = 1X1 + 2 >= 1X1 + 2 = active(f(mark(X1),X2))
        
        mark(a()) = 1 >= 1 = active(a())
        
        mark(b()) = 4 >= 4 = active(b())
        
        f(mark(X1),X2) = 1X1 + 2 >= 1X1 + 2 = f(X1,X2)
        
        f(X1,mark(X2)) = 1X1 + 2 >= 1X1 + 2 = f(X1,X2)
        
        f(active(X1),X2) = 1X1 + 2 >= 1X1 + 2 = f(X1,X2)
        
        f(X1,active(X2)) = 1X1 + 2 >= 1X1 + 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()))
         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)
       Open
    
    DPs:
     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)
    TRS:
     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)
    Subterm Criterion Processor:
     simple projection:
      pi(f#) = 1
     problem:
      DPs:
       f#(mark(X1),X2) -> f#(X1,X2)
       f#(active(X1),X2) -> f#(X1,X2)
      TRS:
       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)
     Subterm Criterion Processor:
      simple projection:
       pi(f#) = 0
      problem:
       DPs:
        
       TRS:
        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)
      Qed