MAYBE

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

Proof:
 Matrix Interpretation Processor: dim=3
  
  interpretation:
               [1 0 1]  
   [top](x0) = [0 0 1]x0
               [0 0 0]  ,
   
              [1 0 0]  
   [ok](x0) = [0 0 0]x0
              [0 0 1]  ,
   
                    
   [proper](x0) = x0
                    ,
   
                [1 0 0]  
   [mark](x0) = [0 0 0]x0
                [0 0 1]  ,
   
         [0]
   [b] = [0]
         [1],
   
         [0]
   [a] = [1]
         [0],
   
                  [1 0 1]     [0]
   [active](x0) = [0 0 0]x0 + [1]
                  [0 0 0]     [0],
   
                 [1 0 0]     [1 0 0]  
   [f](x0, x1) = [0 0 1]x0 + [0 0 0]x1
                 [0 0 1]     [0 0 0]  
  orientation:
                    [2 0 1]    [0]    [0]                   
   active(f(X,X)) = [0 0 0]X + [1] >= [0] = mark(f(a(),b()))
                    [0 0 0]    [0]    [0]                   
   
                 [1]    [0]            
   active(b()) = [1] >= [0] = mark(a())
                 [0]    [0]            
   
                      [1 0 1]     [1 0 0]     [0]    [1 0 1]     [1 0 0]                     
   active(f(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 = f(active(X1),X2)
                      [0 0 0]     [0 0 0]     [0]    [0 0 0]     [0 0 0]                     
   
                    [1 0 0]     [1 0 0]      [1 0 0]     [1 0 0]                   
   f(mark(X1),X2) = [0 0 1]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = mark(f(X1,X2))
                    [0 0 1]     [0 0 0]      [0 0 1]     [0 0 0]                   
   
                      [1 0 0]     [1 0 0]      [1 0 0]     [1 0 0]                             
   proper(f(X1,X2)) = [0 0 1]X1 + [0 0 0]X2 >= [0 0 1]X1 + [0 0 0]X2 = f(proper(X1),proper(X2))
                      [0 0 1]     [0 0 0]      [0 0 1]     [0 0 0]                             
   
                 [0]    [0]          
   proper(a()) = [1] >= [0] = ok(a())
                 [0]    [0]          
   
                 [0]    [0]          
   proper(b()) = [0] >= [0] = ok(b())
                 [1]    [1]          
   
                      [1 0 0]     [1 0 0]      [1 0 0]     [1 0 0]                 
   f(ok(X1),ok(X2)) = [0 0 1]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = ok(f(X1,X2))
                      [0 0 1]     [0 0 0]      [0 0 1]     [0 0 0]                 
   
                  [1 0 1]     [1 0 1]                  
   top(mark(X)) = [0 0 1]X >= [0 0 1]X = top(proper(X))
                  [0 0 0]     [0 0 0]                  
   
                [1 0 1]     [1 0 1]                  
   top(ok(X)) = [0 0 1]X >= [0 0 0]X = top(active(X))
                [0 0 0]     [0 0 0]                  
  problem:
   active(f(X,X)) -> mark(f(a(),b()))
   active(f(X1,X2)) -> f(active(X1),X2)
   f(mark(X1),X2) -> mark(f(X1,X2))
   proper(f(X1,X2)) -> f(proper(X1),proper(X2))
   proper(a()) -> ok(a())
   proper(b()) -> ok(b())
   f(ok(X1),ok(X2)) -> ok(f(X1,X2))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))
  DP Processor:
   DPs:
    active#(f(X,X)) -> f#(a(),b())
    active#(f(X1,X2)) -> active#(X1)
    active#(f(X1,X2)) -> f#(active(X1),X2)
    f#(mark(X1),X2) -> f#(X1,X2)
    proper#(f(X1,X2)) -> proper#(X2)
    proper#(f(X1,X2)) -> proper#(X1)
    proper#(f(X1,X2)) -> f#(proper(X1),proper(X2))
    f#(ok(X1),ok(X2)) -> f#(X1,X2)
    top#(mark(X)) -> proper#(X)
    top#(mark(X)) -> top#(proper(X))
    top#(ok(X)) -> active#(X)
    top#(ok(X)) -> top#(active(X))
   TRS:
    active(f(X,X)) -> mark(f(a(),b()))
    active(f(X1,X2)) -> f(active(X1),X2)
    f(mark(X1),X2) -> mark(f(X1,X2))
    proper(f(X1,X2)) -> f(proper(X1),proper(X2))
    proper(a()) -> ok(a())
    proper(b()) -> ok(b())
    f(ok(X1),ok(X2)) -> ok(f(X1,X2))
    top(mark(X)) -> top(proper(X))
    top(ok(X)) -> top(active(X))
   TDG Processor:
    DPs:
     active#(f(X,X)) -> f#(a(),b())
     active#(f(X1,X2)) -> active#(X1)
     active#(f(X1,X2)) -> f#(active(X1),X2)
     f#(mark(X1),X2) -> f#(X1,X2)
     proper#(f(X1,X2)) -> proper#(X2)
     proper#(f(X1,X2)) -> proper#(X1)
     proper#(f(X1,X2)) -> f#(proper(X1),proper(X2))
     f#(ok(X1),ok(X2)) -> f#(X1,X2)
     top#(mark(X)) -> proper#(X)
     top#(mark(X)) -> top#(proper(X))
     top#(ok(X)) -> active#(X)
     top#(ok(X)) -> top#(active(X))
    TRS:
     active(f(X,X)) -> mark(f(a(),b()))
     active(f(X1,X2)) -> f(active(X1),X2)
     f(mark(X1),X2) -> mark(f(X1,X2))
     proper(f(X1,X2)) -> f(proper(X1),proper(X2))
     proper(a()) -> ok(a())
     proper(b()) -> ok(b())
     f(ok(X1),ok(X2)) -> ok(f(X1,X2))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    graph:
     top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> top#(active(X))
     top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> active#(X)
     top#(ok(X)) -> top#(active(X)) ->
     top#(mark(X)) -> top#(proper(X))
     top#(ok(X)) -> top#(active(X)) -> top#(mark(X)) -> proper#(X)
     top#(ok(X)) -> active#(X) -> active#(f(X1,X2)) -> f#(active(X1),X2)
     top#(ok(X)) -> active#(X) -> active#(f(X1,X2)) -> active#(X1)
     top#(ok(X)) -> active#(X) -> active#(f(X,X)) -> f#(a(),b())
     top#(mark(X)) -> top#(proper(X)) ->
     top#(ok(X)) -> top#(active(X))
     top#(mark(X)) -> top#(proper(X)) -> top#(ok(X)) -> active#(X)
     top#(mark(X)) -> top#(proper(X)) ->
     top#(mark(X)) -> top#(proper(X))
     top#(mark(X)) -> top#(proper(X)) -> top#(mark(X)) -> proper#(X)
     top#(mark(X)) -> proper#(X) ->
     proper#(f(X1,X2)) -> f#(proper(X1),proper(X2))
     top#(mark(X)) -> proper#(X) -> proper#(f(X1,X2)) -> proper#(X1)
     top#(mark(X)) -> proper#(X) ->
     proper#(f(X1,X2)) -> proper#(X2)
     proper#(f(X1,X2)) -> proper#(X2) ->
     proper#(f(X1,X2)) -> f#(proper(X1),proper(X2))
     proper#(f(X1,X2)) -> proper#(X2) ->
     proper#(f(X1,X2)) -> proper#(X1)
     proper#(f(X1,X2)) -> proper#(X2) ->
     proper#(f(X1,X2)) -> proper#(X2)
     proper#(f(X1,X2)) -> proper#(X1) ->
     proper#(f(X1,X2)) -> f#(proper(X1),proper(X2))
     proper#(f(X1,X2)) -> proper#(X1) ->
     proper#(f(X1,X2)) -> proper#(X1)
     proper#(f(X1,X2)) -> proper#(X1) ->
     proper#(f(X1,X2)) -> proper#(X2)
     proper#(f(X1,X2)) -> f#(proper(X1),proper(X2)) ->
     f#(ok(X1),ok(X2)) -> f#(X1,X2)
     proper#(f(X1,X2)) -> f#(proper(X1),proper(X2)) ->
     f#(mark(X1),X2) -> f#(X1,X2)
     f#(ok(X1),ok(X2)) -> f#(X1,X2) -> f#(ok(X1),ok(X2)) -> f#(X1,X2)
     f#(ok(X1),ok(X2)) -> f#(X1,X2) -> f#(mark(X1),X2) -> f#(X1,X2)
     f#(mark(X1),X2) -> f#(X1,X2) -> f#(ok(X1),ok(X2)) -> f#(X1,X2)
     f#(mark(X1),X2) -> f#(X1,X2) ->
     f#(mark(X1),X2) -> f#(X1,X2)
     active#(f(X1,X2)) -> f#(active(X1),X2) ->
     f#(ok(X1),ok(X2)) -> f#(X1,X2)
     active#(f(X1,X2)) -> f#(active(X1),X2) ->
     f#(mark(X1),X2) -> f#(X1,X2)
     active#(f(X1,X2)) -> active#(X1) ->
     active#(f(X1,X2)) -> f#(active(X1),X2)
     active#(f(X1,X2)) -> active#(X1) ->
     active#(f(X1,X2)) -> active#(X1)
     active#(f(X1,X2)) -> active#(X1) ->
     active#(f(X,X)) -> f#(a(),b())
     active#(f(X,X)) -> f#(a(),b()) -> f#(ok(X1),ok(X2)) -> f#(X1,X2)
     active#(f(X,X)) -> f#(a(),b()) -> f#(mark(X1),X2) -> f#(X1,X2)
    SCC Processor:
     #sccs: 4
     #rules: 7
     #arcs: 33/144
     DPs:
      top#(ok(X)) -> top#(active(X))
      top#(mark(X)) -> top#(proper(X))
     TRS:
      active(f(X,X)) -> mark(f(a(),b()))
      active(f(X1,X2)) -> f(active(X1),X2)
      f(mark(X1),X2) -> mark(f(X1,X2))
      proper(f(X1,X2)) -> f(proper(X1),proper(X2))
      proper(a()) -> ok(a())
      proper(b()) -> ok(b())
      f(ok(X1),ok(X2)) -> ok(f(X1,X2))
      top(mark(X)) -> top(proper(X))
      top(ok(X)) -> top(active(X))
     Open
     
     DPs:
      active#(f(X1,X2)) -> active#(X1)
     TRS:
      active(f(X,X)) -> mark(f(a(),b()))
      active(f(X1,X2)) -> f(active(X1),X2)
      f(mark(X1),X2) -> mark(f(X1,X2))
      proper(f(X1,X2)) -> f(proper(X1),proper(X2))
      proper(a()) -> ok(a())
      proper(b()) -> ok(b())
      f(ok(X1),ok(X2)) -> ok(f(X1,X2))
      top(mark(X)) -> top(proper(X))
      top(ok(X)) -> top(active(X))
     Subterm Criterion Processor:
      simple projection:
       pi(active#) = 0
      problem:
       DPs:
        
       TRS:
        active(f(X,X)) -> mark(f(a(),b()))
        active(f(X1,X2)) -> f(active(X1),X2)
        f(mark(X1),X2) -> mark(f(X1,X2))
        proper(f(X1,X2)) -> f(proper(X1),proper(X2))
        proper(a()) -> ok(a())
        proper(b()) -> ok(b())
        f(ok(X1),ok(X2)) -> ok(f(X1,X2))
        top(mark(X)) -> top(proper(X))
        top(ok(X)) -> top(active(X))
      Qed
     
     DPs:
      proper#(f(X1,X2)) -> proper#(X2)
      proper#(f(X1,X2)) -> proper#(X1)
     TRS:
      active(f(X,X)) -> mark(f(a(),b()))
      active(f(X1,X2)) -> f(active(X1),X2)
      f(mark(X1),X2) -> mark(f(X1,X2))
      proper(f(X1,X2)) -> f(proper(X1),proper(X2))
      proper(a()) -> ok(a())
      proper(b()) -> ok(b())
      f(ok(X1),ok(X2)) -> ok(f(X1,X2))
      top(mark(X)) -> top(proper(X))
      top(ok(X)) -> top(active(X))
     Subterm Criterion Processor:
      simple projection:
       pi(proper#) = 0
      problem:
       DPs:
        
       TRS:
        active(f(X,X)) -> mark(f(a(),b()))
        active(f(X1,X2)) -> f(active(X1),X2)
        f(mark(X1),X2) -> mark(f(X1,X2))
        proper(f(X1,X2)) -> f(proper(X1),proper(X2))
        proper(a()) -> ok(a())
        proper(b()) -> ok(b())
        f(ok(X1),ok(X2)) -> ok(f(X1,X2))
        top(mark(X)) -> top(proper(X))
        top(ok(X)) -> top(active(X))
      Qed
     
     DPs:
      f#(mark(X1),X2) -> f#(X1,X2)
      f#(ok(X1),ok(X2)) -> f#(X1,X2)
     TRS:
      active(f(X,X)) -> mark(f(a(),b()))
      active(f(X1,X2)) -> f(active(X1),X2)
      f(mark(X1),X2) -> mark(f(X1,X2))
      proper(f(X1,X2)) -> f(proper(X1),proper(X2))
      proper(a()) -> ok(a())
      proper(b()) -> ok(b())
      f(ok(X1),ok(X2)) -> ok(f(X1,X2))
      top(mark(X)) -> top(proper(X))
      top(ok(X)) -> top(active(X))
     Subterm Criterion Processor:
      simple projection:
       pi(f#) = 1
      problem:
       DPs:
        f#(mark(X1),X2) -> f#(X1,X2)
       TRS:
        active(f(X,X)) -> mark(f(a(),b()))
        active(f(X1,X2)) -> f(active(X1),X2)
        f(mark(X1),X2) -> mark(f(X1,X2))
        proper(f(X1,X2)) -> f(proper(X1),proper(X2))
        proper(a()) -> ok(a())
        proper(b()) -> ok(b())
        f(ok(X1),ok(X2)) -> ok(f(X1,X2))
        top(mark(X)) -> top(proper(X))
        top(ok(X)) -> top(active(X))
      Subterm Criterion Processor:
       simple projection:
        pi(f#) = 0
       problem:
        DPs:
         
        TRS:
         active(f(X,X)) -> mark(f(a(),b()))
         active(f(X1,X2)) -> f(active(X1),X2)
         f(mark(X1),X2) -> mark(f(X1,X2))
         proper(f(X1,X2)) -> f(proper(X1),proper(X2))
         proper(a()) -> ok(a())
         proper(b()) -> ok(b())
         f(ok(X1),ok(X2)) -> ok(f(X1,X2))
         top(mark(X)) -> top(proper(X))
         top(ok(X)) -> top(active(X))
       Qed