WORST_CASE(?,O(n^1))
* Step 1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            goal(xs) -> ordered(xs)
            notEmpty(Cons(x,xs)) -> True()
            notEmpty(Nil()) -> False()
            ordered(Cons(x,Nil())) -> True()
            ordered(Cons(x',Cons(x,xs))) -> ordered[Ite](<(x',x),Cons(x',Cons(x,xs)))
            ordered(Nil()) -> True()
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            ordered[Ite](False(),xs) -> False()
            ordered[Ite](True(),Cons(x',Cons(x,xs))) -> ordered(xs)
        - Signature:
            {</2,goal/1,notEmpty/1,ordered/1,ordered[Ite]/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<,goal,notEmpty,ordered,ordered[Ite]} and constructors {0
            ,Cons,False,Nil,S,True}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(ordered[Ite]) = {1}
        
        Following symbols are considered usable:
          {<,goal,notEmpty,ordered,ordered[Ite]}
        TcT has computed the following interpretation:
                     p(0) = [1]         
                     p(<) = [0]         
                  p(Cons) = [0]         
                 p(False) = [0]         
                   p(Nil) = [0]         
                     p(S) = [1]         
                  p(True) = [0]         
                  p(goal) = [1] x1 + [5]
              p(notEmpty) = [8] x1 + [0]
               p(ordered) = [4]         
          p(ordered[Ite]) = [8] x1 + [4]
        
        Following rules are strictly oriented:
                      goal(xs) = [1] xs + [5]
                               > [4]         
                               = ordered(xs) 
        
        ordered(Cons(x,Nil())) = [4]         
                               > [0]         
                               = True()      
        
                ordered(Nil()) = [4]         
                               > [0]         
                               = True()      
        
        
        Following rules are (at-least) weakly oriented:
                                        <(x,0()) =  [0]                                      
                                                 >= [0]                                      
                                                 =  False()                                  
        
                                     <(0(),S(y)) =  [0]                                      
                                                 >= [0]                                      
                                                 =  True()                                   
        
                                    <(S(x),S(y)) =  [0]                                      
                                                 >= [0]                                      
                                                 =  <(x,y)                                   
        
                            notEmpty(Cons(x,xs)) =  [0]                                      
                                                 >= [0]                                      
                                                 =  True()                                   
        
                                 notEmpty(Nil()) =  [0]                                      
                                                 >= [0]                                      
                                                 =  False()                                  
        
                    ordered(Cons(x',Cons(x,xs))) =  [4]                                      
                                                 >= [4]                                      
                                                 =  ordered[Ite](<(x',x),Cons(x',Cons(x,xs)))
        
                        ordered[Ite](False(),xs) =  [4]                                      
                                                 >= [0]                                      
                                                 =  False()                                  
        
        ordered[Ite](True(),Cons(x',Cons(x,xs))) =  [4]                                      
                                                 >= [4]                                      
                                                 =  ordered(xs)                              
        
* Step 2: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            notEmpty(Cons(x,xs)) -> True()
            notEmpty(Nil()) -> False()
            ordered(Cons(x',Cons(x,xs))) -> ordered[Ite](<(x',x),Cons(x',Cons(x,xs)))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            goal(xs) -> ordered(xs)
            ordered(Cons(x,Nil())) -> True()
            ordered(Nil()) -> True()
            ordered[Ite](False(),xs) -> False()
            ordered[Ite](True(),Cons(x',Cons(x,xs))) -> ordered(xs)
        - Signature:
            {</2,goal/1,notEmpty/1,ordered/1,ordered[Ite]/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<,goal,notEmpty,ordered,ordered[Ite]} and constructors {0
            ,Cons,False,Nil,S,True}
    + Applied Processor:
        NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(linear):
        The following argument positions are considered usable:
          uargs(ordered[Ite]) = {1}
        
        Following symbols are considered usable:
          {<,goal,notEmpty,ordered,ordered[Ite]}
        TcT has computed the following interpretation:
                     p(0) = 0   
                     p(<) = 0   
                  p(Cons) = 0   
                 p(False) = 0   
                   p(Nil) = 0   
                     p(S) = 0   
                  p(True) = 0   
                  p(goal) = 0   
              p(notEmpty) = 8   
               p(ordered) = 0   
          p(ordered[Ite]) = 8*x1
        
        Following rules are strictly oriented:
        notEmpty(Cons(x,xs)) = 8      
                             > 0      
                             = True() 
        
             notEmpty(Nil()) = 8      
                             > 0      
                             = False()
        
        
        Following rules are (at-least) weakly oriented:
                                        <(x,0()) =  0                                        
                                                 >= 0                                        
                                                 =  False()                                  
        
                                     <(0(),S(y)) =  0                                        
                                                 >= 0                                        
                                                 =  True()                                   
        
                                    <(S(x),S(y)) =  0                                        
                                                 >= 0                                        
                                                 =  <(x,y)                                   
        
                                        goal(xs) =  0                                        
                                                 >= 0                                        
                                                 =  ordered(xs)                              
        
                          ordered(Cons(x,Nil())) =  0                                        
                                                 >= 0                                        
                                                 =  True()                                   
        
                    ordered(Cons(x',Cons(x,xs))) =  0                                        
                                                 >= 0                                        
                                                 =  ordered[Ite](<(x',x),Cons(x',Cons(x,xs)))
        
                                  ordered(Nil()) =  0                                        
                                                 >= 0                                        
                                                 =  True()                                   
        
                        ordered[Ite](False(),xs) =  0                                        
                                                 >= 0                                        
                                                 =  False()                                  
        
        ordered[Ite](True(),Cons(x',Cons(x,xs))) =  0                                        
                                                 >= 0                                        
                                                 =  ordered(xs)                              
        
* Step 3: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            ordered(Cons(x',Cons(x,xs))) -> ordered[Ite](<(x',x),Cons(x',Cons(x,xs)))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            goal(xs) -> ordered(xs)
            notEmpty(Cons(x,xs)) -> True()
            notEmpty(Nil()) -> False()
            ordered(Cons(x,Nil())) -> True()
            ordered(Nil()) -> True()
            ordered[Ite](False(),xs) -> False()
            ordered[Ite](True(),Cons(x',Cons(x,xs))) -> ordered(xs)
        - Signature:
            {</2,goal/1,notEmpty/1,ordered/1,ordered[Ite]/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<,goal,notEmpty,ordered,ordered[Ite]} and constructors {0
            ,Cons,False,Nil,S,True}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(ordered[Ite]) = {1}
        
        Following symbols are considered usable:
          {<,goal,notEmpty,ordered,ordered[Ite]}
        TcT has computed the following interpretation:
                     p(0) = [1]                  
                     p(<) = [0]                  
                  p(Cons) = [1] x2 + [5]         
                 p(False) = [0]                  
                   p(Nil) = [3]                  
                     p(S) = [2]                  
                  p(True) = [0]                  
                  p(goal) = [1] x1 + [8]         
              p(notEmpty) = [2]                  
               p(ordered) = [1] x1 + [8]         
          p(ordered[Ite]) = [8] x1 + [1] x2 + [1]
        
        Following rules are strictly oriented:
        ordered(Cons(x',Cons(x,xs))) = [1] xs + [18]                            
                                     > [1] xs + [11]                            
                                     = ordered[Ite](<(x',x),Cons(x',Cons(x,xs)))
        
        
        Following rules are (at-least) weakly oriented:
                                        <(x,0()) =  [0]          
                                                 >= [0]          
                                                 =  False()      
        
                                     <(0(),S(y)) =  [0]          
                                                 >= [0]          
                                                 =  True()       
        
                                    <(S(x),S(y)) =  [0]          
                                                 >= [0]          
                                                 =  <(x,y)       
        
                                        goal(xs) =  [1] xs + [8] 
                                                 >= [1] xs + [8] 
                                                 =  ordered(xs)  
        
                            notEmpty(Cons(x,xs)) =  [2]          
                                                 >= [0]          
                                                 =  True()       
        
                                 notEmpty(Nil()) =  [2]          
                                                 >= [0]          
                                                 =  False()      
        
                          ordered(Cons(x,Nil())) =  [16]         
                                                 >= [0]          
                                                 =  True()       
        
                                  ordered(Nil()) =  [11]         
                                                 >= [0]          
                                                 =  True()       
        
                        ordered[Ite](False(),xs) =  [1] xs + [1] 
                                                 >= [0]          
                                                 =  False()      
        
        ordered[Ite](True(),Cons(x',Cons(x,xs))) =  [1] xs + [11]
                                                 >= [1] xs + [8] 
                                                 =  ordered(xs)  
        
* Step 4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            goal(xs) -> ordered(xs)
            notEmpty(Cons(x,xs)) -> True()
            notEmpty(Nil()) -> False()
            ordered(Cons(x,Nil())) -> True()
            ordered(Cons(x',Cons(x,xs))) -> ordered[Ite](<(x',x),Cons(x',Cons(x,xs)))
            ordered(Nil()) -> True()
            ordered[Ite](False(),xs) -> False()
            ordered[Ite](True(),Cons(x',Cons(x,xs))) -> ordered(xs)
        - Signature:
            {</2,goal/1,notEmpty/1,ordered/1,ordered[Ite]/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<,goal,notEmpty,ordered,ordered[Ite]} and constructors {0
            ,Cons,False,Nil,S,True}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))