WORST_CASE(?,O(n^1))
* Step 1: Sum 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:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 2: NaturalPI 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:
        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) = 1   
              p(notEmpty) = 0   
               p(ordered) = 0   
          p(ordered[Ite]) = 8*x1
        
        Following rules are strictly oriented:
        goal(xs) = 1          
                 > 0          
                 = ordered(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)                                   
        
                            notEmpty(Cons(x,xs)) =  0                                        
                                                 >= 0                                        
                                                 =  True()                                   
        
                                 notEmpty(Nil()) =  0                                        
                                                 >= 0                                        
                                                 =  False()                                  
        
                          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: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            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)
            goal(xs) -> ordered(xs)
            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) = 11
                 p(False) = 0 
                   p(Nil) = 0 
                     p(S) = 0 
                  p(True) = 0 
                  p(goal) = 0 
              p(notEmpty) = 8 
               p(ordered) = 0 
          p(ordered[Ite]) = 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 4: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            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)
            goal(xs) -> ordered(xs)
            notEmpty(Cons(x,xs)) -> True()
            notEmpty(Nil()) -> False()
            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) = 4       
                     p(<) = 0       
                  p(Cons) = 9       
                 p(False) = 0       
                   p(Nil) = 8       
                     p(S) = 0       
                  p(True) = 0       
                  p(goal) = 8       
              p(notEmpty) = 0       
               p(ordered) = 8       
          p(ordered[Ite]) = 8 + 8*x1
        
        Following rules are strictly oriented:
        ordered(Cons(x,Nil())) = 8     
                               > 0     
                               = True()
        
                ordered(Nil()) = 8     
                               > 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)                                   
        
                                        goal(xs) =  8                                        
                                                 >= 8                                        
                                                 =  ordered(xs)                              
        
                            notEmpty(Cons(x,xs)) =  0                                        
                                                 >= 0                                        
                                                 =  True()                                   
        
                                 notEmpty(Nil()) =  0                                        
                                                 >= 0                                        
                                                 =  False()                                  
        
                    ordered(Cons(x',Cons(x,xs))) =  8                                        
                                                 >= 8                                        
                                                 =  ordered[Ite](<(x',x),Cons(x',Cons(x,xs)))
        
                        ordered[Ite](False(),xs) =  8                                        
                                                 >= 0                                        
                                                 =  False()                                  
        
        ordered[Ite](True(),Cons(x',Cons(x,xs))) =  8                                        
                                                 >= 8                                        
                                                 =  ordered(xs)                              
        
* Step 5: NaturalPI 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:
        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(<) = 1            
                  p(Cons) = 1 + x2       
                 p(False) = 1            
                   p(Nil) = 0            
                     p(S) = 2            
                  p(True) = 0            
                  p(goal) = 14 + 9*x1    
              p(notEmpty) = 7 + 8*x1     
               p(ordered) = 12 + 2*x1    
          p(ordered[Ite]) = 8 + x1 + 2*x2
        
        Following rules are strictly oriented:
        ordered(Cons(x',Cons(x,xs))) = 16 + 2*xs                                
                                     > 13 + 2*xs                                
                                     = ordered[Ite](<(x',x),Cons(x',Cons(x,xs)))
        
        
        Following rules are (at-least) weakly oriented:
                                        <(x,0()) =  1          
                                                 >= 1          
                                                 =  False()    
        
                                     <(0(),S(y)) =  1          
                                                 >= 0          
                                                 =  True()     
        
                                    <(S(x),S(y)) =  1          
                                                 >= 1          
                                                 =  <(x,y)     
        
                                        goal(xs) =  14 + 9*xs  
                                                 >= 12 + 2*xs  
                                                 =  ordered(xs)
        
                            notEmpty(Cons(x,xs)) =  15 + 8*xs  
                                                 >= 0          
                                                 =  True()     
        
                                 notEmpty(Nil()) =  7          
                                                 >= 1          
                                                 =  False()    
        
                          ordered(Cons(x,Nil())) =  14         
                                                 >= 0          
                                                 =  True()     
        
                                  ordered(Nil()) =  12         
                                                 >= 0          
                                                 =  True()     
        
                        ordered[Ite](False(),xs) =  9 + 2*xs   
                                                 >= 1          
                                                 =  False()    
        
        ordered[Ite](True(),Cons(x',Cons(x,xs))) =  12 + 2*xs  
                                                 >= 12 + 2*xs  
                                                 =  ordered(xs)
        
* Step 6: 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))