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,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,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) = 8   
              p(notEmpty) = 0   
               p(ordered) = 0   
          p(ordered[Ite]) = 2*x1
        
        Following rules are strictly oriented:
        goal(xs) = 8          
                 > 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,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,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) = 10  
                 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,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,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) = x1        
                 p(False) = 0         
                   p(Nil) = 0         
                     p(S) = 2         
                  p(True) = 0         
                  p(goal) = 15 + 14*x1
              p(notEmpty) = 15        
               p(ordered) = 14        
          p(ordered[Ite]) = 14 + 2*x1 
        
        Following rules are strictly oriented:
        ordered(Cons(x,Nil())) = 14    
                               > 0     
                               = True()
        
                ordered(Nil()) = 14    
                               > 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) =  15 + 14*xs                               
                                        >= 14                                       
                                        =  ordered(xs)                              
        
                   notEmpty(Cons(x,xs)) =  15                                       
                                        >= 0                                        
                                        =  True()                                   
        
                        notEmpty(Nil()) =  15                                       
                                        >= 0                                        
                                        =  False()                                  
        
           ordered(Cons(x',Cons(x,xs))) =  14                                       
                                        >= 14                                       
                                        =  ordered[Ite](<(x',x),Cons(x',Cons(x,xs)))
        
               ordered[Ite](False(),xs) =  14                                       
                                        >= 0                                        
                                        =  False()                                  
        
        ordered[Ite](True(),Cons(x,xs)) =  14                                       
                                        >= 14                                       
                                        =  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,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) = 1 + x2     
                 p(False) = 0          
                   p(Nil) = 1          
                     p(S) = 0          
                  p(True) = 0          
                  p(goal) = 4 + 9*x1   
              p(notEmpty) = 6 + 12*x1  
               p(ordered) = 1 + 8*x1   
          p(ordered[Ite]) = 4*x1 + 8*x2
        
        Following rules are strictly oriented:
        ordered(Cons(x',Cons(x,xs))) = 17 + 8*xs                                
                                     > 16 + 8*xs                                
                                     = 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) =  4 + 9*xs   
                                        >= 1 + 8*xs   
                                        =  ordered(xs)
        
                   notEmpty(Cons(x,xs)) =  18 + 12*xs 
                                        >= 0          
                                        =  True()     
        
                        notEmpty(Nil()) =  18         
                                        >= 0          
                                        =  False()    
        
                 ordered(Cons(x,Nil())) =  17         
                                        >= 0          
                                        =  True()     
        
                         ordered(Nil()) =  9          
                                        >= 0          
                                        =  True()     
        
               ordered[Ite](False(),xs) =  8*xs       
                                        >= 0          
                                        =  False()    
        
        ordered[Ite](True(),Cons(x,xs)) =  8 + 8*xs   
                                        >= 1 + 8*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,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))