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) = [0]         
                     p(<) = [0]         
                  p(Cons) = [0]         
                 p(False) = [0]         
                   p(Nil) = [0]         
                     p(S) = [0]         
                  p(True) = [0]         
                  p(goal) = [4]         
              p(notEmpty) = [0]         
               p(ordered) = [0]         
          p(ordered[Ite]) = [2] x1 + [0]
        
        Following rules are strictly oriented:
        goal(xs) = [4]        
                 > [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 2: WeightGap 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:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          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:
            all
          TcT has computed the following interpretation:
                       p(0) = [0]                  
                       p(<) = [0]                  
                    p(Cons) = [3]                  
                   p(False) = [0]                  
                     p(Nil) = [0]                  
                       p(S) = [1] x1 + [0]         
                    p(True) = [0]                  
                    p(goal) = [9]                  
                p(notEmpty) = [3] x1 + [6]         
                 p(ordered) = [9]                  
            p(ordered[Ite]) = [1] x1 + [3] x2 + [0]
          
          Following rules are strictly oriented:
            notEmpty(Cons(x,xs)) = [15]   
                                 > [0]    
                                 = True() 
          
                 notEmpty(Nil()) = [6]    
                                 > [0]    
                                 = False()
          
          ordered(Cons(x,Nil())) = [9]    
                                 > [0]    
                                 = True() 
          
                  ordered(Nil()) = [9]    
                                 > [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) =  [9]                                      
                                                   >= [9]                                      
                                                   =  ordered(xs)                              
          
                      ordered(Cons(x',Cons(x,xs))) =  [9]                                      
                                                   >= [9]                                      
                                                   =  ordered[Ite](<(x',x),Cons(x',Cons(x,xs)))
          
                          ordered[Ite](False(),xs) =  [3] xs + [0]                             
                                                   >= [0]                                      
                                                   =  False()                                  
          
          ordered[Ite](True(),Cons(x',Cons(x,xs))) =  [9]                                      
                                                   >= [9]                                      
                                                   =  ordered(xs)                              
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: WeightGap 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:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          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:
            all
          TcT has computed the following interpretation:
                       p(0) = [4]                  
                       p(<) = [1]                  
                    p(Cons) = [1] x1 + [1] x2 + [7]
                   p(False) = [0]                  
                     p(Nil) = [1]                  
                       p(S) = [4]                  
                    p(True) = [0]                  
                    p(goal) = [3] x1 + [3]         
                p(notEmpty) = [4] x1 + [0]         
                 p(ordered) = [2] x1 + [3]         
            p(ordered[Ite]) = [1] x1 + [2] x2 + [1]
          
          Following rules are strictly oriented:
          ordered(Cons(x',Cons(x,xs))) = [2] x + [2] x' + [2] xs + [31]           
                                       > [2] x + [2] x' + [2] xs + [30]           
                                       = ordered[Ite](<(x',x),Cons(x',Cons(x,xs)))
          
          
          Following rules are (at-least) weakly oriented:
                                          <(x,0()) =  [1]                           
                                                   >= [0]                           
                                                   =  False()                       
          
                                       <(0(),S(y)) =  [1]                           
                                                   >= [0]                           
                                                   =  True()                        
          
                                      <(S(x),S(y)) =  [1]                           
                                                   >= [1]                           
                                                   =  <(x,y)                        
          
                                          goal(xs) =  [3] xs + [3]                  
                                                   >= [2] xs + [3]                  
                                                   =  ordered(xs)                   
          
                              notEmpty(Cons(x,xs)) =  [4] x + [4] xs + [28]         
                                                   >= [0]                           
                                                   =  True()                        
          
                                   notEmpty(Nil()) =  [4]                           
                                                   >= [0]                           
                                                   =  False()                       
          
                            ordered(Cons(x,Nil())) =  [2] x + [19]                  
                                                   >= [0]                           
                                                   =  True()                        
          
                                    ordered(Nil()) =  [5]                           
                                                   >= [0]                           
                                                   =  True()                        
          
                          ordered[Ite](False(),xs) =  [2] xs + [1]                  
                                                   >= [0]                           
                                                   =  False()                       
          
          ordered[Ite](True(),Cons(x',Cons(x,xs))) =  [2] x + [2] x' + [2] xs + [29]
                                                   >= [2] xs + [3]                  
                                                   =  ordered(xs)                   
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* 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))