WORST_CASE(?,O(n^1))
* Step 1: WeightGap 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:
        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) = [1] x2 + [4]         
                   p(False) = [0]                  
                     p(Nil) = [0]                  
                       p(S) = [1] x1 + [0]         
                    p(True) = [0]                  
                    p(goal) = [2] x1 + [0]         
                p(notEmpty) = [0]                  
                 p(ordered) = [2] x1 + [0]         
            p(ordered[Ite]) = [1] x1 + [2] x2 + [0]
          
          Following rules are strictly oriented:
          ordered(Cons(x,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) =  [2] xs + [0]                             
                                          >= [2] xs + [0]                             
                                          =  ordered(xs)                              
          
                     notEmpty(Cons(x,xs)) =  [0]                                      
                                          >= [0]                                      
                                          =  True()                                   
          
                          notEmpty(Nil()) =  [0]                                      
                                          >= [0]                                      
                                          =  False()                                  
          
             ordered(Cons(x',Cons(x,xs))) =  [2] xs + [16]                            
                                          >= [2] xs + [16]                            
                                          =  ordered[Ite](<(x',x),Cons(x',Cons(x,xs)))
          
                           ordered(Nil()) =  [0]                                      
                                          >= [0]                                      
                                          =  True()                                   
          
                 ordered[Ite](False(),xs) =  [2] xs + [0]                             
                                          >= [0]                                      
                                          =  False()                                  
          
          ordered[Ite](True(),Cons(x,xs)) =  [2] xs + [8]                             
                                          >= [2] xs + [0]                             
                                          =  ordered(xs)                              
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: 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',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(Cons(x,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:
        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) = [8]         
              p(notEmpty) = [0]         
               p(ordered) = [0]         
          p(ordered[Ite]) = [8] x1 + [0]
        
        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: NaturalMI 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)))
            ordered(Nil()) -> True()
        - 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[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:
        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) = [0]         
              p(notEmpty) = [1]         
               p(ordered) = [0]         
          p(ordered[Ite]) = [4] x1 + [0]
        
        Following rules are strictly oriented:
        notEmpty(Cons(x,xs)) = [1]    
                             > [0]    
                             = True() 
        
             notEmpty(Nil()) = [1]    
                             > [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: 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)))
            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(Cons(x,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:
        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(<) = [8]         
                  p(Cons) = [4]         
                 p(False) = [0]         
                   p(Nil) = [0]         
                     p(S) = [0]         
                  p(True) = [8]         
                  p(goal) = [9]         
              p(notEmpty) = [8]         
               p(ordered) = [9]         
          p(ordered[Ite]) = [1] x1 + [1]
        
        Following rules are strictly oriented:
        ordered(Nil()) = [9]   
                       > [8]   
                       = True()
        
        
        Following rules are (at-least) weakly oriented:
                               <(x,0()) =  [8]                                      
                                        >= [0]                                      
                                        =  False()                                  
        
                            <(0(),S(y)) =  [8]                                      
                                        >= [8]                                      
                                        =  True()                                   
        
                           <(S(x),S(y)) =  [8]                                      
                                        >= [8]                                      
                                        =  <(x,y)                                   
        
                               goal(xs) =  [9]                                      
                                        >= [9]                                      
                                        =  ordered(xs)                              
        
                   notEmpty(Cons(x,xs)) =  [8]                                      
                                        >= [8]                                      
                                        =  True()                                   
        
                        notEmpty(Nil()) =  [8]                                      
                                        >= [0]                                      
                                        =  False()                                  
        
                 ordered(Cons(x,Nil())) =  [9]                                      
                                        >= [8]                                      
                                        =  True()                                   
        
           ordered(Cons(x',Cons(x,xs))) =  [9]                                      
                                        >= [9]                                      
                                        =  ordered[Ite](<(x',x),Cons(x',Cons(x,xs)))
        
               ordered[Ite](False(),xs) =  [1]                                      
                                        >= [0]                                      
                                        =  False()                                  
        
        ordered[Ite](True(),Cons(x,xs)) =  [9]                                      
                                        >= [9]                                      
                                        =  ordered(xs)                              
        
* Step 5: 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,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(<) = [12]                 
                    p(Cons) = [1] x2 + [1]         
                   p(False) = [12]                 
                     p(Nil) = [0]                  
                       p(S) = [1]                  
                    p(True) = [5]                  
                    p(goal) = [8] x1 + [15]        
                p(notEmpty) = [12]                 
                 p(ordered) = [8] x1 + [15]        
            p(ordered[Ite]) = [1] x1 + [8] x2 + [2]
          
          Following rules are strictly oriented:
          ordered(Cons(x',Cons(x,xs))) = [8] xs + [31]                            
                                       > [8] xs + [30]                            
                                       = ordered[Ite](<(x',x),Cons(x',Cons(x,xs)))
          
          
          Following rules are (at-least) weakly oriented:
                                 <(x,0()) =  [12]         
                                          >= [12]         
                                          =  False()      
          
                              <(0(),S(y)) =  [12]         
                                          >= [5]          
                                          =  True()       
          
                             <(S(x),S(y)) =  [12]         
                                          >= [12]         
                                          =  <(x,y)       
          
                                 goal(xs) =  [8] xs + [15]
                                          >= [8] xs + [15]
                                          =  ordered(xs)  
          
                     notEmpty(Cons(x,xs)) =  [12]         
                                          >= [5]          
                                          =  True()       
          
                          notEmpty(Nil()) =  [12]         
                                          >= [12]         
                                          =  False()      
          
                   ordered(Cons(x,Nil())) =  [23]         
                                          >= [5]          
                                          =  True()       
          
                           ordered(Nil()) =  [15]         
                                          >= [5]          
                                          =  True()       
          
                 ordered[Ite](False(),xs) =  [8] xs + [14]
                                          >= [12]         
                                          =  False()      
          
          ordered[Ite](True(),Cons(x,xs)) =  [8] xs + [15]
                                          >= [8] xs + [15]
                                          =  ordered(xs)  
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* 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))