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) = [1]          
                       p(<) = [11]         
                    p(Cons) = [6]          
                   p(False) = [1]          
                     p(Nil) = [0]          
                       p(S) = [1] x1 + [8] 
                    p(True) = [0]          
                    p(goal) = [1]          
                p(notEmpty) = [5] x1 + [1] 
                 p(ordered) = [1]          
            p(ordered[Ite]) = [1] x1 + [13]
          
          Following rules are strictly oriented:
            notEmpty(Cons(x,xs)) = [31]  
                                 > [0]   
                                 = True()
          
          ordered(Cons(x,Nil())) = [1]   
                                 > [0]   
                                 = True()
          
                  ordered(Nil()) = [1]   
                                 > [0]   
                                 = True()
          
          
          Following rules are (at-least) weakly oriented:
                                 <(x,0()) =  [11]                                     
                                          >= [1]                                      
                                          =  False()                                  
          
                              <(0(),S(y)) =  [11]                                     
                                          >= [0]                                      
                                          =  True()                                   
          
                             <(S(x),S(y)) =  [11]                                     
                                          >= [11]                                     
                                          =  <(x,y)                                   
          
                                 goal(xs) =  [1]                                      
                                          >= [1]                                      
                                          =  ordered(xs)                              
          
                          notEmpty(Nil()) =  [1]                                      
                                          >= [1]                                      
                                          =  False()                                  
          
             ordered(Cons(x',Cons(x,xs))) =  [1]                                      
                                          >= [24]                                     
                                          =  ordered[Ite](<(x',x),Cons(x',Cons(x,xs)))
          
                 ordered[Ite](False(),xs) =  [14]                                     
                                          >= [1]                                      
                                          =  False()                                  
          
          ordered[Ite](True(),Cons(x,xs)) =  [13]                                     
                                          >= [1]                                      
                                          =  ordered(xs)                              
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            goal(xs) -> ordered(xs)
            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)
            notEmpty(Cons(x,xs)) -> True()
            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(<) = [3]                  
                    p(Cons) = [1] x2 + [2]         
                   p(False) = [1]                  
                     p(Nil) = [2]                  
                       p(S) = [0]                  
                    p(True) = [0]                  
                    p(goal) = [4] x1 + [1]         
                p(notEmpty) = [0]                  
                 p(ordered) = [4] x1 + [9]         
            p(ordered[Ite]) = [1] x1 + [4] x2 + [4]
          
          Following rules are strictly oriented:
          ordered(Cons(x',Cons(x,xs))) = [4] xs + [25]                            
                                       > [4] xs + [23]                            
                                       = ordered[Ite](<(x',x),Cons(x',Cons(x,xs)))
          
          
          Following rules are (at-least) weakly oriented:
                                 <(x,0()) =  [3]          
                                          >= [1]          
                                          =  False()      
          
                              <(0(),S(y)) =  [3]          
                                          >= [0]          
                                          =  True()       
          
                             <(S(x),S(y)) =  [3]          
                                          >= [3]          
                                          =  <(x,y)       
          
                                 goal(xs) =  [4] xs + [1] 
                                          >= [4] xs + [9] 
                                          =  ordered(xs)  
          
                     notEmpty(Cons(x,xs)) =  [0]          
                                          >= [0]          
                                          =  True()       
          
                          notEmpty(Nil()) =  [0]          
                                          >= [1]          
                                          =  False()      
          
                   ordered(Cons(x,Nil())) =  [25]         
                                          >= [0]          
                                          =  True()       
          
                           ordered(Nil()) =  [17]         
                                          >= [0]          
                                          =  True()       
          
                 ordered[Ite](False(),xs) =  [4] xs + [5] 
                                          >= [1]          
                                          =  False()      
          
          ordered[Ite](True(),Cons(x,xs)) =  [4] xs + [12]
                                          >= [4] xs + [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:
            goal(xs) -> ordered(xs)
            notEmpty(Nil()) -> False()
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            notEmpty(Cons(x,xs)) -> True()
            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:
        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(<) = [10]                 
                    p(Cons) = [1] x1 + [1] x2 + [0]
                   p(False) = [10]                 
                     p(Nil) = [1]                  
                       p(S) = [1] x1 + [0]         
                    p(True) = [10]                 
                    p(goal) = [0]                  
                p(notEmpty) = [1] x1 + [10]        
                 p(ordered) = [10]                 
            p(ordered[Ite]) = [1] x1 + [0]         
          
          Following rules are strictly oriented:
          notEmpty(Nil()) = [11]   
                          > [10]   
                          = False()
          
          
          Following rules are (at-least) weakly oriented:
                                 <(x,0()) =  [10]                                     
                                          >= [10]                                     
                                          =  False()                                  
          
                              <(0(),S(y)) =  [10]                                     
                                          >= [10]                                     
                                          =  True()                                   
          
                             <(S(x),S(y)) =  [10]                                     
                                          >= [10]                                     
                                          =  <(x,y)                                   
          
                                 goal(xs) =  [0]                                      
                                          >= [10]                                     
                                          =  ordered(xs)                              
          
                     notEmpty(Cons(x,xs)) =  [1] x + [1] xs + [10]                    
                                          >= [10]                                     
                                          =  True()                                   
          
                   ordered(Cons(x,Nil())) =  [10]                                     
                                          >= [10]                                     
                                          =  True()                                   
          
             ordered(Cons(x',Cons(x,xs))) =  [10]                                     
                                          >= [10]                                     
                                          =  ordered[Ite](<(x',x),Cons(x',Cons(x,xs)))
          
                           ordered(Nil()) =  [10]                                     
                                          >= [10]                                     
                                          =  True()                                   
          
                 ordered[Ite](False(),xs) =  [10]                                     
                                          >= [10]                                     
                                          =  False()                                  
          
          ordered[Ite](True(),Cons(x,xs)) =  [10]                                     
                                          >= [10]                                     
                                          =  ordered(xs)                              
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            goal(xs) -> ordered(xs)
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            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:
        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] x1 + [1] x2 + [0]
                   p(False) = [0]                  
                     p(Nil) = [0]                  
                       p(S) = [1] x1 + [0]         
                    p(True) = [0]                  
                    p(goal) = [1]                  
                p(notEmpty) = [0]                  
                 p(ordered) = [0]                  
            p(ordered[Ite]) = [1] x1 + [0]         
          
          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,xs)) =  [0]                                      
                                          >= [0]                                      
                                          =  ordered(xs)                              
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 5: 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))