WORST_CASE(Omega(n^1),O(n^1))
* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            even(Cons(x,Nil())) -> False()
            even(Cons(x',Cons(x,xs))) -> even(xs)
            even(Nil()) -> True()
            goal(x,y) -> and(lte(x,y),even(x))
            lte(Cons(x,xs),Nil()) -> False()
            lte(Cons(x',xs'),Cons(x,xs)) -> lte(xs',xs)
            lte(Nil(),y) -> True()
            notEmpty(Cons(x,xs)) -> True()
            notEmpty(Nil()) -> False()
        - Weak TRS:
            and(False(),False()) -> False()
            and(False(),True()) -> False()
            and(True(),False()) -> False()
            and(True(),True()) -> True()
        - Signature:
            {and/2,even/1,goal/2,lte/2,notEmpty/1} / {Cons/2,False/0,Nil/0,True/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {and,even,goal,lte,notEmpty} and constructors {Cons,False
            ,Nil,True}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            even(Cons(x,Nil())) -> False()
            even(Cons(x',Cons(x,xs))) -> even(xs)
            even(Nil()) -> True()
            goal(x,y) -> and(lte(x,y),even(x))
            lte(Cons(x,xs),Nil()) -> False()
            lte(Cons(x',xs'),Cons(x,xs)) -> lte(xs',xs)
            lte(Nil(),y) -> True()
            notEmpty(Cons(x,xs)) -> True()
            notEmpty(Nil()) -> False()
        - Weak TRS:
            and(False(),False()) -> False()
            and(False(),True()) -> False()
            and(True(),False()) -> False()
            and(True(),True()) -> True()
        - Signature:
            {and/2,even/1,goal/2,lte/2,notEmpty/1} / {Cons/2,False/0,Nil/0,True/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {and,even,goal,lte,notEmpty} and constructors {Cons,False
            ,Nil,True}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          even(z){z -> Cons(x,Cons(y,z))} =
            even(Cons(x,Cons(y,z))) ->^+ even(z)
              = C[even(z) = even(z){}]

** Step 1.b:1: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            even(Cons(x,Nil())) -> False()
            even(Cons(x',Cons(x,xs))) -> even(xs)
            even(Nil()) -> True()
            goal(x,y) -> and(lte(x,y),even(x))
            lte(Cons(x,xs),Nil()) -> False()
            lte(Cons(x',xs'),Cons(x,xs)) -> lte(xs',xs)
            lte(Nil(),y) -> True()
            notEmpty(Cons(x,xs)) -> True()
            notEmpty(Nil()) -> False()
        - Weak TRS:
            and(False(),False()) -> False()
            and(False(),True()) -> False()
            and(True(),False()) -> False()
            and(True(),True()) -> True()
        - Signature:
            {and/2,even/1,goal/2,lte/2,notEmpty/1} / {Cons/2,False/0,Nil/0,True/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {and,even,goal,lte,notEmpty} and constructors {Cons,False
            ,Nil,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(and) = {1,2}
        
        Following symbols are considered usable:
          {and,even,goal,lte,notEmpty}
        TcT has computed the following interpretation:
              p(Cons) = 2 + x2         
             p(False) = 0              
               p(Nil) = 8              
              p(True) = 2              
               p(and) = 4 + 8*x1 + 4*x2
              p(even) = x1             
              p(goal) = 10 + 12*x1     
               p(lte) = x1             
          p(notEmpty) = x1             
        
        Following rules are strictly oriented:
                 even(Cons(x,Nil())) = 10                   
                                     > 0                    
                                     = False()              
        
           even(Cons(x',Cons(x,xs))) = 4 + xs               
                                     > xs                   
                                     = even(xs)             
        
                         even(Nil()) = 8                    
                                     > 2                    
                                     = True()               
        
                           goal(x,y) = 10 + 12*x            
                                     > 4 + 12*x             
                                     = and(lte(x,y),even(x))
        
               lte(Cons(x,xs),Nil()) = 2 + xs               
                                     > 0                    
                                     = False()              
        
        lte(Cons(x',xs'),Cons(x,xs)) = 2 + xs'              
                                     > xs'                  
                                     = lte(xs',xs)          
        
                        lte(Nil(),y) = 8                    
                                     > 2                    
                                     = True()               
        
                     notEmpty(Nil()) = 8                    
                                     > 0                    
                                     = False()              
        
        
        Following rules are (at-least) weakly oriented:
        and(False(),False()) =  4      
                             >= 0      
                             =  False()
        
         and(False(),True()) =  12     
                             >= 0      
                             =  False()
        
         and(True(),False()) =  20     
                             >= 0      
                             =  False()
        
          and(True(),True()) =  28     
                             >= 2      
                             =  True() 
        
        notEmpty(Cons(x,xs)) =  2 + xs 
                             >= 2      
                             =  True() 
        
** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            notEmpty(Cons(x,xs)) -> True()
        - Weak TRS:
            and(False(),False()) -> False()
            and(False(),True()) -> False()
            and(True(),False()) -> False()
            and(True(),True()) -> True()
            even(Cons(x,Nil())) -> False()
            even(Cons(x',Cons(x,xs))) -> even(xs)
            even(Nil()) -> True()
            goal(x,y) -> and(lte(x,y),even(x))
            lte(Cons(x,xs),Nil()) -> False()
            lte(Cons(x',xs'),Cons(x,xs)) -> lte(xs',xs)
            lte(Nil(),y) -> True()
            notEmpty(Nil()) -> False()
        - Signature:
            {and/2,even/1,goal/2,lte/2,notEmpty/1} / {Cons/2,False/0,Nil/0,True/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {and,even,goal,lte,notEmpty} and constructors {Cons,False
            ,Nil,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(and) = {1,2}
        
        Following symbols are considered usable:
          {and,even,goal,lte,notEmpty}
        TcT has computed the following interpretation:
              p(Cons) = 4        
             p(False) = 0        
               p(Nil) = 1        
              p(True) = 0        
               p(and) = 8*x1 + x2
              p(even) = 0        
              p(goal) = 0        
               p(lte) = 0        
          p(notEmpty) = 4*x1     
        
        Following rules are strictly oriented:
        notEmpty(Cons(x,xs)) = 16    
                             > 0     
                             = True()
        
        
        Following rules are (at-least) weakly oriented:
                and(False(),False()) =  0                    
                                     >= 0                    
                                     =  False()              
        
                 and(False(),True()) =  0                    
                                     >= 0                    
                                     =  False()              
        
                 and(True(),False()) =  0                    
                                     >= 0                    
                                     =  False()              
        
                  and(True(),True()) =  0                    
                                     >= 0                    
                                     =  True()               
        
                 even(Cons(x,Nil())) =  0                    
                                     >= 0                    
                                     =  False()              
        
           even(Cons(x',Cons(x,xs))) =  0                    
                                     >= 0                    
                                     =  even(xs)             
        
                         even(Nil()) =  0                    
                                     >= 0                    
                                     =  True()               
        
                           goal(x,y) =  0                    
                                     >= 0                    
                                     =  and(lte(x,y),even(x))
        
               lte(Cons(x,xs),Nil()) =  0                    
                                     >= 0                    
                                     =  False()              
        
        lte(Cons(x',xs'),Cons(x,xs)) =  0                    
                                     >= 0                    
                                     =  lte(xs',xs)          
        
                        lte(Nil(),y) =  0                    
                                     >= 0                    
                                     =  True()               
        
                     notEmpty(Nil()) =  4                    
                                     >= 0                    
                                     =  False()              
        
** Step 1.b:3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            and(False(),False()) -> False()
            and(False(),True()) -> False()
            and(True(),False()) -> False()
            and(True(),True()) -> True()
            even(Cons(x,Nil())) -> False()
            even(Cons(x',Cons(x,xs))) -> even(xs)
            even(Nil()) -> True()
            goal(x,y) -> and(lte(x,y),even(x))
            lte(Cons(x,xs),Nil()) -> False()
            lte(Cons(x',xs'),Cons(x,xs)) -> lte(xs',xs)
            lte(Nil(),y) -> True()
            notEmpty(Cons(x,xs)) -> True()
            notEmpty(Nil()) -> False()
        - Signature:
            {and/2,even/1,goal/2,lte/2,notEmpty/1} / {Cons/2,False/0,Nil/0,True/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {and,even,goal,lte,notEmpty} and constructors {Cons,False
            ,Nil,True}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(Omega(n^1),O(n^1))