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: Sum.  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:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:2: 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) = 10       
             p(False) = 0        
               p(Nil) = 1        
              p(True) = 0        
               p(and) = 4*x1 + x2
              p(even) = 4        
              p(goal) = 4        
               p(lte) = 0        
          p(notEmpty) = 8        
        
        Following rules are strictly oriented:
         even(Cons(x,Nil())) = 4      
                             > 0      
                             = False()
        
                 even(Nil()) = 4      
                             > 0      
                             = True() 
        
        notEmpty(Cons(x,xs)) = 8      
                             > 0      
                             = True() 
        
             notEmpty(Nil()) = 8      
                             > 0      
                             = False()
        
        
        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',Cons(x,xs))) =  4                    
                                     >= 4                    
                                     =  even(xs)             
        
                           goal(x,y) =  4                    
                                     >= 4                    
                                     =  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()               
        
** Step 1.b:2: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            even(Cons(x',Cons(x,xs))) -> even(xs)
            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()
        - Weak TRS:
            and(False(),False()) -> False()
            and(False(),True()) -> False()
            and(True(),False()) -> False()
            and(True(),True()) -> True()
            even(Cons(x,Nil())) -> False()
            even(Nil()) -> 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:
        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(and) = {1,2}
        
        Following symbols are considered usable:
          {and,even,goal,lte,notEmpty}
        TcT has computed the following interpretation:
              p(Cons) = [1] x2 + [4]         
             p(False) = [2]                  
               p(Nil) = [4]                  
              p(True) = [9]                  
               p(and) = [1] x1 + [1] x2 + [6]
              p(even) = [2] x1 + [1]         
              p(goal) = [6] x1 + [9]         
               p(lte) = [4] x1 + [0]         
          p(notEmpty) = [1] x1 + [12]        
        
        Following rules are strictly oriented:
           even(Cons(x',Cons(x,xs))) = [2] xs + [17]        
                                     > [2] xs + [1]         
                                     = even(xs)             
        
                           goal(x,y) = [6] x + [9]          
                                     > [6] x + [7]          
                                     = and(lte(x,y),even(x))
        
               lte(Cons(x,xs),Nil()) = [4] xs + [16]        
                                     > [2]                  
                                     = False()              
        
        lte(Cons(x',xs'),Cons(x,xs)) = [4] xs' + [16]       
                                     > [4] xs' + [0]        
                                     = lte(xs',xs)          
        
                        lte(Nil(),y) = [16]                 
                                     > [9]                  
                                     = True()               
        
        
        Following rules are (at-least) weakly oriented:
        and(False(),False()) =  [10]         
                             >= [2]          
                             =  False()      
        
         and(False(),True()) =  [17]         
                             >= [2]          
                             =  False()      
        
         and(True(),False()) =  [17]         
                             >= [2]          
                             =  False()      
        
          and(True(),True()) =  [24]         
                             >= [9]          
                             =  True()       
        
         even(Cons(x,Nil())) =  [17]         
                             >= [2]          
                             =  False()      
        
                 even(Nil()) =  [9]          
                             >= [9]          
                             =  True()       
        
        notEmpty(Cons(x,xs)) =  [1] xs + [16]
                             >= [9]          
                             =  True()       
        
             notEmpty(Nil()) =  [16]         
                             >= [2]          
                             =  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))