WORST_CASE(?,O(n^1))
* Step 1: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            add0(x,Nil()) -> x
            add0(x',Cons(x,xs)) -> add0(Cons(Cons(Nil(),Nil()),x'),xs)
            goal(x,y) -> add0(x,y)
            notEmpty(Cons(x,xs)) -> True()
            notEmpty(Nil()) -> False()
        - Signature:
            {add0/2,goal/2,notEmpty/1} / {Cons/2,False/0,Nil/0,True/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add0,goal,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:
          none
        
        Following symbols are considered usable:
          {add0,goal,notEmpty}
        TcT has computed the following interpretation:
              p(Cons) = x1 + x2 
             p(False) = 0       
               p(Nil) = 0       
              p(True) = 0       
              p(add0) = 2 + 8*x1
              p(goal) = 2 + 8*x1
          p(notEmpty) = 0       
        
        Following rules are strictly oriented:
        add0(x,Nil()) = 2 + 8*x
                      > x      
                      = x      
        
        
        Following rules are (at-least) weakly oriented:
         add0(x',Cons(x,xs)) =  2 + 8*x'                           
                             >= 2 + 8*x'                           
                             =  add0(Cons(Cons(Nil(),Nil()),x'),xs)
        
                   goal(x,y) =  2 + 8*x                            
                             >= 2 + 8*x                            
                             =  add0(x,y)                          
        
        notEmpty(Cons(x,xs)) =  0                                  
                             >= 0                                  
                             =  True()                             
        
             notEmpty(Nil()) =  0                                  
                             >= 0                                  
                             =  False()                            
        
* Step 2: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            add0(x',Cons(x,xs)) -> add0(Cons(Cons(Nil(),Nil()),x'),xs)
            goal(x,y) -> add0(x,y)
            notEmpty(Cons(x,xs)) -> True()
            notEmpty(Nil()) -> False()
        - Weak TRS:
            add0(x,Nil()) -> x
        - Signature:
            {add0/2,goal/2,notEmpty/1} / {Cons/2,False/0,Nil/0,True/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add0,goal,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:
          none
        
        Following symbols are considered usable:
          {add0,goal,notEmpty}
        TcT has computed the following interpretation:
              p(Cons) = [1] x1 + [0] 
             p(False) = [13]         
               p(Nil) = [0]          
              p(True) = [13]         
              p(add0) = [2] x1 + [7] 
              p(goal) = [8] x1 + [11]
          p(notEmpty) = [13]         
        
        Following rules are strictly oriented:
        goal(x,y) = [8] x + [11]
                  > [2] x + [7] 
                  = add0(x,y)   
        
        
        Following rules are (at-least) weakly oriented:
               add0(x,Nil()) =  [2] x + [7]                        
                             >= [1] x + [0]                        
                             =  x                                  
        
         add0(x',Cons(x,xs)) =  [2] x' + [7]                       
                             >= [7]                                
                             =  add0(Cons(Cons(Nil(),Nil()),x'),xs)
        
        notEmpty(Cons(x,xs)) =  [13]                               
                             >= [13]                               
                             =  True()                             
        
             notEmpty(Nil()) =  [13]                               
                             >= [13]                               
                             =  False()                            
        
* Step 3: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            add0(x',Cons(x,xs)) -> add0(Cons(Cons(Nil(),Nil()),x'),xs)
            notEmpty(Cons(x,xs)) -> True()
            notEmpty(Nil()) -> False()
        - Weak TRS:
            add0(x,Nil()) -> x
            goal(x,y) -> add0(x,y)
        - Signature:
            {add0/2,goal/2,notEmpty/1} / {Cons/2,False/0,Nil/0,True/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add0,goal,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:
          none
        
        Following symbols are considered usable:
          {add0,goal,notEmpty}
        TcT has computed the following interpretation:
              p(Cons) = [1] x2 + [8]          
             p(False) = [5]                   
               p(Nil) = [0]                   
              p(True) = [1]                   
              p(add0) = [1] x1 + [2] x2 + [8] 
              p(goal) = [1] x1 + [2] x2 + [12]
          p(notEmpty) = [15]                  
        
        Following rules are strictly oriented:
         add0(x',Cons(x,xs)) = [1] x' + [2] xs + [24]             
                             > [1] x' + [2] xs + [16]             
                             = add0(Cons(Cons(Nil(),Nil()),x'),xs)
        
        notEmpty(Cons(x,xs)) = [15]                               
                             > [1]                                
                             = True()                             
        
             notEmpty(Nil()) = [15]                               
                             > [5]                                
                             = False()                            
        
        
        Following rules are (at-least) weakly oriented:
        add0(x,Nil()) =  [1] x + [8]         
                      >= [1] x + [0]         
                      =  x                   
        
            goal(x,y) =  [1] x + [2] y + [12]
                      >= [1] x + [2] y + [8] 
                      =  add0(x,y)           
        
* Step 4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            add0(x,Nil()) -> x
            add0(x',Cons(x,xs)) -> add0(Cons(Cons(Nil(),Nil()),x'),xs)
            goal(x,y) -> add0(x,y)
            notEmpty(Cons(x,xs)) -> True()
            notEmpty(Nil()) -> False()
        - Signature:
            {add0/2,goal/2,notEmpty/1} / {Cons/2,False/0,Nil/0,True/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add0,goal,notEmpty} and constructors {Cons,False,Nil
            ,True}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))