WORST_CASE(Omega(n^1),O(n^1))
* Step 1: Sum WORST_CASE(Omega(n^1),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:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(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:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          add0(x,z){z -> Cons(y,z)} =
            add0(x,Cons(y,z)) ->^+ add0(Cons(Cons(Nil(),Nil()),x),z)
              = C[add0(Cons(Cons(Nil(),Nil()),x),z) = add0(x,z){x -> Cons(Cons(Nil(),Nil()),x)}]

** Step 1.b: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 1.b:2: NaturalPI 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:
        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) = 4 + x2          
             p(False) = 0               
               p(Nil) = 1               
              p(True) = 0               
              p(add0) = 2 + 2*x1 + 4*x2 
              p(goal) = 11 + 3*x1 + 4*x2
          p(notEmpty) = 10 + x1         
        
        Following rules are strictly oriented:
         add0(x',Cons(x,xs)) = 18 + 2*x' + 4*xs                   
                             > 10 + 2*x' + 4*xs                   
                             = add0(Cons(Cons(Nil(),Nil()),x'),xs)
        
                   goal(x,y) = 11 + 3*x + 4*y                     
                             > 2 + 2*x + 4*y                      
                             = add0(x,y)                          
        
        notEmpty(Cons(x,xs)) = 14 + xs                            
                             > 0                                  
                             = True()                             
        
             notEmpty(Nil()) = 11                                 
                             > 0                                  
                             = False()                            
        
        
        Following rules are (at-least) weakly oriented:
        add0(x,Nil()) =  6 + 2*x
                      >= x      
                      =  x      
        
** Step 1.b:3: 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(Omega(n^1),O(n^1))