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

** Step 1.b:1: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            goal(x) -> list(x)
            list(Cons(x,xs)) -> list(xs)
            list(Nil()) -> True()
            list(Nil()) -> isEmpty[Match](Nil())
            notEmpty(Cons(x,xs)) -> True()
            notEmpty(Nil()) -> False()
        - Signature:
            {goal/1,list/1,notEmpty/1} / {Cons/2,False/0,Nil/0,True/0,isEmpty[Match]/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal,list,notEmpty} and constructors {Cons,False,Nil,True
            ,isEmpty[Match]}
    + 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:
          {goal,list,notEmpty}
        TcT has computed the following interpretation:
                    p(Cons) = 0
                   p(False) = 0
                     p(Nil) = 0
                    p(True) = 0
                    p(goal) = 8
          p(isEmpty[Match]) = 0
                    p(list) = 0
                p(notEmpty) = 0
        
        Following rules are strictly oriented:
        goal(x) = 8      
                > 0      
                = list(x)
        
        
        Following rules are (at-least) weakly oriented:
            list(Cons(x,xs)) =  0                    
                             >= 0                    
                             =  list(xs)             
        
                 list(Nil()) =  0                    
                             >= 0                    
                             =  True()               
        
                 list(Nil()) =  0                    
                             >= 0                    
                             =  isEmpty[Match](Nil())
        
        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:
            list(Cons(x,xs)) -> list(xs)
            list(Nil()) -> True()
            list(Nil()) -> isEmpty[Match](Nil())
            notEmpty(Cons(x,xs)) -> True()
            notEmpty(Nil()) -> False()
        - Weak TRS:
            goal(x) -> list(x)
        - Signature:
            {goal/1,list/1,notEmpty/1} / {Cons/2,False/0,Nil/0,True/0,isEmpty[Match]/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal,list,notEmpty} and constructors {Cons,False,Nil,True
            ,isEmpty[Match]}
    + 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:
          {goal,list,notEmpty}
        TcT has computed the following interpretation:
                    p(Cons) = 1 + x2   
                   p(False) = 0        
                     p(Nil) = 0        
                    p(True) = 0        
                    p(goal) = 11 + 9*x1
          p(isEmpty[Match]) = 0        
                    p(list) = 10 + 9*x1
                p(notEmpty) = 13 + 4*x1
        
        Following rules are strictly oriented:
            list(Cons(x,xs)) = 19 + 9*xs            
                             > 10 + 9*xs            
                             = list(xs)             
        
                 list(Nil()) = 10                   
                             > 0                    
                             = True()               
        
                 list(Nil()) = 10                   
                             > 0                    
                             = isEmpty[Match](Nil())
        
        notEmpty(Cons(x,xs)) = 17 + 4*xs            
                             > 0                    
                             = True()               
        
             notEmpty(Nil()) = 13                   
                             > 0                    
                             = False()              
        
        
        Following rules are (at-least) weakly oriented:
        goal(x) =  11 + 9*x
                >= 10 + 9*x
                =  list(x) 
        
** Step 1.b:3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            goal(x) -> list(x)
            list(Cons(x,xs)) -> list(xs)
            list(Nil()) -> True()
            list(Nil()) -> isEmpty[Match](Nil())
            notEmpty(Cons(x,xs)) -> True()
            notEmpty(Nil()) -> False()
        - Signature:
            {goal/1,list/1,notEmpty/1} / {Cons/2,False/0,Nil/0,True/0,isEmpty[Match]/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal,list,notEmpty} and constructors {Cons,False,Nil,True
            ,isEmpty[Match]}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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