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

** Step 1.b:1: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            duplicate(Cons(x,xs)) -> Cons(x,Cons(x,duplicate(xs)))
            duplicate(Nil()) -> Nil()
            goal(x) -> duplicate(x)
        - Signature:
            {duplicate/1,goal/1} / {Cons/2,Nil/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {duplicate,goal} and constructors {Cons,Nil}
    + 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(Cons) = {2}
        
        Following symbols are considered usable:
          {duplicate,goal}
        TcT has computed the following interpretation:
               p(Cons) = x2      
                p(Nil) = 0       
          p(duplicate) = 0       
               p(goal) = 5 + 2*x1
        
        Following rules are strictly oriented:
        goal(x) = 5 + 2*x     
                > 0           
                = duplicate(x)
        
        
        Following rules are (at-least) weakly oriented:
        duplicate(Cons(x,xs)) =  0                            
                              >= 0                            
                              =  Cons(x,Cons(x,duplicate(xs)))
        
             duplicate(Nil()) =  0                            
                              >= 0                            
                              =  Nil()                        
        
** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            duplicate(Cons(x,xs)) -> Cons(x,Cons(x,duplicate(xs)))
            duplicate(Nil()) -> Nil()
        - Weak TRS:
            goal(x) -> duplicate(x)
        - Signature:
            {duplicate/1,goal/1} / {Cons/2,Nil/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {duplicate,goal} and constructors {Cons,Nil}
    + 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(Cons) = {2}
        
        Following symbols are considered usable:
          {duplicate,goal}
        TcT has computed the following interpretation:
               p(Cons) = x2       
                p(Nil) = 2        
          p(duplicate) = 6        
               p(goal) = 10 + 2*x1
        
        Following rules are strictly oriented:
        duplicate(Nil()) = 6    
                         > 2    
                         = Nil()
        
        
        Following rules are (at-least) weakly oriented:
        duplicate(Cons(x,xs)) =  6                            
                              >= 6                            
                              =  Cons(x,Cons(x,duplicate(xs)))
        
                      goal(x) =  10 + 2*x                     
                              >= 6                            
                              =  duplicate(x)                 
        
** Step 1.b:3: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            duplicate(Cons(x,xs)) -> Cons(x,Cons(x,duplicate(xs)))
        - Weak TRS:
            duplicate(Nil()) -> Nil()
            goal(x) -> duplicate(x)
        - Signature:
            {duplicate/1,goal/1} / {Cons/2,Nil/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {duplicate,goal} and constructors {Cons,Nil}
    + 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(Cons) = {2}
        
        Following symbols are considered usable:
          {duplicate,goal}
        TcT has computed the following interpretation:
               p(Cons) = 1 + x2   
                p(Nil) = 0        
          p(duplicate) = 1 + 10*x1
               p(goal) = 8 + 12*x1
        
        Following rules are strictly oriented:
        duplicate(Cons(x,xs)) = 11 + 10*xs                   
                              > 3 + 10*xs                    
                              = Cons(x,Cons(x,duplicate(xs)))
        
        
        Following rules are (at-least) weakly oriented:
        duplicate(Nil()) =  1           
                         >= 0           
                         =  Nil()       
        
                 goal(x) =  8 + 12*x    
                         >= 1 + 10*x    
                         =  duplicate(x)
        
** Step 1.b:4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            duplicate(Cons(x,xs)) -> Cons(x,Cons(x,duplicate(xs)))
            duplicate(Nil()) -> Nil()
            goal(x) -> duplicate(x)
        - Signature:
            {duplicate/1,goal/1} / {Cons/2,Nil/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {duplicate,goal} and constructors {Cons,Nil}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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