* 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: MI 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:
        MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 1, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)):
        
        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] x_2 + [0]
                p(Nil) = [0]          
          p(duplicate) = [0]          
               p(goal) = [8]          
        
        Following rules are strictly oriented:
        goal(x) = [8]         
                > [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: WeightGap 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:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(Cons) = {2}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                 p(Cons) = [1] x2 + [0]
                  p(Nil) = [0]         
            p(duplicate) = [5]         
                 p(goal) = [5]         
          
          Following rules are strictly oriented:
          duplicate(Nil()) = [5]  
                           > [0]  
                           = Nil()
          
          
          Following rules are (at-least) weakly oriented:
          duplicate(Cons(x,xs)) =  [5]                          
                                >= [5]                          
                                =  Cons(x,Cons(x,duplicate(xs)))
          
                        goal(x) =  [5]                          
                                >= [5]                          
                                =  duplicate(x)                 
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:3: WeightGap 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:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(Cons) = {2}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                 p(Cons) = [1] x2 + [1]
                  p(Nil) = [10]        
            p(duplicate) = [3] x1 + [0]
                 p(goal) = [9] x1 + [0]
          
          Following rules are strictly oriented:
          duplicate(Cons(x,xs)) = [3] xs + [3]                 
                                > [3] xs + [2]                 
                                = Cons(x,Cons(x,duplicate(xs)))
          
          
          Following rules are (at-least) weakly oriented:
          duplicate(Nil()) =  [30]        
                           >= [10]        
                           =  Nil()       
          
                   goal(x) =  [9] x + [0] 
                           >= [3] x + [0] 
                           =  duplicate(x)
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** 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))