WORST_CASE(?,O(n^1))
* Step 1: DependencyPairs 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:
        DependencyPairs {dpKind_ = WIDP}
    + Details:
        We add the following weak innermost dependency pairs:
        
        Strict DPs
          duplicate#(Cons(x,xs)) -> c_1(duplicate#(xs))
          duplicate#(Nil()) -> c_2()
          goal#(x) -> c_3(duplicate#(x))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            duplicate#(Cons(x,xs)) -> c_1(duplicate#(xs))
            duplicate#(Nil()) -> c_2()
            goal#(x) -> c_3(duplicate#(x))
        - Strict TRS:
            duplicate(Cons(x,xs)) -> Cons(x,Cons(x,duplicate(xs)))
            duplicate(Nil()) -> Nil()
            goal(x) -> duplicate(x)
        - Signature:
            {duplicate/1,goal/1,duplicate#/1,goal#/1} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {duplicate#,goal#} and constructors {Cons,Nil}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          duplicate#(Cons(x,xs)) -> c_1(duplicate#(xs))
          duplicate#(Nil()) -> c_2()
          goal#(x) -> c_3(duplicate#(x))
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            duplicate#(Cons(x,xs)) -> c_1(duplicate#(xs))
            duplicate#(Nil()) -> c_2()
            goal#(x) -> c_3(duplicate#(x))
        - Signature:
            {duplicate/1,goal/1,duplicate#/1,goal#/1} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {duplicate#,goal#} and constructors {Cons,Nil}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2}
        by application of
          Pre({2}) = {1,3}.
        Here rules are labelled as follows:
          1: duplicate#(Cons(x,xs)) -> c_1(duplicate#(xs))
          2: duplicate#(Nil()) -> c_2()
          3: goal#(x) -> c_3(duplicate#(x))
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            duplicate#(Cons(x,xs)) -> c_1(duplicate#(xs))
            goal#(x) -> c_3(duplicate#(x))
        - Weak DPs:
            duplicate#(Nil()) -> c_2()
        - Signature:
            {duplicate/1,goal/1,duplicate#/1,goal#/1} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {duplicate#,goal#} and constructors {Cons,Nil}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:duplicate#(Cons(x,xs)) -> c_1(duplicate#(xs))
             -->_1 duplicate#(Nil()) -> c_2():3
             -->_1 duplicate#(Cons(x,xs)) -> c_1(duplicate#(xs)):1
          
          2:S:goal#(x) -> c_3(duplicate#(x))
             -->_1 duplicate#(Nil()) -> c_2():3
             -->_1 duplicate#(Cons(x,xs)) -> c_1(duplicate#(xs)):1
          
          3:W:duplicate#(Nil()) -> c_2()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: duplicate#(Nil()) -> c_2()
* Step 5: RemoveHeads WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            duplicate#(Cons(x,xs)) -> c_1(duplicate#(xs))
            goal#(x) -> c_3(duplicate#(x))
        - Signature:
            {duplicate/1,goal/1,duplicate#/1,goal#/1} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {duplicate#,goal#} and constructors {Cons,Nil}
    + Applied Processor:
        RemoveHeads
    + Details:
        Consider the dependency graph
        
        1:S:duplicate#(Cons(x,xs)) -> c_1(duplicate#(xs))
           -->_1 duplicate#(Cons(x,xs)) -> c_1(duplicate#(xs)):1
        
        2:S:goal#(x) -> c_3(duplicate#(x))
           -->_1 duplicate#(Cons(x,xs)) -> c_1(duplicate#(xs)):1
        
        
        Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts).
        
        [(2,goal#(x) -> c_3(duplicate#(x)))]
* Step 6: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            duplicate#(Cons(x,xs)) -> c_1(duplicate#(xs))
        - Signature:
            {duplicate/1,goal/1,duplicate#/1,goal#/1} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {duplicate#,goal#} and constructors {Cons,Nil}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: duplicate#(Cons(x,xs)) -> c_1(duplicate#(xs))
          
        The strictly oriented rules are moved into the weak component.
** Step 6.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            duplicate#(Cons(x,xs)) -> c_1(duplicate#(xs))
        - Signature:
            {duplicate/1,goal/1,duplicate#/1,goal#/1} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {duplicate#,goal#} and constructors {Cons,Nil}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_1) = {1}
        
        Following symbols are considered usable:
          {duplicate#,goal#}
        TcT has computed the following interpretation:
                p(Cons) = [1] x2 + [8]
                 p(Nil) = [8]         
           p(duplicate) = [1]         
                p(goal) = [1] x1 + [1]
          p(duplicate#) = [1] x1 + [0]
               p(goal#) = [1] x1 + [8]
                 p(c_1) = [1] x1 + [0]
                 p(c_2) = [0]         
                 p(c_3) = [1] x1 + [0]
        
        Following rules are strictly oriented:
        duplicate#(Cons(x,xs)) = [1] xs + [8]       
                               > [1] xs + [0]       
                               = c_1(duplicate#(xs))
        
        
        Following rules are (at-least) weakly oriented:
        
** Step 6.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            duplicate#(Cons(x,xs)) -> c_1(duplicate#(xs))
        - Signature:
            {duplicate/1,goal/1,duplicate#/1,goal#/1} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {duplicate#,goal#} and constructors {Cons,Nil}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

** Step 6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            duplicate#(Cons(x,xs)) -> c_1(duplicate#(xs))
        - Signature:
            {duplicate/1,goal/1,duplicate#/1,goal#/1} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {duplicate#,goal#} and constructors {Cons,Nil}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:duplicate#(Cons(x,xs)) -> c_1(duplicate#(xs))
             -->_1 duplicate#(Cons(x,xs)) -> c_1(duplicate#(xs)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: duplicate#(Cons(x,xs)) -> c_1(duplicate#(xs))
** Step 6.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {duplicate/1,goal/1,duplicate#/1,goal#/1} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1}
        - 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(?,O(n^1))