WORST_CASE(?,O(n^1))
* Step 1: InnermostRuleRemoval WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            2nd(cons(X,XS)) -> head(activate(XS))
            activate(X) -> X
            activate(n__from(X)) -> from(activate(X))
            activate(n__s(X)) -> s(activate(X))
            activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
            from(X) -> cons(X,n__from(n__s(X)))
            from(X) -> n__from(X)
            head(cons(X,XS)) -> X
            s(X) -> n__s(X)
            sel(0(),cons(X,XS)) -> X
            sel(s(N),cons(X,XS)) -> sel(N,activate(XS))
            take(X1,X2) -> n__take(X1,X2)
            take(0(),XS) -> nil()
            take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
        - Signature:
            {2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd,activate,from,head,s,sel,take} and constructors {0
            ,cons,n__from,n__s,n__take,nil}
    + Applied Processor:
        InnermostRuleRemoval
    + Details:
        Arguments of following rules are not normal-forms.
          sel(s(N),cons(X,XS)) -> sel(N,activate(XS))
          take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
        All above mentioned rules can be savely removed.
* Step 2: DependencyPairs WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            2nd(cons(X,XS)) -> head(activate(XS))
            activate(X) -> X
            activate(n__from(X)) -> from(activate(X))
            activate(n__s(X)) -> s(activate(X))
            activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
            from(X) -> cons(X,n__from(n__s(X)))
            from(X) -> n__from(X)
            head(cons(X,XS)) -> X
            s(X) -> n__s(X)
            sel(0(),cons(X,XS)) -> X
            take(X1,X2) -> n__take(X1,X2)
            take(0(),XS) -> nil()
        - Signature:
            {2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd,activate,from,head,s,sel,take} and constructors {0
            ,cons,n__from,n__s,n__take,nil}
    + Applied Processor:
        DependencyPairs {dpKind_ = WIDP}
    + Details:
        We add the following weak innermost dependency pairs:
        
        Strict DPs
          2nd#(cons(X,XS)) -> c_1(head#(activate(XS)))
          activate#(X) -> c_2()
          activate#(n__from(X)) -> c_3(from#(activate(X)))
          activate#(n__s(X)) -> c_4(s#(activate(X)))
          activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)))
          from#(X) -> c_6()
          from#(X) -> c_7()
          head#(cons(X,XS)) -> c_8()
          s#(X) -> c_9()
          sel#(0(),cons(X,XS)) -> c_10()
          take#(X1,X2) -> c_11()
          take#(0(),XS) -> c_12()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 3: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            2nd#(cons(X,XS)) -> c_1(head#(activate(XS)))
            activate#(X) -> c_2()
            activate#(n__from(X)) -> c_3(from#(activate(X)))
            activate#(n__s(X)) -> c_4(s#(activate(X)))
            activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)))
            from#(X) -> c_6()
            from#(X) -> c_7()
            head#(cons(X,XS)) -> c_8()
            s#(X) -> c_9()
            sel#(0(),cons(X,XS)) -> c_10()
            take#(X1,X2) -> c_11()
            take#(0(),XS) -> c_12()
        - Strict TRS:
            2nd(cons(X,XS)) -> head(activate(XS))
            activate(X) -> X
            activate(n__from(X)) -> from(activate(X))
            activate(n__s(X)) -> s(activate(X))
            activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
            from(X) -> cons(X,n__from(n__s(X)))
            from(X) -> n__from(X)
            head(cons(X,XS)) -> X
            s(X) -> n__s(X)
            sel(0(),cons(X,XS)) -> X
            take(X1,X2) -> n__take(X1,X2)
            take(0(),XS) -> nil()
        - Signature:
            {2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2
            ,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0
            ,c_9/0,c_10/0,c_11/0,c_12/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,s#,sel#
            ,take#} and constructors {0,cons,n__from,n__s,n__take,nil}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          activate(X) -> X
          activate(n__from(X)) -> from(activate(X))
          activate(n__s(X)) -> s(activate(X))
          activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
          from(X) -> cons(X,n__from(n__s(X)))
          from(X) -> n__from(X)
          s(X) -> n__s(X)
          take(X1,X2) -> n__take(X1,X2)
          take(0(),XS) -> nil()
          2nd#(cons(X,XS)) -> c_1(head#(activate(XS)))
          activate#(X) -> c_2()
          activate#(n__from(X)) -> c_3(from#(activate(X)))
          activate#(n__s(X)) -> c_4(s#(activate(X)))
          activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)))
          from#(X) -> c_6()
          from#(X) -> c_7()
          head#(cons(X,XS)) -> c_8()
          s#(X) -> c_9()
          sel#(0(),cons(X,XS)) -> c_10()
          take#(X1,X2) -> c_11()
          take#(0(),XS) -> c_12()
* Step 4: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            2nd#(cons(X,XS)) -> c_1(head#(activate(XS)))
            activate#(X) -> c_2()
            activate#(n__from(X)) -> c_3(from#(activate(X)))
            activate#(n__s(X)) -> c_4(s#(activate(X)))
            activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)))
            from#(X) -> c_6()
            from#(X) -> c_7()
            head#(cons(X,XS)) -> c_8()
            s#(X) -> c_9()
            sel#(0(),cons(X,XS)) -> c_10()
            take#(X1,X2) -> c_11()
            take#(0(),XS) -> c_12()
        - Strict TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(activate(X))
            activate(n__s(X)) -> s(activate(X))
            activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
            from(X) -> cons(X,n__from(n__s(X)))
            from(X) -> n__from(X)
            s(X) -> n__s(X)
            take(X1,X2) -> n__take(X1,X2)
            take(0(),XS) -> nil()
        - Signature:
            {2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2
            ,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0
            ,c_9/0,c_10/0,c_11/0,c_12/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,s#,sel#
            ,take#} and constructors {0,cons,n__from,n__s,n__take,nil}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnTrs}
    + Details:
        The weightgap principle applies using the following constant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(from) = {1},
            uargs(s) = {1},
            uargs(take) = {1,2},
            uargs(from#) = {1},
            uargs(head#) = {1},
            uargs(s#) = {1},
            uargs(take#) = {1,2},
            uargs(c_1) = {1},
            uargs(c_3) = {1},
            uargs(c_4) = {1},
            uargs(c_5) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                    p(0) = [5]                  
                  p(2nd) = [0]                  
             p(activate) = [4] x1 + [1]         
                 p(cons) = [1] x2 + [1]         
                 p(from) = [1] x1 + [8]         
                 p(head) = [1]                  
              p(n__from) = [1] x1 + [5]         
                 p(n__s) = [1] x1 + [1]         
              p(n__take) = [1] x1 + [1] x2 + [4]
                  p(nil) = [0]                  
                    p(s) = [1] x1 + [3]         
                  p(sel) = [0]                  
                 p(take) = [1] x1 + [1] x2 + [7]
                 p(2nd#) = [8] x1 + [0]         
            p(activate#) = [4] x1 + [9]         
                p(from#) = [1] x1 + [0]         
                p(head#) = [1] x1 + [0]         
                   p(s#) = [1] x1 + [0]         
                 p(sel#) = [0]                  
                p(take#) = [1] x1 + [1] x2 + [0]
                  p(c_1) = [1] x1 + [0]         
                  p(c_2) = [0]                  
                  p(c_3) = [1] x1 + [0]         
                  p(c_4) = [1] x1 + [0]         
                  p(c_5) = [1] x1 + [0]         
                  p(c_6) = [0]                  
                  p(c_7) = [0]                  
                  p(c_8) = [0]                  
                  p(c_9) = [0]                  
                 p(c_10) = [0]                  
                 p(c_11) = [0]                  
                 p(c_12) = [0]                  
          
          Following rules are strictly oriented:
                   2nd#(cons(X,XS)) = [8] XS + [8]                         
                                    > [4] XS + [1]                         
                                    = c_1(head#(activate(XS)))             
          
                       activate#(X) = [4] X + [9]                          
                                    > [0]                                  
                                    = c_2()                                
          
              activate#(n__from(X)) = [4] X + [29]                         
                                    > [4] X + [1]                          
                                    = c_3(from#(activate(X)))              
          
                 activate#(n__s(X)) = [4] X + [13]                         
                                    > [4] X + [1]                          
                                    = c_4(s#(activate(X)))                 
          
          activate#(n__take(X1,X2)) = [4] X1 + [4] X2 + [25]               
                                    > [4] X1 + [4] X2 + [2]                
                                    = c_5(take#(activate(X1),activate(X2)))
          
                  head#(cons(X,XS)) = [1] XS + [1]                         
                                    > [0]                                  
                                    = c_8()                                
          
                      take#(0(),XS) = [1] XS + [5]                         
                                    > [0]                                  
                                    = c_12()                               
          
                        activate(X) = [4] X + [1]                          
                                    > [1] X + [0]                          
                                    = X                                    
          
               activate(n__from(X)) = [4] X + [21]                         
                                    > [4] X + [9]                          
                                    = from(activate(X))                    
          
                  activate(n__s(X)) = [4] X + [5]                          
                                    > [4] X + [4]                          
                                    = s(activate(X))                       
          
           activate(n__take(X1,X2)) = [4] X1 + [4] X2 + [17]               
                                    > [4] X1 + [4] X2 + [9]                
                                    = take(activate(X1),activate(X2))      
          
                            from(X) = [1] X + [8]                          
                                    > [1] X + [7]                          
                                    = cons(X,n__from(n__s(X)))             
          
                            from(X) = [1] X + [8]                          
                                    > [1] X + [5]                          
                                    = n__from(X)                           
          
                               s(X) = [1] X + [3]                          
                                    > [1] X + [1]                          
                                    = n__s(X)                              
          
                        take(X1,X2) = [1] X1 + [1] X2 + [7]                
                                    > [1] X1 + [1] X2 + [4]                
                                    = n__take(X1,X2)                       
          
                       take(0(),XS) = [1] XS + [12]                        
                                    > [0]                                  
                                    = nil()                                
          
          
          Following rules are (at-least) weakly oriented:
                      from#(X) =  [1] X + [0]          
                               >= [0]                  
                               =  c_6()                
          
                      from#(X) =  [1] X + [0]          
                               >= [0]                  
                               =  c_7()                
          
                         s#(X) =  [1] X + [0]          
                               >= [0]                  
                               =  c_9()                
          
          sel#(0(),cons(X,XS)) =  [0]                  
                               >= [0]                  
                               =  c_10()               
          
                  take#(X1,X2) =  [1] X1 + [1] X2 + [0]
                               >= [0]                  
                               =  c_11()               
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 5: PredecessorEstimation WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            from#(X) -> c_6()
            from#(X) -> c_7()
            s#(X) -> c_9()
            sel#(0(),cons(X,XS)) -> c_10()
            take#(X1,X2) -> c_11()
        - Weak DPs:
            2nd#(cons(X,XS)) -> c_1(head#(activate(XS)))
            activate#(X) -> c_2()
            activate#(n__from(X)) -> c_3(from#(activate(X)))
            activate#(n__s(X)) -> c_4(s#(activate(X)))
            activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)))
            head#(cons(X,XS)) -> c_8()
            take#(0(),XS) -> c_12()
        - Weak TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(activate(X))
            activate(n__s(X)) -> s(activate(X))
            activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
            from(X) -> cons(X,n__from(n__s(X)))
            from(X) -> n__from(X)
            s(X) -> n__s(X)
            take(X1,X2) -> n__take(X1,X2)
            take(0(),XS) -> nil()
        - Signature:
            {2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2
            ,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0
            ,c_9/0,c_10/0,c_11/0,c_12/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,s#,sel#
            ,take#} and constructors {0,cons,n__from,n__s,n__take,nil}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {4}
        by application of
          Pre({4}) = {}.
        Here rules are labelled as follows:
          1: from#(X) -> c_6()
          2: from#(X) -> c_7()
          3: s#(X) -> c_9()
          4: sel#(0(),cons(X,XS)) -> c_10()
          5: take#(X1,X2) -> c_11()
          6: 2nd#(cons(X,XS)) -> c_1(head#(activate(XS)))
          7: activate#(X) -> c_2()
          8: activate#(n__from(X)) -> c_3(from#(activate(X)))
          9: activate#(n__s(X)) -> c_4(s#(activate(X)))
          10: activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)))
          11: head#(cons(X,XS)) -> c_8()
          12: take#(0(),XS) -> c_12()
* Step 6: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            from#(X) -> c_6()
            from#(X) -> c_7()
            s#(X) -> c_9()
            take#(X1,X2) -> c_11()
        - Weak DPs:
            2nd#(cons(X,XS)) -> c_1(head#(activate(XS)))
            activate#(X) -> c_2()
            activate#(n__from(X)) -> c_3(from#(activate(X)))
            activate#(n__s(X)) -> c_4(s#(activate(X)))
            activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)))
            head#(cons(X,XS)) -> c_8()
            sel#(0(),cons(X,XS)) -> c_10()
            take#(0(),XS) -> c_12()
        - Weak TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(activate(X))
            activate(n__s(X)) -> s(activate(X))
            activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
            from(X) -> cons(X,n__from(n__s(X)))
            from(X) -> n__from(X)
            s(X) -> n__s(X)
            take(X1,X2) -> n__take(X1,X2)
            take(0(),XS) -> nil()
        - Signature:
            {2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2
            ,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0
            ,c_9/0,c_10/0,c_11/0,c_12/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,s#,sel#
            ,take#} and constructors {0,cons,n__from,n__s,n__take,nil}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:from#(X) -> c_6()
             
          
          2:S:from#(X) -> c_7()
             
          
          3:S:s#(X) -> c_9()
             
          
          4:S:take#(X1,X2) -> c_11()
             
          
          5:W:2nd#(cons(X,XS)) -> c_1(head#(activate(XS)))
             -->_1 head#(cons(X,XS)) -> c_8():10
          
          6:W:activate#(X) -> c_2()
             
          
          7:W:activate#(n__from(X)) -> c_3(from#(activate(X)))
             -->_1 from#(X) -> c_7():2
             -->_1 from#(X) -> c_6():1
          
          8:W:activate#(n__s(X)) -> c_4(s#(activate(X)))
             -->_1 s#(X) -> c_9():3
          
          9:W:activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)))
             -->_1 take#(0(),XS) -> c_12():12
             -->_1 take#(X1,X2) -> c_11():4
          
          10:W:head#(cons(X,XS)) -> c_8()
             
          
          11:W:sel#(0(),cons(X,XS)) -> c_10()
             
          
          12:W:take#(0(),XS) -> c_12()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          11: sel#(0(),cons(X,XS)) -> c_10()
          12: take#(0(),XS) -> c_12()
          6: activate#(X) -> c_2()
          5: 2nd#(cons(X,XS)) -> c_1(head#(activate(XS)))
          10: head#(cons(X,XS)) -> c_8()
* Step 7: Decompose WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            from#(X) -> c_6()
            from#(X) -> c_7()
            s#(X) -> c_9()
            take#(X1,X2) -> c_11()
        - Weak DPs:
            activate#(n__from(X)) -> c_3(from#(activate(X)))
            activate#(n__s(X)) -> c_4(s#(activate(X)))
            activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)))
        - Weak TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(activate(X))
            activate(n__s(X)) -> s(activate(X))
            activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
            from(X) -> cons(X,n__from(n__s(X)))
            from(X) -> n__from(X)
            s(X) -> n__s(X)
            take(X1,X2) -> n__take(X1,X2)
            take(0(),XS) -> nil()
        - Signature:
            {2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2
            ,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0
            ,c_9/0,c_10/0,c_11/0,c_12/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,s#,sel#
            ,take#} and constructors {0,cons,n__from,n__s,n__take,nil}
    + Applied Processor:
        Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
    + Details:
        We analyse the complexity of following sub-problems (R) and (S).
        Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
        
        Problem (R)
          - Strict DPs:
              from#(X) -> c_6()
          - Weak DPs:
              activate#(n__from(X)) -> c_3(from#(activate(X)))
              activate#(n__s(X)) -> c_4(s#(activate(X)))
              activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)))
              from#(X) -> c_7()
              s#(X) -> c_9()
              take#(X1,X2) -> c_11()
          - Weak TRS:
              activate(X) -> X
              activate(n__from(X)) -> from(activate(X))
              activate(n__s(X)) -> s(activate(X))
              activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
              from(X) -> cons(X,n__from(n__s(X)))
              from(X) -> n__from(X)
              s(X) -> n__s(X)
              take(X1,X2) -> n__take(X1,X2)
              take(0(),XS) -> nil()
          - Signature:
              {2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2
              ,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0
              ,c_9/0,c_10/0,c_11/0,c_12/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,s#,sel#
              ,take#} and constructors {0,cons,n__from,n__s,n__take,nil}
        
        Problem (S)
          - Strict DPs:
              from#(X) -> c_7()
              s#(X) -> c_9()
              take#(X1,X2) -> c_11()
          - Weak DPs:
              activate#(n__from(X)) -> c_3(from#(activate(X)))
              activate#(n__s(X)) -> c_4(s#(activate(X)))
              activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)))
              from#(X) -> c_6()
          - Weak TRS:
              activate(X) -> X
              activate(n__from(X)) -> from(activate(X))
              activate(n__s(X)) -> s(activate(X))
              activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
              from(X) -> cons(X,n__from(n__s(X)))
              from(X) -> n__from(X)
              s(X) -> n__s(X)
              take(X1,X2) -> n__take(X1,X2)
              take(0(),XS) -> nil()
          - Signature:
              {2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2
              ,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0
              ,c_9/0,c_10/0,c_11/0,c_12/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,s#,sel#
              ,take#} and constructors {0,cons,n__from,n__s,n__take,nil}
** Step 7.a:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            from#(X) -> c_6()
        - Weak DPs:
            activate#(n__from(X)) -> c_3(from#(activate(X)))
            activate#(n__s(X)) -> c_4(s#(activate(X)))
            activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)))
            from#(X) -> c_7()
            s#(X) -> c_9()
            take#(X1,X2) -> c_11()
        - Weak TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(activate(X))
            activate(n__s(X)) -> s(activate(X))
            activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
            from(X) -> cons(X,n__from(n__s(X)))
            from(X) -> n__from(X)
            s(X) -> n__s(X)
            take(X1,X2) -> n__take(X1,X2)
            take(0(),XS) -> nil()
        - Signature:
            {2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2
            ,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0
            ,c_9/0,c_10/0,c_11/0,c_12/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,s#,sel#
            ,take#} and constructors {0,cons,n__from,n__s,n__take,nil}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:from#(X) -> c_6()
             
          
          2:W:from#(X) -> c_7()
             
          
          3:W:s#(X) -> c_9()
             
          
          4:W:take#(X1,X2) -> c_11()
             
          
          7:W:activate#(n__from(X)) -> c_3(from#(activate(X)))
             -->_1 from#(X) -> c_6():1
             -->_1 from#(X) -> c_7():2
          
          8:W:activate#(n__s(X)) -> c_4(s#(activate(X)))
             -->_1 s#(X) -> c_9():3
          
          9:W:activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)))
             -->_1 take#(X1,X2) -> c_11():4
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          9: activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)))
          8: activate#(n__s(X)) -> c_4(s#(activate(X)))
          4: take#(X1,X2) -> c_11()
          3: s#(X) -> c_9()
          2: from#(X) -> c_7()
** Step 7.a:2: Trivial WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            from#(X) -> c_6()
        - Weak DPs:
            activate#(n__from(X)) -> c_3(from#(activate(X)))
        - Weak TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(activate(X))
            activate(n__s(X)) -> s(activate(X))
            activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
            from(X) -> cons(X,n__from(n__s(X)))
            from(X) -> n__from(X)
            s(X) -> n__s(X)
            take(X1,X2) -> n__take(X1,X2)
            take(0(),XS) -> nil()
        - Signature:
            {2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2
            ,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0
            ,c_9/0,c_10/0,c_11/0,c_12/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,s#,sel#
            ,take#} and constructors {0,cons,n__from,n__s,n__take,nil}
    + Applied Processor:
        Trivial
    + Details:
        Consider the dependency graph
          1:S:from#(X) -> c_6()
             
          
          7:W:activate#(n__from(X)) -> c_3(from#(activate(X)))
             -->_1 from#(X) -> c_6():1
          
        The dependency graph contains no loops, we remove all dependency pairs.
** Step 7.a:3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(activate(X))
            activate(n__s(X)) -> s(activate(X))
            activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
            from(X) -> cons(X,n__from(n__s(X)))
            from(X) -> n__from(X)
            s(X) -> n__s(X)
            take(X1,X2) -> n__take(X1,X2)
            take(0(),XS) -> nil()
        - Signature:
            {2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2
            ,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0
            ,c_9/0,c_10/0,c_11/0,c_12/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,s#,sel#
            ,take#} and constructors {0,cons,n__from,n__s,n__take,nil}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

** Step 7.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            from#(X) -> c_7()
            s#(X) -> c_9()
            take#(X1,X2) -> c_11()
        - Weak DPs:
            activate#(n__from(X)) -> c_3(from#(activate(X)))
            activate#(n__s(X)) -> c_4(s#(activate(X)))
            activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)))
            from#(X) -> c_6()
        - Weak TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(activate(X))
            activate(n__s(X)) -> s(activate(X))
            activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
            from(X) -> cons(X,n__from(n__s(X)))
            from(X) -> n__from(X)
            s(X) -> n__s(X)
            take(X1,X2) -> n__take(X1,X2)
            take(0(),XS) -> nil()
        - Signature:
            {2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2
            ,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0
            ,c_9/0,c_10/0,c_11/0,c_12/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,s#,sel#
            ,take#} and constructors {0,cons,n__from,n__s,n__take,nil}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:from#(X) -> c_7()
             
          
          2:S:s#(X) -> c_9()
             
          
          3:S:take#(X1,X2) -> c_11()
             
          
          4:W:activate#(n__from(X)) -> c_3(from#(activate(X)))
             -->_1 from#(X) -> c_6():7
             -->_1 from#(X) -> c_7():1
          
          5:W:activate#(n__s(X)) -> c_4(s#(activate(X)))
             -->_1 s#(X) -> c_9():2
          
          6:W:activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)))
             -->_1 take#(X1,X2) -> c_11():3
          
          7:W:from#(X) -> c_6()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          7: from#(X) -> c_6()
** Step 7.b:2: Decompose WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            from#(X) -> c_7()
            s#(X) -> c_9()
            take#(X1,X2) -> c_11()
        - Weak DPs:
            activate#(n__from(X)) -> c_3(from#(activate(X)))
            activate#(n__s(X)) -> c_4(s#(activate(X)))
            activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)))
        - Weak TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(activate(X))
            activate(n__s(X)) -> s(activate(X))
            activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
            from(X) -> cons(X,n__from(n__s(X)))
            from(X) -> n__from(X)
            s(X) -> n__s(X)
            take(X1,X2) -> n__take(X1,X2)
            take(0(),XS) -> nil()
        - Signature:
            {2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2
            ,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0
            ,c_9/0,c_10/0,c_11/0,c_12/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,s#,sel#
            ,take#} and constructors {0,cons,n__from,n__s,n__take,nil}
    + Applied Processor:
        Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
    + Details:
        We analyse the complexity of following sub-problems (R) and (S).
        Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
        
        Problem (R)
          - Strict DPs:
              from#(X) -> c_7()
          - Weak DPs:
              activate#(n__from(X)) -> c_3(from#(activate(X)))
              activate#(n__s(X)) -> c_4(s#(activate(X)))
              activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)))
              s#(X) -> c_9()
              take#(X1,X2) -> c_11()
          - Weak TRS:
              activate(X) -> X
              activate(n__from(X)) -> from(activate(X))
              activate(n__s(X)) -> s(activate(X))
              activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
              from(X) -> cons(X,n__from(n__s(X)))
              from(X) -> n__from(X)
              s(X) -> n__s(X)
              take(X1,X2) -> n__take(X1,X2)
              take(0(),XS) -> nil()
          - Signature:
              {2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2
              ,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0
              ,c_9/0,c_10/0,c_11/0,c_12/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,s#,sel#
              ,take#} and constructors {0,cons,n__from,n__s,n__take,nil}
        
        Problem (S)
          - Strict DPs:
              s#(X) -> c_9()
              take#(X1,X2) -> c_11()
          - Weak DPs:
              activate#(n__from(X)) -> c_3(from#(activate(X)))
              activate#(n__s(X)) -> c_4(s#(activate(X)))
              activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)))
              from#(X) -> c_7()
          - Weak TRS:
              activate(X) -> X
              activate(n__from(X)) -> from(activate(X))
              activate(n__s(X)) -> s(activate(X))
              activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
              from(X) -> cons(X,n__from(n__s(X)))
              from(X) -> n__from(X)
              s(X) -> n__s(X)
              take(X1,X2) -> n__take(X1,X2)
              take(0(),XS) -> nil()
          - Signature:
              {2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2
              ,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0
              ,c_9/0,c_10/0,c_11/0,c_12/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,s#,sel#
              ,take#} and constructors {0,cons,n__from,n__s,n__take,nil}
*** Step 7.b:2.a:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            from#(X) -> c_7()
        - Weak DPs:
            activate#(n__from(X)) -> c_3(from#(activate(X)))
            activate#(n__s(X)) -> c_4(s#(activate(X)))
            activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)))
            s#(X) -> c_9()
            take#(X1,X2) -> c_11()
        - Weak TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(activate(X))
            activate(n__s(X)) -> s(activate(X))
            activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
            from(X) -> cons(X,n__from(n__s(X)))
            from(X) -> n__from(X)
            s(X) -> n__s(X)
            take(X1,X2) -> n__take(X1,X2)
            take(0(),XS) -> nil()
        - Signature:
            {2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2
            ,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0
            ,c_9/0,c_10/0,c_11/0,c_12/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,s#,sel#
            ,take#} and constructors {0,cons,n__from,n__s,n__take,nil}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:from#(X) -> c_7()
             
          
          2:W:s#(X) -> c_9()
             
          
          3:W:take#(X1,X2) -> c_11()
             
          
          4:W:activate#(n__from(X)) -> c_3(from#(activate(X)))
             -->_1 from#(X) -> c_7():1
          
          5:W:activate#(n__s(X)) -> c_4(s#(activate(X)))
             -->_1 s#(X) -> c_9():2
          
          6:W:activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)))
             -->_1 take#(X1,X2) -> c_11():3
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          6: activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)))
          5: activate#(n__s(X)) -> c_4(s#(activate(X)))
          3: take#(X1,X2) -> c_11()
          2: s#(X) -> c_9()
*** Step 7.b:2.a:2: Trivial WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            from#(X) -> c_7()
        - Weak DPs:
            activate#(n__from(X)) -> c_3(from#(activate(X)))
        - Weak TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(activate(X))
            activate(n__s(X)) -> s(activate(X))
            activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
            from(X) -> cons(X,n__from(n__s(X)))
            from(X) -> n__from(X)
            s(X) -> n__s(X)
            take(X1,X2) -> n__take(X1,X2)
            take(0(),XS) -> nil()
        - Signature:
            {2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2
            ,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0
            ,c_9/0,c_10/0,c_11/0,c_12/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,s#,sel#
            ,take#} and constructors {0,cons,n__from,n__s,n__take,nil}
    + Applied Processor:
        Trivial
    + Details:
        Consider the dependency graph
          1:S:from#(X) -> c_7()
             
          
          4:W:activate#(n__from(X)) -> c_3(from#(activate(X)))
             -->_1 from#(X) -> c_7():1
          
        The dependency graph contains no loops, we remove all dependency pairs.
*** Step 7.b:2.a:3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(activate(X))
            activate(n__s(X)) -> s(activate(X))
            activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
            from(X) -> cons(X,n__from(n__s(X)))
            from(X) -> n__from(X)
            s(X) -> n__s(X)
            take(X1,X2) -> n__take(X1,X2)
            take(0(),XS) -> nil()
        - Signature:
            {2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2
            ,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0
            ,c_9/0,c_10/0,c_11/0,c_12/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,s#,sel#
            ,take#} and constructors {0,cons,n__from,n__s,n__take,nil}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

*** Step 7.b:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            s#(X) -> c_9()
            take#(X1,X2) -> c_11()
        - Weak DPs:
            activate#(n__from(X)) -> c_3(from#(activate(X)))
            activate#(n__s(X)) -> c_4(s#(activate(X)))
            activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)))
            from#(X) -> c_7()
        - Weak TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(activate(X))
            activate(n__s(X)) -> s(activate(X))
            activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
            from(X) -> cons(X,n__from(n__s(X)))
            from(X) -> n__from(X)
            s(X) -> n__s(X)
            take(X1,X2) -> n__take(X1,X2)
            take(0(),XS) -> nil()
        - Signature:
            {2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2
            ,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0
            ,c_9/0,c_10/0,c_11/0,c_12/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,s#,sel#
            ,take#} and constructors {0,cons,n__from,n__s,n__take,nil}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:s#(X) -> c_9()
             
          
          2:S:take#(X1,X2) -> c_11()
             
          
          3:W:activate#(n__from(X)) -> c_3(from#(activate(X)))
             -->_1 from#(X) -> c_7():6
          
          4:W:activate#(n__s(X)) -> c_4(s#(activate(X)))
             -->_1 s#(X) -> c_9():1
          
          5:W:activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)))
             -->_1 take#(X1,X2) -> c_11():2
          
          6:W:from#(X) -> c_7()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: activate#(n__from(X)) -> c_3(from#(activate(X)))
          6: from#(X) -> c_7()
*** Step 7.b:2.b:2: Decompose WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            s#(X) -> c_9()
            take#(X1,X2) -> c_11()
        - Weak DPs:
            activate#(n__s(X)) -> c_4(s#(activate(X)))
            activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)))
        - Weak TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(activate(X))
            activate(n__s(X)) -> s(activate(X))
            activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
            from(X) -> cons(X,n__from(n__s(X)))
            from(X) -> n__from(X)
            s(X) -> n__s(X)
            take(X1,X2) -> n__take(X1,X2)
            take(0(),XS) -> nil()
        - Signature:
            {2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2
            ,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0
            ,c_9/0,c_10/0,c_11/0,c_12/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,s#,sel#
            ,take#} and constructors {0,cons,n__from,n__s,n__take,nil}
    + Applied Processor:
        Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
    + Details:
        We analyse the complexity of following sub-problems (R) and (S).
        Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
        
        Problem (R)
          - Strict DPs:
              s#(X) -> c_9()
          - Weak DPs:
              activate#(n__s(X)) -> c_4(s#(activate(X)))
              activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)))
              take#(X1,X2) -> c_11()
          - Weak TRS:
              activate(X) -> X
              activate(n__from(X)) -> from(activate(X))
              activate(n__s(X)) -> s(activate(X))
              activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
              from(X) -> cons(X,n__from(n__s(X)))
              from(X) -> n__from(X)
              s(X) -> n__s(X)
              take(X1,X2) -> n__take(X1,X2)
              take(0(),XS) -> nil()
          - Signature:
              {2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2
              ,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0
              ,c_9/0,c_10/0,c_11/0,c_12/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,s#,sel#
              ,take#} and constructors {0,cons,n__from,n__s,n__take,nil}
        
        Problem (S)
          - Strict DPs:
              take#(X1,X2) -> c_11()
          - Weak DPs:
              activate#(n__s(X)) -> c_4(s#(activate(X)))
              activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)))
              s#(X) -> c_9()
          - Weak TRS:
              activate(X) -> X
              activate(n__from(X)) -> from(activate(X))
              activate(n__s(X)) -> s(activate(X))
              activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
              from(X) -> cons(X,n__from(n__s(X)))
              from(X) -> n__from(X)
              s(X) -> n__s(X)
              take(X1,X2) -> n__take(X1,X2)
              take(0(),XS) -> nil()
          - Signature:
              {2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2
              ,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0
              ,c_9/0,c_10/0,c_11/0,c_12/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,s#,sel#
              ,take#} and constructors {0,cons,n__from,n__s,n__take,nil}
**** Step 7.b:2.b:2.a:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            s#(X) -> c_9()
        - Weak DPs:
            activate#(n__s(X)) -> c_4(s#(activate(X)))
            activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)))
            take#(X1,X2) -> c_11()
        - Weak TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(activate(X))
            activate(n__s(X)) -> s(activate(X))
            activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
            from(X) -> cons(X,n__from(n__s(X)))
            from(X) -> n__from(X)
            s(X) -> n__s(X)
            take(X1,X2) -> n__take(X1,X2)
            take(0(),XS) -> nil()
        - Signature:
            {2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2
            ,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0
            ,c_9/0,c_10/0,c_11/0,c_12/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,s#,sel#
            ,take#} and constructors {0,cons,n__from,n__s,n__take,nil}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:s#(X) -> c_9()
             
          
          2:W:take#(X1,X2) -> c_11()
             
          
          4:W:activate#(n__s(X)) -> c_4(s#(activate(X)))
             -->_1 s#(X) -> c_9():1
          
          5:W:activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)))
             -->_1 take#(X1,X2) -> c_11():2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          5: activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)))
          2: take#(X1,X2) -> c_11()
**** Step 7.b:2.b:2.a:2: Trivial WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            s#(X) -> c_9()
        - Weak DPs:
            activate#(n__s(X)) -> c_4(s#(activate(X)))
        - Weak TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(activate(X))
            activate(n__s(X)) -> s(activate(X))
            activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
            from(X) -> cons(X,n__from(n__s(X)))
            from(X) -> n__from(X)
            s(X) -> n__s(X)
            take(X1,X2) -> n__take(X1,X2)
            take(0(),XS) -> nil()
        - Signature:
            {2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2
            ,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0
            ,c_9/0,c_10/0,c_11/0,c_12/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,s#,sel#
            ,take#} and constructors {0,cons,n__from,n__s,n__take,nil}
    + Applied Processor:
        Trivial
    + Details:
        Consider the dependency graph
          1:S:s#(X) -> c_9()
             
          
          4:W:activate#(n__s(X)) -> c_4(s#(activate(X)))
             -->_1 s#(X) -> c_9():1
          
        The dependency graph contains no loops, we remove all dependency pairs.
**** Step 7.b:2.b:2.a:3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(activate(X))
            activate(n__s(X)) -> s(activate(X))
            activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
            from(X) -> cons(X,n__from(n__s(X)))
            from(X) -> n__from(X)
            s(X) -> n__s(X)
            take(X1,X2) -> n__take(X1,X2)
            take(0(),XS) -> nil()
        - Signature:
            {2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2
            ,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0
            ,c_9/0,c_10/0,c_11/0,c_12/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,s#,sel#
            ,take#} and constructors {0,cons,n__from,n__s,n__take,nil}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

**** Step 7.b:2.b:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            take#(X1,X2) -> c_11()
        - Weak DPs:
            activate#(n__s(X)) -> c_4(s#(activate(X)))
            activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)))
            s#(X) -> c_9()
        - Weak TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(activate(X))
            activate(n__s(X)) -> s(activate(X))
            activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
            from(X) -> cons(X,n__from(n__s(X)))
            from(X) -> n__from(X)
            s(X) -> n__s(X)
            take(X1,X2) -> n__take(X1,X2)
            take(0(),XS) -> nil()
        - Signature:
            {2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2
            ,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0
            ,c_9/0,c_10/0,c_11/0,c_12/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,s#,sel#
            ,take#} and constructors {0,cons,n__from,n__s,n__take,nil}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:take#(X1,X2) -> c_11()
             
          
          2:W:activate#(n__s(X)) -> c_4(s#(activate(X)))
             -->_1 s#(X) -> c_9():4
          
          3:W:activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)))
             -->_1 take#(X1,X2) -> c_11():1
          
          4:W:s#(X) -> c_9()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: activate#(n__s(X)) -> c_4(s#(activate(X)))
          4: s#(X) -> c_9()
**** Step 7.b:2.b:2.b:2: Trivial WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            take#(X1,X2) -> c_11()
        - Weak DPs:
            activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)))
        - Weak TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(activate(X))
            activate(n__s(X)) -> s(activate(X))
            activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
            from(X) -> cons(X,n__from(n__s(X)))
            from(X) -> n__from(X)
            s(X) -> n__s(X)
            take(X1,X2) -> n__take(X1,X2)
            take(0(),XS) -> nil()
        - Signature:
            {2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2
            ,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0
            ,c_9/0,c_10/0,c_11/0,c_12/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,s#,sel#
            ,take#} and constructors {0,cons,n__from,n__s,n__take,nil}
    + Applied Processor:
        Trivial
    + Details:
        Consider the dependency graph
          1:S:take#(X1,X2) -> c_11()
             
          
          3:W:activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)))
             -->_1 take#(X1,X2) -> c_11():1
          
        The dependency graph contains no loops, we remove all dependency pairs.
**** Step 7.b:2.b:2.b:3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(activate(X))
            activate(n__s(X)) -> s(activate(X))
            activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
            from(X) -> cons(X,n__from(n__s(X)))
            from(X) -> n__from(X)
            s(X) -> n__s(X)
            take(X1,X2) -> n__take(X1,X2)
            take(0(),XS) -> nil()
        - Signature:
            {2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2
            ,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0
            ,c_9/0,c_10/0,c_11/0,c_12/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,s#,sel#
            ,take#} and constructors {0,cons,n__from,n__s,n__take,nil}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))