WORST_CASE(?,O(n^1))
* Step 1: 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(X)
            activate(n__take(X1,X2)) -> take(X1,X2)
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
            head(cons(X,XS)) -> 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,sel/2,take/2} / {0/0,cons/2,n__from/1,n__take/2,nil/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd,activate,from,head,sel,take} and constructors {0,cons
            ,n__from,n__take,nil,s}
    + 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#(X))
          activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
          from#(X) -> c_5()
          from#(X) -> c_6()
          head#(cons(X,XS)) -> c_7()
          sel#(0(),cons(X,XS)) -> c_8()
          sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)))
          take#(X1,X2) -> c_10()
          take#(0(),XS) -> c_11()
          take#(s(N),cons(X,XS)) -> c_12(activate#(XS))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: 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#(X))
            activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
            from#(X) -> c_5()
            from#(X) -> c_6()
            head#(cons(X,XS)) -> c_7()
            sel#(0(),cons(X,XS)) -> c_8()
            sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)))
            take#(X1,X2) -> c_10()
            take#(0(),XS) -> c_11()
            take#(s(N),cons(X,XS)) -> c_12(activate#(XS))
        - Strict TRS:
            2nd(cons(X,XS)) -> head(activate(XS))
            activate(X) -> X
            activate(n__from(X)) -> from(X)
            activate(n__take(X1,X2)) -> take(X1,X2)
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
            head(cons(X,XS)) -> 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,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0
            ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0
            ,c_12/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel#
            ,take#} and constructors {0,cons,n__from,n__take,nil,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          activate(X) -> X
          activate(n__from(X)) -> from(X)
          activate(n__take(X1,X2)) -> take(X1,X2)
          from(X) -> cons(X,n__from(s(X)))
          from(X) -> n__from(X)
          take(X1,X2) -> n__take(X1,X2)
          take(0(),XS) -> nil()
          take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
          2nd#(cons(X,XS)) -> c_1(head#(activate(XS)))
          activate#(X) -> c_2()
          activate#(n__from(X)) -> c_3(from#(X))
          activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
          from#(X) -> c_5()
          from#(X) -> c_6()
          head#(cons(X,XS)) -> c_7()
          sel#(0(),cons(X,XS)) -> c_8()
          sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)))
          take#(X1,X2) -> c_10()
          take#(0(),XS) -> c_11()
          take#(s(N),cons(X,XS)) -> c_12(activate#(XS))
* Step 3: 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#(X))
            activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
            from#(X) -> c_5()
            from#(X) -> c_6()
            head#(cons(X,XS)) -> c_7()
            sel#(0(),cons(X,XS)) -> c_8()
            sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)))
            take#(X1,X2) -> c_10()
            take#(0(),XS) -> c_11()
            take#(s(N),cons(X,XS)) -> c_12(activate#(XS))
        - Strict TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(X)
            activate(n__take(X1,X2)) -> take(X1,X2)
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
            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,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0
            ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0
            ,c_12/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel#
            ,take#} and constructors {0,cons,n__from,n__take,nil,s}
    + 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(cons) = {2},
            uargs(n__take) = {2},
            uargs(head#) = {1},
            uargs(sel#) = {2},
            uargs(c_1) = {1},
            uargs(c_3) = {1},
            uargs(c_4) = {1},
            uargs(c_9) = {1},
            uargs(c_12) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                    p(0) = [1]                  
                  p(2nd) = [0]                  
             p(activate) = [1] x1 + [3]         
                 p(cons) = [1] x2 + [1]         
                 p(from) = [7]                  
                 p(head) = [1] x1 + [1]         
              p(n__from) = [5]                  
              p(n__take) = [1] x1 + [1] x2 + [1]
                  p(nil) = [0]                  
                    p(s) = [1] x1 + [2]         
                  p(sel) = [0]                  
                 p(take) = [1] x1 + [1] x2 + [3]
                 p(2nd#) = [2] x1 + [0]         
            p(activate#) = [0]                  
                p(from#) = [0]                  
                p(head#) = [1] x1 + [0]         
                 p(sel#) = [5] x1 + [1] x2 + [0]
                p(take#) = [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) = [0]                  
                  p(c_6) = [0]                  
                  p(c_7) = [0]                  
                  p(c_8) = [0]                  
                  p(c_9) = [1] x1 + [0]         
                 p(c_10) = [0]                  
                 p(c_11) = [0]                  
                 p(c_12) = [1] x1 + [0]         
          
          Following rules are strictly oriented:
                 head#(cons(X,XS)) = [1] XS + [1]                   
                                   > [0]                            
                                   = c_7()                          
          
              sel#(0(),cons(X,XS)) = [1] XS + [6]                   
                                   > [0]                            
                                   = c_8()                          
          
             sel#(s(N),cons(X,XS)) = [5] N + [1] XS + [11]          
                                   > [5] N + [1] XS + [3]           
                                   = c_9(sel#(N,activate(XS)))      
          
                       activate(X) = [1] X + [3]                    
                                   > [1] X + [0]                    
                                   = X                              
          
              activate(n__from(X)) = [8]                            
                                   > [7]                            
                                   = from(X)                        
          
          activate(n__take(X1,X2)) = [1] X1 + [1] X2 + [4]          
                                   > [1] X1 + [1] X2 + [3]          
                                   = take(X1,X2)                    
          
                           from(X) = [7]                            
                                   > [6]                            
                                   = cons(X,n__from(s(X)))          
          
                           from(X) = [7]                            
                                   > [5]                            
                                   = n__from(X)                     
          
                       take(X1,X2) = [1] X1 + [1] X2 + [3]          
                                   > [1] X1 + [1] X2 + [1]          
                                   = n__take(X1,X2)                 
          
                      take(0(),XS) = [1] XS + [4]                   
                                   > [0]                            
                                   = nil()                          
          
             take(s(N),cons(X,XS)) = [1] N + [1] XS + [6]           
                                   > [1] N + [1] XS + [5]           
                                   = cons(X,n__take(N,activate(XS)))
          
          
          Following rules are (at-least) weakly oriented:
                   2nd#(cons(X,XS)) =  [2] XS + [2]            
                                    >= [1] XS + [3]            
                                    =  c_1(head#(activate(XS)))
          
                       activate#(X) =  [0]                     
                                    >= [0]                     
                                    =  c_2()                   
          
              activate#(n__from(X)) =  [0]                     
                                    >= [0]                     
                                    =  c_3(from#(X))           
          
          activate#(n__take(X1,X2)) =  [0]                     
                                    >= [0]                     
                                    =  c_4(take#(X1,X2))       
          
                           from#(X) =  [0]                     
                                    >= [0]                     
                                    =  c_5()                   
          
                           from#(X) =  [0]                     
                                    >= [0]                     
                                    =  c_6()                   
          
                       take#(X1,X2) =  [0]                     
                                    >= [0]                     
                                    =  c_10()                  
          
                      take#(0(),XS) =  [0]                     
                                    >= [0]                     
                                    =  c_11()                  
          
             take#(s(N),cons(X,XS)) =  [0]                     
                                    >= [0]                     
                                    =  c_12(activate#(XS))     
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: PredecessorEstimation 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#(X))
            activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
            from#(X) -> c_5()
            from#(X) -> c_6()
            take#(X1,X2) -> c_10()
            take#(0(),XS) -> c_11()
            take#(s(N),cons(X,XS)) -> c_12(activate#(XS))
        - Weak DPs:
            head#(cons(X,XS)) -> c_7()
            sel#(0(),cons(X,XS)) -> c_8()
            sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)))
        - Weak TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(X)
            activate(n__take(X1,X2)) -> take(X1,X2)
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
            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,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0
            ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0
            ,c_12/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel#
            ,take#} and constructors {0,cons,n__from,n__take,nil,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,2,5,6,7,8}
        by application of
          Pre({1,2,5,6,7,8}) = {3,4,9}.
        Here rules are labelled as follows:
          1: 2nd#(cons(X,XS)) -> c_1(head#(activate(XS)))
          2: activate#(X) -> c_2()
          3: activate#(n__from(X)) -> c_3(from#(X))
          4: activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
          5: from#(X) -> c_5()
          6: from#(X) -> c_6()
          7: take#(X1,X2) -> c_10()
          8: take#(0(),XS) -> c_11()
          9: take#(s(N),cons(X,XS)) -> c_12(activate#(XS))
          10: head#(cons(X,XS)) -> c_7()
          11: sel#(0(),cons(X,XS)) -> c_8()
          12: sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)))
* Step 5: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(n__from(X)) -> c_3(from#(X))
            activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
            take#(s(N),cons(X,XS)) -> c_12(activate#(XS))
        - Weak DPs:
            2nd#(cons(X,XS)) -> c_1(head#(activate(XS)))
            activate#(X) -> c_2()
            from#(X) -> c_5()
            from#(X) -> c_6()
            head#(cons(X,XS)) -> c_7()
            sel#(0(),cons(X,XS)) -> c_8()
            sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)))
            take#(X1,X2) -> c_10()
            take#(0(),XS) -> c_11()
        - Weak TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(X)
            activate(n__take(X1,X2)) -> take(X1,X2)
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
            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,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0
            ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0
            ,c_12/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel#
            ,take#} and constructors {0,cons,n__from,n__take,nil,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1}
        by application of
          Pre({1}) = {3}.
        Here rules are labelled as follows:
          1: activate#(n__from(X)) -> c_3(from#(X))
          2: activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
          3: take#(s(N),cons(X,XS)) -> c_12(activate#(XS))
          4: 2nd#(cons(X,XS)) -> c_1(head#(activate(XS)))
          5: activate#(X) -> c_2()
          6: from#(X) -> c_5()
          7: from#(X) -> c_6()
          8: head#(cons(X,XS)) -> c_7()
          9: sel#(0(),cons(X,XS)) -> c_8()
          10: sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)))
          11: take#(X1,X2) -> c_10()
          12: take#(0(),XS) -> c_11()
* Step 6: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
            take#(s(N),cons(X,XS)) -> c_12(activate#(XS))
        - Weak DPs:
            2nd#(cons(X,XS)) -> c_1(head#(activate(XS)))
            activate#(X) -> c_2()
            activate#(n__from(X)) -> c_3(from#(X))
            from#(X) -> c_5()
            from#(X) -> c_6()
            head#(cons(X,XS)) -> c_7()
            sel#(0(),cons(X,XS)) -> c_8()
            sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)))
            take#(X1,X2) -> c_10()
            take#(0(),XS) -> c_11()
        - Weak TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(X)
            activate(n__take(X1,X2)) -> take(X1,X2)
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
            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,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0
            ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0
            ,c_12/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel#
            ,take#} and constructors {0,cons,n__from,n__take,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
             -->_1 take#(s(N),cons(X,XS)) -> c_12(activate#(XS)):2
             -->_1 take#(0(),XS) -> c_11():12
             -->_1 take#(X1,X2) -> c_10():11
          
          2:S:take#(s(N),cons(X,XS)) -> c_12(activate#(XS))
             -->_1 activate#(n__from(X)) -> c_3(from#(X)):5
             -->_1 activate#(X) -> c_2():4
             -->_1 activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)):1
          
          3:W:2nd#(cons(X,XS)) -> c_1(head#(activate(XS)))
             -->_1 head#(cons(X,XS)) -> c_7():8
          
          4:W:activate#(X) -> c_2()
             
          
          5:W:activate#(n__from(X)) -> c_3(from#(X))
             -->_1 from#(X) -> c_6():7
             -->_1 from#(X) -> c_5():6
          
          6:W:from#(X) -> c_5()
             
          
          7:W:from#(X) -> c_6()
             
          
          8:W:head#(cons(X,XS)) -> c_7()
             
          
          9:W:sel#(0(),cons(X,XS)) -> c_8()
             
          
          10:W:sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)))
             -->_1 sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))):10
             -->_1 sel#(0(),cons(X,XS)) -> c_8():9
          
          11:W:take#(X1,X2) -> c_10()
             
          
          12:W:take#(0(),XS) -> c_11()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          10: sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)))
          9: sel#(0(),cons(X,XS)) -> c_8()
          3: 2nd#(cons(X,XS)) -> c_1(head#(activate(XS)))
          8: head#(cons(X,XS)) -> c_7()
          11: take#(X1,X2) -> c_10()
          12: take#(0(),XS) -> c_11()
          4: activate#(X) -> c_2()
          5: activate#(n__from(X)) -> c_3(from#(X))
          6: from#(X) -> c_5()
          7: from#(X) -> c_6()
* Step 7: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
            take#(s(N),cons(X,XS)) -> c_12(activate#(XS))
        - Weak TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(X)
            activate(n__take(X1,X2)) -> take(X1,X2)
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
            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,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0
            ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0
            ,c_12/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel#
            ,take#} and constructors {0,cons,n__from,n__take,nil,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
          take#(s(N),cons(X,XS)) -> c_12(activate#(XS))
* Step 8: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
            take#(s(N),cons(X,XS)) -> c_12(activate#(XS))
        - Signature:
            {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0
            ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0
            ,c_12/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel#
            ,take#} and constructors {0,cons,n__from,n__take,nil,s}
    + 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:
          2: take#(s(N),cons(X,XS)) -> c_12(activate#(XS))
          
        Consider the set of all dependency pairs
          1: activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
          2: take#(s(N),cons(X,XS)) -> c_12(activate#(XS))
        Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1))
        SPACE(?,?)on application of the dependency pairs
          {2}
        These cover all (indirect) predecessors of dependency pairs
          {1,2}
        their number of applications is equally bounded.
        The dependency pairs are shifted into the weak component.
** Step 8.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
            take#(s(N),cons(X,XS)) -> c_12(activate#(XS))
        - Signature:
            {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0
            ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0
            ,c_12/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel#
            ,take#} and constructors {0,cons,n__from,n__take,nil,s}
    + 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_4) = {1},
          uargs(c_12) = {1}
        
        Following symbols are considered usable:
          {2nd#,activate#,from#,head#,sel#,take#}
        TcT has computed the following interpretation:
                  p(0) = [0]                   
                p(2nd) = [2] x1 + [1]          
           p(activate) = [8] x1 + [1]          
               p(cons) = [1] x2 + [5]          
               p(from) = [1] x1 + [0]          
               p(head) = [8] x1 + [1]          
            p(n__from) = [1]                   
            p(n__take) = [1] x1 + [1] x2 + [10]
                p(nil) = [8]                   
                  p(s) = [1] x1 + [8]          
                p(sel) = [0]                   
               p(take) = [1]                   
               p(2nd#) = [1]                   
          p(activate#) = [1] x1 + [2]          
              p(from#) = [1] x1 + [1]          
              p(head#) = [1]                   
               p(sel#) = [1]                   
              p(take#) = [1] x1 + [1] x2 + [8] 
                p(c_1) = [8] x1 + [1]          
                p(c_2) = [0]                   
                p(c_3) = [1] x1 + [1]          
                p(c_4) = [1] x1 + [4]          
                p(c_5) = [0]                   
                p(c_6) = [1]                   
                p(c_7) = [1]                   
                p(c_8) = [1]                   
                p(c_9) = [1]                   
               p(c_10) = [2]                   
               p(c_11) = [0]                   
               p(c_12) = [1] x1 + [0]          
        
        Following rules are strictly oriented:
        take#(s(N),cons(X,XS)) = [1] N + [1] XS + [21]
                               > [1] XS + [2]         
                               = c_12(activate#(XS))  
        
        
        Following rules are (at-least) weakly oriented:
        activate#(n__take(X1,X2)) =  [1] X1 + [1] X2 + [12]
                                  >= [1] X1 + [1] X2 + [12]
                                  =  c_4(take#(X1,X2))     
        
** Step 8.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
        - Weak DPs:
            take#(s(N),cons(X,XS)) -> c_12(activate#(XS))
        - Signature:
            {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0
            ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0
            ,c_12/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel#
            ,take#} and constructors {0,cons,n__from,n__take,nil,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

** Step 8.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
            take#(s(N),cons(X,XS)) -> c_12(activate#(XS))
        - Signature:
            {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0
            ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0
            ,c_12/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel#
            ,take#} and constructors {0,cons,n__from,n__take,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
             -->_1 take#(s(N),cons(X,XS)) -> c_12(activate#(XS)):2
          
          2:W:take#(s(N),cons(X,XS)) -> c_12(activate#(XS))
             -->_1 activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: activate#(n__take(X1,X2)) -> c_4(take#(X1,X2))
          2: take#(s(N),cons(X,XS)) -> c_12(activate#(XS))
** Step 8.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0
            ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0
            ,c_12/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel#
            ,take#} and constructors {0,cons,n__from,n__take,nil,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))