* Step 1: Sum WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            activate(X) -> X
            activate(n__natsFrom(X)) -> natsFrom(X)
            afterNth(N,XS) -> snd(splitAt(N,XS))
            fst(pair(XS,YS)) -> XS
            head(cons(N,XS)) -> N
            natsFrom(N) -> cons(N,n__natsFrom(s(N)))
            natsFrom(X) -> n__natsFrom(X)
            sel(N,XS) -> head(afterNth(N,XS))
            snd(pair(XS,YS)) -> YS
            splitAt(0(),XS) -> pair(nil(),XS)
            splitAt(s(N),cons(X,XS)) -> u(splitAt(N,activate(XS)),N,X,activate(XS))
            tail(cons(N,XS)) -> activate(XS)
            take(N,XS) -> fst(splitAt(N,XS))
            u(pair(YS,ZS),N,X,XS) -> pair(cons(activate(X),YS),ZS)
        - Signature:
            {activate/1,afterNth/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4} / {0/0,cons/2
            ,n__natsFrom/1,nil/0,pair/2,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,afterNth,fst,head,natsFrom,sel,snd,splitAt,tail
            ,take,u} and constructors {0,cons,n__natsFrom,nil,pair,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 2: DependencyPairs WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            activate(X) -> X
            activate(n__natsFrom(X)) -> natsFrom(X)
            afterNth(N,XS) -> snd(splitAt(N,XS))
            fst(pair(XS,YS)) -> XS
            head(cons(N,XS)) -> N
            natsFrom(N) -> cons(N,n__natsFrom(s(N)))
            natsFrom(X) -> n__natsFrom(X)
            sel(N,XS) -> head(afterNth(N,XS))
            snd(pair(XS,YS)) -> YS
            splitAt(0(),XS) -> pair(nil(),XS)
            splitAt(s(N),cons(X,XS)) -> u(splitAt(N,activate(XS)),N,X,activate(XS))
            tail(cons(N,XS)) -> activate(XS)
            take(N,XS) -> fst(splitAt(N,XS))
            u(pair(YS,ZS),N,X,XS) -> pair(cons(activate(X),YS),ZS)
        - Signature:
            {activate/1,afterNth/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4} / {0/0,cons/2
            ,n__natsFrom/1,nil/0,pair/2,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,afterNth,fst,head,natsFrom,sel,snd,splitAt,tail
            ,take,u} and constructors {0,cons,n__natsFrom,nil,pair,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          activate#(X) -> c_1()
          activate#(n__natsFrom(X)) -> c_2(natsFrom#(X))
          afterNth#(N,XS) -> c_3(snd#(splitAt(N,XS)),splitAt#(N,XS))
          fst#(pair(XS,YS)) -> c_4()
          head#(cons(N,XS)) -> c_5()
          natsFrom#(N) -> c_6()
          natsFrom#(X) -> c_7()
          sel#(N,XS) -> c_8(head#(afterNth(N,XS)),afterNth#(N,XS))
          snd#(pair(XS,YS)) -> c_9()
          splitAt#(0(),XS) -> c_10()
          splitAt#(s(N),cons(X,XS)) -> c_11(u#(splitAt(N,activate(XS)),N,X,activate(XS))
                                           ,splitAt#(N,activate(XS))
                                           ,activate#(XS)
                                           ,activate#(XS))
          tail#(cons(N,XS)) -> c_12(activate#(XS))
          take#(N,XS) -> c_13(fst#(splitAt(N,XS)),splitAt#(N,XS))
          u#(pair(YS,ZS),N,X,XS) -> c_14(activate#(X))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 3: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(X) -> c_1()
            activate#(n__natsFrom(X)) -> c_2(natsFrom#(X))
            afterNth#(N,XS) -> c_3(snd#(splitAt(N,XS)),splitAt#(N,XS))
            fst#(pair(XS,YS)) -> c_4()
            head#(cons(N,XS)) -> c_5()
            natsFrom#(N) -> c_6()
            natsFrom#(X) -> c_7()
            sel#(N,XS) -> c_8(head#(afterNth(N,XS)),afterNth#(N,XS))
            snd#(pair(XS,YS)) -> c_9()
            splitAt#(0(),XS) -> c_10()
            splitAt#(s(N),cons(X,XS)) -> c_11(u#(splitAt(N,activate(XS)),N,X,activate(XS))
                                             ,splitAt#(N,activate(XS))
                                             ,activate#(XS)
                                             ,activate#(XS))
            tail#(cons(N,XS)) -> c_12(activate#(XS))
            take#(N,XS) -> c_13(fst#(splitAt(N,XS)),splitAt#(N,XS))
            u#(pair(YS,ZS),N,X,XS) -> c_14(activate#(X))
        - Weak TRS:
            activate(X) -> X
            activate(n__natsFrom(X)) -> natsFrom(X)
            afterNth(N,XS) -> snd(splitAt(N,XS))
            fst(pair(XS,YS)) -> XS
            head(cons(N,XS)) -> N
            natsFrom(N) -> cons(N,n__natsFrom(s(N)))
            natsFrom(X) -> n__natsFrom(X)
            sel(N,XS) -> head(afterNth(N,XS))
            snd(pair(XS,YS)) -> YS
            splitAt(0(),XS) -> pair(nil(),XS)
            splitAt(s(N),cons(X,XS)) -> u(splitAt(N,activate(XS)),N,X,activate(XS))
            tail(cons(N,XS)) -> activate(XS)
            take(N,XS) -> fst(splitAt(N,XS))
            u(pair(YS,ZS),N,X,XS) -> pair(cons(activate(X),YS),ZS)
        - Signature:
            {activate/1,afterNth/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1
            ,afterNth#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2
            ,n__natsFrom/1,nil/0,pair/2,s/1,c_1/0,c_2/1,c_3/2,c_4/0,c_5/0,c_6/0,c_7/0,c_8/2,c_9/0,c_10/0,c_11/4,c_12/1
            ,c_13/2,c_14/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,afterNth#,fst#,head#,natsFrom#,sel#,snd#
            ,splitAt#,tail#,take#,u#} and constructors {0,cons,n__natsFrom,nil,pair,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          activate(X) -> X
          activate(n__natsFrom(X)) -> natsFrom(X)
          afterNth(N,XS) -> snd(splitAt(N,XS))
          natsFrom(N) -> cons(N,n__natsFrom(s(N)))
          natsFrom(X) -> n__natsFrom(X)
          snd(pair(XS,YS)) -> YS
          splitAt(0(),XS) -> pair(nil(),XS)
          splitAt(s(N),cons(X,XS)) -> u(splitAt(N,activate(XS)),N,X,activate(XS))
          u(pair(YS,ZS),N,X,XS) -> pair(cons(activate(X),YS),ZS)
          activate#(X) -> c_1()
          activate#(n__natsFrom(X)) -> c_2(natsFrom#(X))
          afterNth#(N,XS) -> c_3(snd#(splitAt(N,XS)),splitAt#(N,XS))
          fst#(pair(XS,YS)) -> c_4()
          head#(cons(N,XS)) -> c_5()
          natsFrom#(N) -> c_6()
          natsFrom#(X) -> c_7()
          sel#(N,XS) -> c_8(head#(afterNth(N,XS)),afterNth#(N,XS))
          snd#(pair(XS,YS)) -> c_9()
          splitAt#(0(),XS) -> c_10()
          splitAt#(s(N),cons(X,XS)) -> c_11(u#(splitAt(N,activate(XS)),N,X,activate(XS))
                                           ,splitAt#(N,activate(XS))
                                           ,activate#(XS)
                                           ,activate#(XS))
          tail#(cons(N,XS)) -> c_12(activate#(XS))
          take#(N,XS) -> c_13(fst#(splitAt(N,XS)),splitAt#(N,XS))
          u#(pair(YS,ZS),N,X,XS) -> c_14(activate#(X))
* Step 4: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(X) -> c_1()
            activate#(n__natsFrom(X)) -> c_2(natsFrom#(X))
            afterNth#(N,XS) -> c_3(snd#(splitAt(N,XS)),splitAt#(N,XS))
            fst#(pair(XS,YS)) -> c_4()
            head#(cons(N,XS)) -> c_5()
            natsFrom#(N) -> c_6()
            natsFrom#(X) -> c_7()
            sel#(N,XS) -> c_8(head#(afterNth(N,XS)),afterNth#(N,XS))
            snd#(pair(XS,YS)) -> c_9()
            splitAt#(0(),XS) -> c_10()
            splitAt#(s(N),cons(X,XS)) -> c_11(u#(splitAt(N,activate(XS)),N,X,activate(XS))
                                             ,splitAt#(N,activate(XS))
                                             ,activate#(XS)
                                             ,activate#(XS))
            tail#(cons(N,XS)) -> c_12(activate#(XS))
            take#(N,XS) -> c_13(fst#(splitAt(N,XS)),splitAt#(N,XS))
            u#(pair(YS,ZS),N,X,XS) -> c_14(activate#(X))
        - Weak TRS:
            activate(X) -> X
            activate(n__natsFrom(X)) -> natsFrom(X)
            afterNth(N,XS) -> snd(splitAt(N,XS))
            natsFrom(N) -> cons(N,n__natsFrom(s(N)))
            natsFrom(X) -> n__natsFrom(X)
            snd(pair(XS,YS)) -> YS
            splitAt(0(),XS) -> pair(nil(),XS)
            splitAt(s(N),cons(X,XS)) -> u(splitAt(N,activate(XS)),N,X,activate(XS))
            u(pair(YS,ZS),N,X,XS) -> pair(cons(activate(X),YS),ZS)
        - Signature:
            {activate/1,afterNth/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1
            ,afterNth#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2
            ,n__natsFrom/1,nil/0,pair/2,s/1,c_1/0,c_2/1,c_3/2,c_4/0,c_5/0,c_6/0,c_7/0,c_8/2,c_9/0,c_10/0,c_11/4,c_12/1
            ,c_13/2,c_14/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,afterNth#,fst#,head#,natsFrom#,sel#,snd#
            ,splitAt#,tail#,take#,u#} and constructors {0,cons,n__natsFrom,nil,pair,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,4,5,6,7,9,10}
        by application of
          Pre({1,4,5,6,7,9,10}) = {2,3,8,11,12,13,14}.
        Here rules are labelled as follows:
          1: activate#(X) -> c_1()
          2: activate#(n__natsFrom(X)) -> c_2(natsFrom#(X))
          3: afterNth#(N,XS) -> c_3(snd#(splitAt(N,XS)),splitAt#(N,XS))
          4: fst#(pair(XS,YS)) -> c_4()
          5: head#(cons(N,XS)) -> c_5()
          6: natsFrom#(N) -> c_6()
          7: natsFrom#(X) -> c_7()
          8: sel#(N,XS) -> c_8(head#(afterNth(N,XS)),afterNth#(N,XS))
          9: snd#(pair(XS,YS)) -> c_9()
          10: splitAt#(0(),XS) -> c_10()
          11: splitAt#(s(N),cons(X,XS)) -> c_11(u#(splitAt(N,activate(XS)),N,X,activate(XS))
                                               ,splitAt#(N,activate(XS))
                                               ,activate#(XS)
                                               ,activate#(XS))
          12: tail#(cons(N,XS)) -> c_12(activate#(XS))
          13: take#(N,XS) -> c_13(fst#(splitAt(N,XS)),splitAt#(N,XS))
          14: u#(pair(YS,ZS),N,X,XS) -> c_14(activate#(X))
* Step 5: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(n__natsFrom(X)) -> c_2(natsFrom#(X))
            afterNth#(N,XS) -> c_3(snd#(splitAt(N,XS)),splitAt#(N,XS))
            sel#(N,XS) -> c_8(head#(afterNth(N,XS)),afterNth#(N,XS))
            splitAt#(s(N),cons(X,XS)) -> c_11(u#(splitAt(N,activate(XS)),N,X,activate(XS))
                                             ,splitAt#(N,activate(XS))
                                             ,activate#(XS)
                                             ,activate#(XS))
            tail#(cons(N,XS)) -> c_12(activate#(XS))
            take#(N,XS) -> c_13(fst#(splitAt(N,XS)),splitAt#(N,XS))
            u#(pair(YS,ZS),N,X,XS) -> c_14(activate#(X))
        - Weak DPs:
            activate#(X) -> c_1()
            fst#(pair(XS,YS)) -> c_4()
            head#(cons(N,XS)) -> c_5()
            natsFrom#(N) -> c_6()
            natsFrom#(X) -> c_7()
            snd#(pair(XS,YS)) -> c_9()
            splitAt#(0(),XS) -> c_10()
        - Weak TRS:
            activate(X) -> X
            activate(n__natsFrom(X)) -> natsFrom(X)
            afterNth(N,XS) -> snd(splitAt(N,XS))
            natsFrom(N) -> cons(N,n__natsFrom(s(N)))
            natsFrom(X) -> n__natsFrom(X)
            snd(pair(XS,YS)) -> YS
            splitAt(0(),XS) -> pair(nil(),XS)
            splitAt(s(N),cons(X,XS)) -> u(splitAt(N,activate(XS)),N,X,activate(XS))
            u(pair(YS,ZS),N,X,XS) -> pair(cons(activate(X),YS),ZS)
        - Signature:
            {activate/1,afterNth/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1
            ,afterNth#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2
            ,n__natsFrom/1,nil/0,pair/2,s/1,c_1/0,c_2/1,c_3/2,c_4/0,c_5/0,c_6/0,c_7/0,c_8/2,c_9/0,c_10/0,c_11/4,c_12/1
            ,c_13/2,c_14/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,afterNth#,fst#,head#,natsFrom#,sel#,snd#
            ,splitAt#,tail#,take#,u#} and constructors {0,cons,n__natsFrom,nil,pair,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1}
        by application of
          Pre({1}) = {4,5,7}.
        Here rules are labelled as follows:
          1: activate#(n__natsFrom(X)) -> c_2(natsFrom#(X))
          2: afterNth#(N,XS) -> c_3(snd#(splitAt(N,XS)),splitAt#(N,XS))
          3: sel#(N,XS) -> c_8(head#(afterNth(N,XS)),afterNth#(N,XS))
          4: splitAt#(s(N),cons(X,XS)) -> c_11(u#(splitAt(N,activate(XS)),N,X,activate(XS))
                                              ,splitAt#(N,activate(XS))
                                              ,activate#(XS)
                                              ,activate#(XS))
          5: tail#(cons(N,XS)) -> c_12(activate#(XS))
          6: take#(N,XS) -> c_13(fst#(splitAt(N,XS)),splitAt#(N,XS))
          7: u#(pair(YS,ZS),N,X,XS) -> c_14(activate#(X))
          8: activate#(X) -> c_1()
          9: fst#(pair(XS,YS)) -> c_4()
          10: head#(cons(N,XS)) -> c_5()
          11: natsFrom#(N) -> c_6()
          12: natsFrom#(X) -> c_7()
          13: snd#(pair(XS,YS)) -> c_9()
          14: splitAt#(0(),XS) -> c_10()
* Step 6: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            afterNth#(N,XS) -> c_3(snd#(splitAt(N,XS)),splitAt#(N,XS))
            sel#(N,XS) -> c_8(head#(afterNth(N,XS)),afterNth#(N,XS))
            splitAt#(s(N),cons(X,XS)) -> c_11(u#(splitAt(N,activate(XS)),N,X,activate(XS))
                                             ,splitAt#(N,activate(XS))
                                             ,activate#(XS)
                                             ,activate#(XS))
            tail#(cons(N,XS)) -> c_12(activate#(XS))
            take#(N,XS) -> c_13(fst#(splitAt(N,XS)),splitAt#(N,XS))
            u#(pair(YS,ZS),N,X,XS) -> c_14(activate#(X))
        - Weak DPs:
            activate#(X) -> c_1()
            activate#(n__natsFrom(X)) -> c_2(natsFrom#(X))
            fst#(pair(XS,YS)) -> c_4()
            head#(cons(N,XS)) -> c_5()
            natsFrom#(N) -> c_6()
            natsFrom#(X) -> c_7()
            snd#(pair(XS,YS)) -> c_9()
            splitAt#(0(),XS) -> c_10()
        - Weak TRS:
            activate(X) -> X
            activate(n__natsFrom(X)) -> natsFrom(X)
            afterNth(N,XS) -> snd(splitAt(N,XS))
            natsFrom(N) -> cons(N,n__natsFrom(s(N)))
            natsFrom(X) -> n__natsFrom(X)
            snd(pair(XS,YS)) -> YS
            splitAt(0(),XS) -> pair(nil(),XS)
            splitAt(s(N),cons(X,XS)) -> u(splitAt(N,activate(XS)),N,X,activate(XS))
            u(pair(YS,ZS),N,X,XS) -> pair(cons(activate(X),YS),ZS)
        - Signature:
            {activate/1,afterNth/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1
            ,afterNth#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2
            ,n__natsFrom/1,nil/0,pair/2,s/1,c_1/0,c_2/1,c_3/2,c_4/0,c_5/0,c_6/0,c_7/0,c_8/2,c_9/0,c_10/0,c_11/4,c_12/1
            ,c_13/2,c_14/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,afterNth#,fst#,head#,natsFrom#,sel#,snd#
            ,splitAt#,tail#,take#,u#} and constructors {0,cons,n__natsFrom,nil,pair,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {4,6}
        by application of
          Pre({4,6}) = {3}.
        Here rules are labelled as follows:
          1: afterNth#(N,XS) -> c_3(snd#(splitAt(N,XS)),splitAt#(N,XS))
          2: sel#(N,XS) -> c_8(head#(afterNth(N,XS)),afterNth#(N,XS))
          3: splitAt#(s(N),cons(X,XS)) -> c_11(u#(splitAt(N,activate(XS)),N,X,activate(XS))
                                              ,splitAt#(N,activate(XS))
                                              ,activate#(XS)
                                              ,activate#(XS))
          4: tail#(cons(N,XS)) -> c_12(activate#(XS))
          5: take#(N,XS) -> c_13(fst#(splitAt(N,XS)),splitAt#(N,XS))
          6: u#(pair(YS,ZS),N,X,XS) -> c_14(activate#(X))
          7: activate#(X) -> c_1()
          8: activate#(n__natsFrom(X)) -> c_2(natsFrom#(X))
          9: fst#(pair(XS,YS)) -> c_4()
          10: head#(cons(N,XS)) -> c_5()
          11: natsFrom#(N) -> c_6()
          12: natsFrom#(X) -> c_7()
          13: snd#(pair(XS,YS)) -> c_9()
          14: splitAt#(0(),XS) -> c_10()
* Step 7: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            afterNth#(N,XS) -> c_3(snd#(splitAt(N,XS)),splitAt#(N,XS))
            sel#(N,XS) -> c_8(head#(afterNth(N,XS)),afterNth#(N,XS))
            splitAt#(s(N),cons(X,XS)) -> c_11(u#(splitAt(N,activate(XS)),N,X,activate(XS))
                                             ,splitAt#(N,activate(XS))
                                             ,activate#(XS)
                                             ,activate#(XS))
            take#(N,XS) -> c_13(fst#(splitAt(N,XS)),splitAt#(N,XS))
        - Weak DPs:
            activate#(X) -> c_1()
            activate#(n__natsFrom(X)) -> c_2(natsFrom#(X))
            fst#(pair(XS,YS)) -> c_4()
            head#(cons(N,XS)) -> c_5()
            natsFrom#(N) -> c_6()
            natsFrom#(X) -> c_7()
            snd#(pair(XS,YS)) -> c_9()
            splitAt#(0(),XS) -> c_10()
            tail#(cons(N,XS)) -> c_12(activate#(XS))
            u#(pair(YS,ZS),N,X,XS) -> c_14(activate#(X))
        - Weak TRS:
            activate(X) -> X
            activate(n__natsFrom(X)) -> natsFrom(X)
            afterNth(N,XS) -> snd(splitAt(N,XS))
            natsFrom(N) -> cons(N,n__natsFrom(s(N)))
            natsFrom(X) -> n__natsFrom(X)
            snd(pair(XS,YS)) -> YS
            splitAt(0(),XS) -> pair(nil(),XS)
            splitAt(s(N),cons(X,XS)) -> u(splitAt(N,activate(XS)),N,X,activate(XS))
            u(pair(YS,ZS),N,X,XS) -> pair(cons(activate(X),YS),ZS)
        - Signature:
            {activate/1,afterNth/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1
            ,afterNth#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2
            ,n__natsFrom/1,nil/0,pair/2,s/1,c_1/0,c_2/1,c_3/2,c_4/0,c_5/0,c_6/0,c_7/0,c_8/2,c_9/0,c_10/0,c_11/4,c_12/1
            ,c_13/2,c_14/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,afterNth#,fst#,head#,natsFrom#,sel#,snd#
            ,splitAt#,tail#,take#,u#} and constructors {0,cons,n__natsFrom,nil,pair,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:afterNth#(N,XS) -> c_3(snd#(splitAt(N,XS)),splitAt#(N,XS))
             -->_2 splitAt#(s(N),cons(X,XS)) -> c_11(u#(splitAt(N,activate(XS)),N,X,activate(XS))
                                                    ,splitAt#(N,activate(XS))
                                                    ,activate#(XS)
                                                    ,activate#(XS)):3
             -->_2 splitAt#(0(),XS) -> c_10():12
             -->_1 snd#(pair(XS,YS)) -> c_9():11
          
          2:S:sel#(N,XS) -> c_8(head#(afterNth(N,XS)),afterNth#(N,XS))
             -->_1 head#(cons(N,XS)) -> c_5():8
             -->_2 afterNth#(N,XS) -> c_3(snd#(splitAt(N,XS)),splitAt#(N,XS)):1
          
          3:S:splitAt#(s(N),cons(X,XS)) -> c_11(u#(splitAt(N,activate(XS)),N,X,activate(XS))
                                               ,splitAt#(N,activate(XS))
                                               ,activate#(XS)
                                               ,activate#(XS))
             -->_1 u#(pair(YS,ZS),N,X,XS) -> c_14(activate#(X)):14
             -->_4 activate#(n__natsFrom(X)) -> c_2(natsFrom#(X)):6
             -->_3 activate#(n__natsFrom(X)) -> c_2(natsFrom#(X)):6
             -->_2 splitAt#(0(),XS) -> c_10():12
             -->_4 activate#(X) -> c_1():5
             -->_3 activate#(X) -> c_1():5
             -->_2 splitAt#(s(N),cons(X,XS)) -> c_11(u#(splitAt(N,activate(XS)),N,X,activate(XS))
                                                    ,splitAt#(N,activate(XS))
                                                    ,activate#(XS)
                                                    ,activate#(XS)):3
          
          4:S:take#(N,XS) -> c_13(fst#(splitAt(N,XS)),splitAt#(N,XS))
             -->_2 splitAt#(0(),XS) -> c_10():12
             -->_1 fst#(pair(XS,YS)) -> c_4():7
             -->_2 splitAt#(s(N),cons(X,XS)) -> c_11(u#(splitAt(N,activate(XS)),N,X,activate(XS))
                                                    ,splitAt#(N,activate(XS))
                                                    ,activate#(XS)
                                                    ,activate#(XS)):3
          
          5:W:activate#(X) -> c_1()
             
          
          6:W:activate#(n__natsFrom(X)) -> c_2(natsFrom#(X))
             -->_1 natsFrom#(X) -> c_7():10
             -->_1 natsFrom#(N) -> c_6():9
          
          7:W:fst#(pair(XS,YS)) -> c_4()
             
          
          8:W:head#(cons(N,XS)) -> c_5()
             
          
          9:W:natsFrom#(N) -> c_6()
             
          
          10:W:natsFrom#(X) -> c_7()
             
          
          11:W:snd#(pair(XS,YS)) -> c_9()
             
          
          12:W:splitAt#(0(),XS) -> c_10()
             
          
          13:W:tail#(cons(N,XS)) -> c_12(activate#(XS))
             -->_1 activate#(n__natsFrom(X)) -> c_2(natsFrom#(X)):6
             -->_1 activate#(X) -> c_1():5
          
          14:W:u#(pair(YS,ZS),N,X,XS) -> c_14(activate#(X))
             -->_1 activate#(n__natsFrom(X)) -> c_2(natsFrom#(X)):6
             -->_1 activate#(X) -> c_1():5
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          13: tail#(cons(N,XS)) -> c_12(activate#(XS))
          7: fst#(pair(XS,YS)) -> c_4()
          8: head#(cons(N,XS)) -> c_5()
          11: snd#(pair(XS,YS)) -> c_9()
          12: splitAt#(0(),XS) -> c_10()
          14: u#(pair(YS,ZS),N,X,XS) -> c_14(activate#(X))
          5: activate#(X) -> c_1()
          6: activate#(n__natsFrom(X)) -> c_2(natsFrom#(X))
          9: natsFrom#(N) -> c_6()
          10: natsFrom#(X) -> c_7()
* Step 8: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            afterNth#(N,XS) -> c_3(snd#(splitAt(N,XS)),splitAt#(N,XS))
            sel#(N,XS) -> c_8(head#(afterNth(N,XS)),afterNth#(N,XS))
            splitAt#(s(N),cons(X,XS)) -> c_11(u#(splitAt(N,activate(XS)),N,X,activate(XS))
                                             ,splitAt#(N,activate(XS))
                                             ,activate#(XS)
                                             ,activate#(XS))
            take#(N,XS) -> c_13(fst#(splitAt(N,XS)),splitAt#(N,XS))
        - Weak TRS:
            activate(X) -> X
            activate(n__natsFrom(X)) -> natsFrom(X)
            afterNth(N,XS) -> snd(splitAt(N,XS))
            natsFrom(N) -> cons(N,n__natsFrom(s(N)))
            natsFrom(X) -> n__natsFrom(X)
            snd(pair(XS,YS)) -> YS
            splitAt(0(),XS) -> pair(nil(),XS)
            splitAt(s(N),cons(X,XS)) -> u(splitAt(N,activate(XS)),N,X,activate(XS))
            u(pair(YS,ZS),N,X,XS) -> pair(cons(activate(X),YS),ZS)
        - Signature:
            {activate/1,afterNth/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1
            ,afterNth#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2
            ,n__natsFrom/1,nil/0,pair/2,s/1,c_1/0,c_2/1,c_3/2,c_4/0,c_5/0,c_6/0,c_7/0,c_8/2,c_9/0,c_10/0,c_11/4,c_12/1
            ,c_13/2,c_14/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,afterNth#,fst#,head#,natsFrom#,sel#,snd#
            ,splitAt#,tail#,take#,u#} and constructors {0,cons,n__natsFrom,nil,pair,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:afterNth#(N,XS) -> c_3(snd#(splitAt(N,XS)),splitAt#(N,XS))
             -->_2 splitAt#(s(N),cons(X,XS)) -> c_11(u#(splitAt(N,activate(XS)),N,X,activate(XS))
                                                    ,splitAt#(N,activate(XS))
                                                    ,activate#(XS)
                                                    ,activate#(XS)):3
          
          2:S:sel#(N,XS) -> c_8(head#(afterNth(N,XS)),afterNth#(N,XS))
             -->_2 afterNth#(N,XS) -> c_3(snd#(splitAt(N,XS)),splitAt#(N,XS)):1
          
          3:S:splitAt#(s(N),cons(X,XS)) -> c_11(u#(splitAt(N,activate(XS)),N,X,activate(XS))
                                               ,splitAt#(N,activate(XS))
                                               ,activate#(XS)
                                               ,activate#(XS))
             -->_2 splitAt#(s(N),cons(X,XS)) -> c_11(u#(splitAt(N,activate(XS)),N,X,activate(XS))
                                                    ,splitAt#(N,activate(XS))
                                                    ,activate#(XS)
                                                    ,activate#(XS)):3
          
          4:S:take#(N,XS) -> c_13(fst#(splitAt(N,XS)),splitAt#(N,XS))
             -->_2 splitAt#(s(N),cons(X,XS)) -> c_11(u#(splitAt(N,activate(XS)),N,X,activate(XS))
                                                    ,splitAt#(N,activate(XS))
                                                    ,activate#(XS)
                                                    ,activate#(XS)):3
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          afterNth#(N,XS) -> c_3(splitAt#(N,XS))
          sel#(N,XS) -> c_8(afterNth#(N,XS))
          splitAt#(s(N),cons(X,XS)) -> c_11(splitAt#(N,activate(XS)))
          take#(N,XS) -> c_13(splitAt#(N,XS))
* Step 9: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            afterNth#(N,XS) -> c_3(splitAt#(N,XS))
            sel#(N,XS) -> c_8(afterNth#(N,XS))
            splitAt#(s(N),cons(X,XS)) -> c_11(splitAt#(N,activate(XS)))
            take#(N,XS) -> c_13(splitAt#(N,XS))
        - Weak TRS:
            activate(X) -> X
            activate(n__natsFrom(X)) -> natsFrom(X)
            afterNth(N,XS) -> snd(splitAt(N,XS))
            natsFrom(N) -> cons(N,n__natsFrom(s(N)))
            natsFrom(X) -> n__natsFrom(X)
            snd(pair(XS,YS)) -> YS
            splitAt(0(),XS) -> pair(nil(),XS)
            splitAt(s(N),cons(X,XS)) -> u(splitAt(N,activate(XS)),N,X,activate(XS))
            u(pair(YS,ZS),N,X,XS) -> pair(cons(activate(X),YS),ZS)
        - Signature:
            {activate/1,afterNth/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1
            ,afterNth#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2
            ,n__natsFrom/1,nil/0,pair/2,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/1
            ,c_13/1,c_14/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,afterNth#,fst#,head#,natsFrom#,sel#,snd#
            ,splitAt#,tail#,take#,u#} and constructors {0,cons,n__natsFrom,nil,pair,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          activate(X) -> X
          activate(n__natsFrom(X)) -> natsFrom(X)
          natsFrom(N) -> cons(N,n__natsFrom(s(N)))
          natsFrom(X) -> n__natsFrom(X)
          afterNth#(N,XS) -> c_3(splitAt#(N,XS))
          sel#(N,XS) -> c_8(afterNth#(N,XS))
          splitAt#(s(N),cons(X,XS)) -> c_11(splitAt#(N,activate(XS)))
          take#(N,XS) -> c_13(splitAt#(N,XS))
* Step 10: RemoveHeads WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            afterNth#(N,XS) -> c_3(splitAt#(N,XS))
            sel#(N,XS) -> c_8(afterNth#(N,XS))
            splitAt#(s(N),cons(X,XS)) -> c_11(splitAt#(N,activate(XS)))
            take#(N,XS) -> c_13(splitAt#(N,XS))
        - Weak TRS:
            activate(X) -> X
            activate(n__natsFrom(X)) -> natsFrom(X)
            natsFrom(N) -> cons(N,n__natsFrom(s(N)))
            natsFrom(X) -> n__natsFrom(X)
        - Signature:
            {activate/1,afterNth/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1
            ,afterNth#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2
            ,n__natsFrom/1,nil/0,pair/2,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/1
            ,c_13/1,c_14/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,afterNth#,fst#,head#,natsFrom#,sel#,snd#
            ,splitAt#,tail#,take#,u#} and constructors {0,cons,n__natsFrom,nil,pair,s}
    + Applied Processor:
        RemoveHeads
    + Details:
        Consider the dependency graph
        
        1:S:afterNth#(N,XS) -> c_3(splitAt#(N,XS))
           -->_1 splitAt#(s(N),cons(X,XS)) -> c_11(splitAt#(N,activate(XS))):3
        
        2:S:sel#(N,XS) -> c_8(afterNth#(N,XS))
           -->_1 afterNth#(N,XS) -> c_3(splitAt#(N,XS)):1
        
        3:S:splitAt#(s(N),cons(X,XS)) -> c_11(splitAt#(N,activate(XS)))
           -->_1 splitAt#(s(N),cons(X,XS)) -> c_11(splitAt#(N,activate(XS))):3
        
        4:S:take#(N,XS) -> c_13(splitAt#(N,XS))
           -->_1 splitAt#(s(N),cons(X,XS)) -> c_11(splitAt#(N,activate(XS))):3
        
        
        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,sel#(N,XS) -> c_8(afterNth#(N,XS))),(4,take#(N,XS) -> c_13(splitAt#(N,XS)))]
* Step 11: RemoveHeads WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            afterNth#(N,XS) -> c_3(splitAt#(N,XS))
            splitAt#(s(N),cons(X,XS)) -> c_11(splitAt#(N,activate(XS)))
        - Weak TRS:
            activate(X) -> X
            activate(n__natsFrom(X)) -> natsFrom(X)
            natsFrom(N) -> cons(N,n__natsFrom(s(N)))
            natsFrom(X) -> n__natsFrom(X)
        - Signature:
            {activate/1,afterNth/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1
            ,afterNth#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2
            ,n__natsFrom/1,nil/0,pair/2,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/1
            ,c_13/1,c_14/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,afterNth#,fst#,head#,natsFrom#,sel#,snd#
            ,splitAt#,tail#,take#,u#} and constructors {0,cons,n__natsFrom,nil,pair,s}
    + Applied Processor:
        RemoveHeads
    + Details:
        Consider the dependency graph
        
        1:S:afterNth#(N,XS) -> c_3(splitAt#(N,XS))
           -->_1 splitAt#(s(N),cons(X,XS)) -> c_11(splitAt#(N,activate(XS))):3
        
        3:S:splitAt#(s(N),cons(X,XS)) -> c_11(splitAt#(N,activate(XS)))
           -->_1 splitAt#(s(N),cons(X,XS)) -> c_11(splitAt#(N,activate(XS))):3
        
        
        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).
        
        [(1,afterNth#(N,XS) -> c_3(splitAt#(N,XS)))]
* Step 12: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            splitAt#(s(N),cons(X,XS)) -> c_11(splitAt#(N,activate(XS)))
        - Weak TRS:
            activate(X) -> X
            activate(n__natsFrom(X)) -> natsFrom(X)
            natsFrom(N) -> cons(N,n__natsFrom(s(N)))
            natsFrom(X) -> n__natsFrom(X)
        - Signature:
            {activate/1,afterNth/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1
            ,afterNth#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2
            ,n__natsFrom/1,nil/0,pair/2,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/1
            ,c_13/1,c_14/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,afterNth#,fst#,head#,natsFrom#,sel#,snd#
            ,splitAt#,tail#,take#,u#} and constructors {0,cons,n__natsFrom,nil,pair,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:
          3: splitAt#(s(N),cons(X,XS)) -> c_11(splitAt#(N,activate(XS)))
          
        The strictly oriented rules are moved into the weak component.
** Step 12.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            splitAt#(s(N),cons(X,XS)) -> c_11(splitAt#(N,activate(XS)))
        - Weak TRS:
            activate(X) -> X
            activate(n__natsFrom(X)) -> natsFrom(X)
            natsFrom(N) -> cons(N,n__natsFrom(s(N)))
            natsFrom(X) -> n__natsFrom(X)
        - Signature:
            {activate/1,afterNth/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1
            ,afterNth#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2
            ,n__natsFrom/1,nil/0,pair/2,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/1
            ,c_13/1,c_14/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,afterNth#,fst#,head#,natsFrom#,sel#,snd#
            ,splitAt#,tail#,take#,u#} and constructors {0,cons,n__natsFrom,nil,pair,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_11) = {1}
        
        Following symbols are considered usable:
          {activate#,afterNth#,fst#,head#,natsFrom#,sel#,snd#,splitAt#,tail#,take#,u#}
        TcT has computed the following interpretation:
                    p(0) = [1]                           
             p(activate) = [0]                           
             p(afterNth) = [1] x1 + [0]                  
                 p(cons) = [1] x1 + [0]                  
                  p(fst) = [1]                           
                 p(head) = [1] x1 + [2]                  
          p(n__natsFrom) = [0]                           
             p(natsFrom) = [4] x1 + [1]                  
                  p(nil) = [2]                           
                 p(pair) = [8]                           
                    p(s) = [1] x1 + [4]                  
                  p(sel) = [1] x1 + [1] x2 + [2]         
                  p(snd) = [2] x1 + [1]                  
              p(splitAt) = [2] x1 + [2]                  
                 p(tail) = [1]                           
                 p(take) = [8] x2 + [1]                  
                    p(u) = [0]                           
            p(activate#) = [2]                           
            p(afterNth#) = [1] x1 + [8]                  
                 p(fst#) = [1] x1 + [4]                  
                p(head#) = [8] x1 + [0]                  
            p(natsFrom#) = [1]                           
                 p(sel#) = [1] x1 + [2] x2 + [1]         
                 p(snd#) = [0]                           
             p(splitAt#) = [2] x1 + [0]                  
                p(tail#) = [2] x1 + [1]                  
                p(take#) = [1] x1 + [2] x2 + [0]         
                   p(u#) = [1] x1 + [1] x2 + [4] x4 + [0]
                  p(c_1) = [0]                           
                  p(c_2) = [1]                           
                  p(c_3) = [1] x1 + [0]                  
                  p(c_4) = [1]                           
                  p(c_5) = [1]                           
                  p(c_6) = [1]                           
                  p(c_7) = [2]                           
                  p(c_8) = [0]                           
                  p(c_9) = [4]                           
                 p(c_10) = [4]                           
                 p(c_11) = [1] x1 + [4]                  
                 p(c_12) = [2]                           
                 p(c_13) = [2] x1 + [2]                  
                 p(c_14) = [1]                           
        
        Following rules are strictly oriented:
        splitAt#(s(N),cons(X,XS)) = [2] N + [8]                   
                                  > [2] N + [4]                   
                                  = c_11(splitAt#(N,activate(XS)))
        
        
        Following rules are (at-least) weakly oriented:
        
** Step 12.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            splitAt#(s(N),cons(X,XS)) -> c_11(splitAt#(N,activate(XS)))
        - Weak TRS:
            activate(X) -> X
            activate(n__natsFrom(X)) -> natsFrom(X)
            natsFrom(N) -> cons(N,n__natsFrom(s(N)))
            natsFrom(X) -> n__natsFrom(X)
        - Signature:
            {activate/1,afterNth/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1
            ,afterNth#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2
            ,n__natsFrom/1,nil/0,pair/2,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/1
            ,c_13/1,c_14/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,afterNth#,fst#,head#,natsFrom#,sel#,snd#
            ,splitAt#,tail#,take#,u#} and constructors {0,cons,n__natsFrom,nil,pair,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

** Step 12.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            splitAt#(s(N),cons(X,XS)) -> c_11(splitAt#(N,activate(XS)))
        - Weak TRS:
            activate(X) -> X
            activate(n__natsFrom(X)) -> natsFrom(X)
            natsFrom(N) -> cons(N,n__natsFrom(s(N)))
            natsFrom(X) -> n__natsFrom(X)
        - Signature:
            {activate/1,afterNth/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1
            ,afterNth#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2
            ,n__natsFrom/1,nil/0,pair/2,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/1
            ,c_13/1,c_14/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,afterNth#,fst#,head#,natsFrom#,sel#,snd#
            ,splitAt#,tail#,take#,u#} and constructors {0,cons,n__natsFrom,nil,pair,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:splitAt#(s(N),cons(X,XS)) -> c_11(splitAt#(N,activate(XS)))
             -->_1 splitAt#(s(N),cons(X,XS)) -> c_11(splitAt#(N,activate(XS))):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: splitAt#(s(N),cons(X,XS)) -> c_11(splitAt#(N,activate(XS)))
** Step 12.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            activate(X) -> X
            activate(n__natsFrom(X)) -> natsFrom(X)
            natsFrom(N) -> cons(N,n__natsFrom(s(N)))
            natsFrom(X) -> n__natsFrom(X)
        - Signature:
            {activate/1,afterNth/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1
            ,afterNth#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2
            ,n__natsFrom/1,nil/0,pair/2,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/1
            ,c_13/1,c_14/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,afterNth#,fst#,head#,natsFrom#,sel#,snd#
            ,splitAt#,tail#,take#,u#} and constructors {0,cons,n__natsFrom,nil,pair,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))