* Step 1: Sum WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
            U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
            U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
            U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
            U21(mark(X1),X2) -> mark(U21(X1,X2))
            U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
            U31(mark(X1),X2) -> mark(U31(X1,X2))
            U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
            U41(mark(X1),X2) -> mark(U41(X1,X2))
            U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
            U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
            U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
            U61(mark(X1),X2) -> mark(U61(X1,X2))
            U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
            U71(mark(X1),X2) -> mark(U71(X1,X2))
            U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
            U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4))
            U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4))
            U82(mark(X1),X2) -> mark(U82(X1,X2))
            U82(ok(X1),ok(X2)) -> ok(U82(X1,X2))
            U91(mark(X1),X2) -> mark(U91(X1,X2))
            U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
            active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
            active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
            active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
            active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
            active(U21(X1,X2)) -> U21(active(X1),X2)
            active(U21(tt(),X)) -> mark(X)
            active(U31(X1,X2)) -> U31(active(X1),X2)
            active(U31(tt(),N)) -> mark(N)
            active(U41(X1,X2)) -> U41(active(X1),X2)
            active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
            active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
            active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
            active(U61(X1,X2)) -> U61(active(X1),X2)
            active(U61(tt(),Y)) -> mark(Y)
            active(U71(X1,X2)) -> U71(active(X1),X2)
            active(U71(tt(),XS)) -> mark(pair(nil(),XS))
            active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4)
            active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X))
            active(U82(X1,X2)) -> U82(active(X1),X2)
            active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
            active(U91(X1,X2)) -> U91(active(X1),X2)
            active(U91(tt(),XS)) -> mark(XS)
            active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS))
            active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
            active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
            active(and(X1,X2)) -> and(active(X1),X2)
            active(and(tt(),X)) -> mark(X)
            active(cons(X1,X2)) -> cons(active(X1),X2)
            active(fst(X)) -> fst(active(X))
            active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X))
            active(head(X)) -> head(active(X))
            active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N))
            active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2)))
            active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2)))
            active(isLNat(fst(V1))) -> mark(isPLNat(V1))
            active(isLNat(natsFrom(V1))) -> mark(isNatural(V1))
            active(isLNat(nil())) -> mark(tt())
            active(isLNat(snd(V1))) -> mark(isPLNat(V1))
            active(isLNat(tail(V1))) -> mark(isLNat(V1))
            active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2)))
            active(isNatural(0())) -> mark(tt())
            active(isNatural(head(V1))) -> mark(isLNat(V1))
            active(isNatural(s(V1))) -> mark(isNatural(V1))
            active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2)))
            active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2)))
            active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2)))
            active(natsFrom(N)) -> mark(U41(isNatural(N),N))
            active(natsFrom(X)) -> natsFrom(active(X))
            active(pair(X1,X2)) -> pair(X1,active(X2))
            active(pair(X1,X2)) -> pair(active(X1),X2)
            active(s(X)) -> s(active(X))
            active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS))
            active(sel(X1,X2)) -> sel(X1,active(X2))
            active(sel(X1,X2)) -> sel(active(X1),X2)
            active(snd(X)) -> snd(active(X))
            active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y))
            active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
            active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
            active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS))
            active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS))
            active(tail(X)) -> tail(active(X))
            active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS))
            active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS))
            active(take(X1,X2)) -> take(X1,active(X2))
            active(take(X1,X2)) -> take(active(X1),X2)
            afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
            afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
            afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
            and(mark(X1),X2) -> mark(and(X1,X2))
            and(ok(X1),ok(X2)) -> ok(and(X1,X2))
            cons(mark(X1),X2) -> mark(cons(X1,X2))
            cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
            fst(mark(X)) -> mark(fst(X))
            fst(ok(X)) -> ok(fst(X))
            head(mark(X)) -> mark(head(X))
            head(ok(X)) -> ok(head(X))
            isLNat(ok(X)) -> ok(isLNat(X))
            isNatural(ok(X)) -> ok(isNatural(X))
            isPLNat(ok(X)) -> ok(isPLNat(X))
            natsFrom(mark(X)) -> mark(natsFrom(X))
            natsFrom(ok(X)) -> ok(natsFrom(X))
            pair(X1,mark(X2)) -> mark(pair(X1,X2))
            pair(mark(X1),X2) -> mark(pair(X1,X2))
            pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
            proper(0()) -> ok(0())
            proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
            proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
            proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
            proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
            proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
            proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
            proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
            proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
            proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4))
            proper(U82(X1,X2)) -> U82(proper(X1),proper(X2))
            proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
            proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
            proper(and(X1,X2)) -> and(proper(X1),proper(X2))
            proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
            proper(fst(X)) -> fst(proper(X))
            proper(head(X)) -> head(proper(X))
            proper(isLNat(X)) -> isLNat(proper(X))
            proper(isNatural(X)) -> isNatural(proper(X))
            proper(isPLNat(X)) -> isPLNat(proper(X))
            proper(natsFrom(X)) -> natsFrom(proper(X))
            proper(nil()) -> ok(nil())
            proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
            proper(s(X)) -> s(proper(X))
            proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
            proper(snd(X)) -> snd(proper(X))
            proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
            proper(tail(X)) -> tail(proper(X))
            proper(take(X1,X2)) -> take(proper(X1),proper(X2))
            proper(tt()) -> ok(tt())
            s(mark(X)) -> mark(s(X))
            s(ok(X)) -> ok(s(X))
            sel(X1,mark(X2)) -> mark(sel(X1,X2))
            sel(mark(X1),X2) -> mark(sel(X1,X2))
            sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
            snd(mark(X)) -> mark(snd(X))
            snd(ok(X)) -> ok(snd(X))
            splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
            splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
            splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
            tail(mark(X)) -> mark(tail(X))
            tail(ok(X)) -> ok(tail(X))
            take(X1,mark(X2)) -> mark(take(X1,X2))
            take(mark(X1),X2) -> mark(take(X1,X2))
            take(ok(X1),ok(X2)) -> ok(take(X1,X2))
            top(mark(X)) -> top(proper(X))
            top(ok(X)) -> top(active(X))
        - Signature:
            {U101/3,U11/3,U21/2,U31/2,U41/2,U51/3,U61/2,U71/2,U81/4,U82/2,U91/2,active/1,afterNth/2,and/2,cons/2,fst/1
            ,head/1,isLNat/1,isNatural/1,isPLNat/1,natsFrom/1,pair/2,proper/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2
            ,top/1} / {0/0,mark/1,nil/0,ok/1,tt/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {U101,U11,U21,U31,U41,U51,U61,U71,U81,U82,U91,active
            ,afterNth,and,cons,fst,head,isLNat,isNatural,isPLNat,natsFrom,pair,proper,s,sel,snd,splitAt,tail,take
            ,top} and constructors {0,mark,nil,ok,tt}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
            U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
            U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
            U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
            U21(mark(X1),X2) -> mark(U21(X1,X2))
            U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
            U31(mark(X1),X2) -> mark(U31(X1,X2))
            U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
            U41(mark(X1),X2) -> mark(U41(X1,X2))
            U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
            U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
            U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
            U61(mark(X1),X2) -> mark(U61(X1,X2))
            U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
            U71(mark(X1),X2) -> mark(U71(X1,X2))
            U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
            U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4))
            U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4))
            U82(mark(X1),X2) -> mark(U82(X1,X2))
            U82(ok(X1),ok(X2)) -> ok(U82(X1,X2))
            U91(mark(X1),X2) -> mark(U91(X1,X2))
            U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
            active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
            active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
            active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
            active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
            active(U21(X1,X2)) -> U21(active(X1),X2)
            active(U21(tt(),X)) -> mark(X)
            active(U31(X1,X2)) -> U31(active(X1),X2)
            active(U31(tt(),N)) -> mark(N)
            active(U41(X1,X2)) -> U41(active(X1),X2)
            active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
            active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
            active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
            active(U61(X1,X2)) -> U61(active(X1),X2)
            active(U61(tt(),Y)) -> mark(Y)
            active(U71(X1,X2)) -> U71(active(X1),X2)
            active(U71(tt(),XS)) -> mark(pair(nil(),XS))
            active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4)
            active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X))
            active(U82(X1,X2)) -> U82(active(X1),X2)
            active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
            active(U91(X1,X2)) -> U91(active(X1),X2)
            active(U91(tt(),XS)) -> mark(XS)
            active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS))
            active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
            active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
            active(and(X1,X2)) -> and(active(X1),X2)
            active(and(tt(),X)) -> mark(X)
            active(cons(X1,X2)) -> cons(active(X1),X2)
            active(fst(X)) -> fst(active(X))
            active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X))
            active(head(X)) -> head(active(X))
            active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N))
            active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2)))
            active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2)))
            active(isLNat(fst(V1))) -> mark(isPLNat(V1))
            active(isLNat(natsFrom(V1))) -> mark(isNatural(V1))
            active(isLNat(nil())) -> mark(tt())
            active(isLNat(snd(V1))) -> mark(isPLNat(V1))
            active(isLNat(tail(V1))) -> mark(isLNat(V1))
            active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2)))
            active(isNatural(0())) -> mark(tt())
            active(isNatural(head(V1))) -> mark(isLNat(V1))
            active(isNatural(s(V1))) -> mark(isNatural(V1))
            active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2)))
            active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2)))
            active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2)))
            active(natsFrom(N)) -> mark(U41(isNatural(N),N))
            active(natsFrom(X)) -> natsFrom(active(X))
            active(pair(X1,X2)) -> pair(X1,active(X2))
            active(pair(X1,X2)) -> pair(active(X1),X2)
            active(s(X)) -> s(active(X))
            active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS))
            active(sel(X1,X2)) -> sel(X1,active(X2))
            active(sel(X1,X2)) -> sel(active(X1),X2)
            active(snd(X)) -> snd(active(X))
            active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y))
            active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
            active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
            active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS))
            active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS))
            active(tail(X)) -> tail(active(X))
            active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS))
            active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS))
            active(take(X1,X2)) -> take(X1,active(X2))
            active(take(X1,X2)) -> take(active(X1),X2)
            afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
            afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
            afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
            and(mark(X1),X2) -> mark(and(X1,X2))
            and(ok(X1),ok(X2)) -> ok(and(X1,X2))
            cons(mark(X1),X2) -> mark(cons(X1,X2))
            cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
            fst(mark(X)) -> mark(fst(X))
            fst(ok(X)) -> ok(fst(X))
            head(mark(X)) -> mark(head(X))
            head(ok(X)) -> ok(head(X))
            isLNat(ok(X)) -> ok(isLNat(X))
            isNatural(ok(X)) -> ok(isNatural(X))
            isPLNat(ok(X)) -> ok(isPLNat(X))
            natsFrom(mark(X)) -> mark(natsFrom(X))
            natsFrom(ok(X)) -> ok(natsFrom(X))
            pair(X1,mark(X2)) -> mark(pair(X1,X2))
            pair(mark(X1),X2) -> mark(pair(X1,X2))
            pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
            proper(0()) -> ok(0())
            proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
            proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
            proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
            proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
            proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
            proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
            proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
            proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
            proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4))
            proper(U82(X1,X2)) -> U82(proper(X1),proper(X2))
            proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
            proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
            proper(and(X1,X2)) -> and(proper(X1),proper(X2))
            proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
            proper(fst(X)) -> fst(proper(X))
            proper(head(X)) -> head(proper(X))
            proper(isLNat(X)) -> isLNat(proper(X))
            proper(isNatural(X)) -> isNatural(proper(X))
            proper(isPLNat(X)) -> isPLNat(proper(X))
            proper(natsFrom(X)) -> natsFrom(proper(X))
            proper(nil()) -> ok(nil())
            proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
            proper(s(X)) -> s(proper(X))
            proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
            proper(snd(X)) -> snd(proper(X))
            proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
            proper(tail(X)) -> tail(proper(X))
            proper(take(X1,X2)) -> take(proper(X1),proper(X2))
            proper(tt()) -> ok(tt())
            s(mark(X)) -> mark(s(X))
            s(ok(X)) -> ok(s(X))
            sel(X1,mark(X2)) -> mark(sel(X1,X2))
            sel(mark(X1),X2) -> mark(sel(X1,X2))
            sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
            snd(mark(X)) -> mark(snd(X))
            snd(ok(X)) -> ok(snd(X))
            splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
            splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
            splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
            tail(mark(X)) -> mark(tail(X))
            tail(ok(X)) -> ok(tail(X))
            take(X1,mark(X2)) -> mark(take(X1,X2))
            take(mark(X1),X2) -> mark(take(X1,X2))
            take(ok(X1),ok(X2)) -> ok(take(X1,X2))
            top(mark(X)) -> top(proper(X))
            top(ok(X)) -> top(active(X))
        - Signature:
            {U101/3,U11/3,U21/2,U31/2,U41/2,U51/3,U61/2,U71/2,U81/4,U82/2,U91/2,active/1,afterNth/2,and/2,cons/2,fst/1
            ,head/1,isLNat/1,isNatural/1,isPLNat/1,natsFrom/1,pair/2,proper/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2
            ,top/1} / {0/0,mark/1,nil/0,ok/1,tt/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {U101,U11,U21,U31,U41,U51,U61,U71,U81,U82,U91,active
            ,afterNth,and,cons,fst,head,isLNat,isNatural,isPLNat,natsFrom,pair,proper,s,sel,snd,splitAt,tail,take
            ,top} and constructors {0,mark,nil,ok,tt}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          U101(x,y,z){x -> mark(x)} =
            U101(mark(x),y,z) ->^+ mark(U101(x,y,z))
              = C[U101(x,y,z) = U101(x,y,z){}]

WORST_CASE(Omega(n^1),?)