* Step 1: Sum WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            U11(mark(X1),X2) -> mark(U11(X1,X2))
            U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
            U12(mark(X)) -> mark(U12(X))
            U12(ok(X)) -> ok(U12(X))
            U21(mark(X1),X2) -> mark(U21(X1,X2))
            U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
            U22(mark(X)) -> mark(U22(X))
            U22(ok(X)) -> ok(U22(X))
            U31(mark(X1),X2) -> mark(U31(X1,X2))
            U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
            U32(mark(X)) -> mark(U32(X))
            U32(ok(X)) -> ok(U32(X))
            U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
            U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
            U42(mark(X1),X2) -> mark(U42(X1,X2))
            U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
            U43(mark(X)) -> mark(U43(X))
            U43(ok(X)) -> ok(U43(X))
            U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
            U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
            U52(mark(X1),X2) -> mark(U52(X1,X2))
            U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
            U53(mark(X)) -> mark(U53(X))
            U53(ok(X)) -> ok(U53(X))
            U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3))
            U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3))
            U62(mark(X1),X2) -> mark(U62(X1,X2))
            U62(ok(X1),ok(X2)) -> ok(U62(X1,X2))
            U63(mark(X)) -> mark(U63(X))
            U63(ok(X)) -> ok(U63(X))
            U71(mark(X1),X2) -> mark(U71(X1,X2))
            U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
            U81(mark(X)) -> mark(U81(X))
            U81(ok(X)) -> ok(U81(X))
            U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4))
            U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4))
            active(U11(X1,X2)) -> U11(active(X1),X2)
            active(U11(tt(),V1)) -> mark(U12(isNatList(V1)))
            active(U12(X)) -> U12(active(X))
            active(U12(tt())) -> mark(tt())
            active(U21(X1,X2)) -> U21(active(X1),X2)
            active(U21(tt(),V1)) -> mark(U22(isNat(V1)))
            active(U22(X)) -> U22(active(X))
            active(U22(tt())) -> mark(tt())
            active(U31(X1,X2)) -> U31(active(X1),X2)
            active(U31(tt(),V)) -> mark(U32(isNatList(V)))
            active(U32(X)) -> U32(active(X))
            active(U32(tt())) -> mark(tt())
            active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
            active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2))
            active(U42(X1,X2)) -> U42(active(X1),X2)
            active(U42(tt(),V2)) -> mark(U43(isNatIList(V2)))
            active(U43(X)) -> U43(active(X))
            active(U43(tt())) -> mark(tt())
            active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
            active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2))
            active(U52(X1,X2)) -> U52(active(X1),X2)
            active(U52(tt(),V2)) -> mark(U53(isNatList(V2)))
            active(U53(X)) -> U53(active(X))
            active(U53(tt())) -> mark(tt())
            active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3)
            active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2))
            active(U62(X1,X2)) -> U62(active(X1),X2)
            active(U62(tt(),V2)) -> mark(U63(isNatIList(V2)))
            active(U63(X)) -> U63(active(X))
            active(U63(tt())) -> mark(tt())
            active(U71(X1,X2)) -> U71(active(X1),X2)
            active(U71(tt(),L)) -> mark(s(length(L)))
            active(U81(X)) -> U81(active(X))
            active(U81(tt())) -> mark(nil())
            active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4)
            active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
            active(and(X1,X2)) -> and(active(X1),X2)
            active(and(tt(),X)) -> mark(X)
            active(cons(X1,X2)) -> cons(active(X1),X2)
            active(isNat(0())) -> mark(tt())
            active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
            active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
            active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
            active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2))
            active(isNatIList(zeros())) -> mark(tt())
            active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2)))
            active(isNatIListKind(nil())) -> mark(tt())
            active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2)))
            active(isNatIListKind(zeros())) -> mark(tt())
            active(isNatKind(0())) -> mark(tt())
            active(isNatKind(length(V1))) -> mark(isNatIListKind(V1))
            active(isNatKind(s(V1))) -> mark(isNatKind(V1))
            active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2))
            active(isNatList(nil())) -> mark(tt())
            active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2))
            active(length(X)) -> length(active(X))
            active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N)))
                                                 ,L))
            active(length(nil())) -> mark(0())
            active(s(X)) -> s(active(X))
            active(take(X1,X2)) -> take(X1,active(X2))
            active(take(X1,X2)) -> take(active(X1),X2)
            active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL))))
            active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL))
                                                         ,and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))))
                                                     ,IL
                                                     ,M
                                                     ,N))
            active(zeros()) -> mark(cons(0(),zeros()))
            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))
            isNat(ok(X)) -> ok(isNat(X))
            isNatIList(ok(X)) -> ok(isNatIList(X))
            isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
            isNatKind(ok(X)) -> ok(isNatKind(X))
            isNatList(ok(X)) -> ok(isNatList(X))
            length(mark(X)) -> mark(length(X))
            length(ok(X)) -> ok(length(X))
            proper(0()) -> ok(0())
            proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
            proper(U12(X)) -> U12(proper(X))
            proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
            proper(U22(X)) -> U22(proper(X))
            proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
            proper(U32(X)) -> U32(proper(X))
            proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
            proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
            proper(U43(X)) -> U43(proper(X))
            proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
            proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
            proper(U53(X)) -> U53(proper(X))
            proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3))
            proper(U62(X1,X2)) -> U62(proper(X1),proper(X2))
            proper(U63(X)) -> U63(proper(X))
            proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
            proper(U81(X)) -> U81(proper(X))
            proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4))
            proper(and(X1,X2)) -> and(proper(X1),proper(X2))
            proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
            proper(isNat(X)) -> isNat(proper(X))
            proper(isNatIList(X)) -> isNatIList(proper(X))
            proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
            proper(isNatKind(X)) -> isNatKind(proper(X))
            proper(isNatList(X)) -> isNatList(proper(X))
            proper(length(X)) -> length(proper(X))
            proper(nil()) -> ok(nil())
            proper(s(X)) -> s(proper(X))
            proper(take(X1,X2)) -> take(proper(X1),proper(X2))
            proper(tt()) -> ok(tt())
            proper(zeros()) -> ok(zeros())
            s(mark(X)) -> mark(s(X))
            s(ok(X)) -> ok(s(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:
            {U11/2,U12/1,U21/2,U22/1,U31/2,U32/1,U41/3,U42/2,U43/1,U51/3,U52/2,U53/1,U61/3,U62/2,U63/1,U71/2,U81/1,U91/4
            ,active/1,and/2,cons/2,isNat/1,isNatIList/1,isNatIListKind/1,isNatKind/1,isNatList/1,length/1,proper/1,s/1
            ,take/2,top/1} / {0/0,mark/1,nil/0,ok/1,tt/0,zeros/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {U11,U12,U21,U22,U31,U32,U41,U42,U43,U51,U52,U53,U61,U62
            ,U63,U71,U81,U91,active,and,cons,isNat,isNatIList,isNatIListKind,isNatKind,isNatList,length,proper,s,take
            ,top} and constructors {0,mark,nil,ok,tt,zeros}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            U11(mark(X1),X2) -> mark(U11(X1,X2))
            U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
            U12(mark(X)) -> mark(U12(X))
            U12(ok(X)) -> ok(U12(X))
            U21(mark(X1),X2) -> mark(U21(X1,X2))
            U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
            U22(mark(X)) -> mark(U22(X))
            U22(ok(X)) -> ok(U22(X))
            U31(mark(X1),X2) -> mark(U31(X1,X2))
            U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
            U32(mark(X)) -> mark(U32(X))
            U32(ok(X)) -> ok(U32(X))
            U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
            U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
            U42(mark(X1),X2) -> mark(U42(X1,X2))
            U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
            U43(mark(X)) -> mark(U43(X))
            U43(ok(X)) -> ok(U43(X))
            U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
            U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
            U52(mark(X1),X2) -> mark(U52(X1,X2))
            U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
            U53(mark(X)) -> mark(U53(X))
            U53(ok(X)) -> ok(U53(X))
            U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3))
            U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3))
            U62(mark(X1),X2) -> mark(U62(X1,X2))
            U62(ok(X1),ok(X2)) -> ok(U62(X1,X2))
            U63(mark(X)) -> mark(U63(X))
            U63(ok(X)) -> ok(U63(X))
            U71(mark(X1),X2) -> mark(U71(X1,X2))
            U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
            U81(mark(X)) -> mark(U81(X))
            U81(ok(X)) -> ok(U81(X))
            U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4))
            U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4))
            active(U11(X1,X2)) -> U11(active(X1),X2)
            active(U11(tt(),V1)) -> mark(U12(isNatList(V1)))
            active(U12(X)) -> U12(active(X))
            active(U12(tt())) -> mark(tt())
            active(U21(X1,X2)) -> U21(active(X1),X2)
            active(U21(tt(),V1)) -> mark(U22(isNat(V1)))
            active(U22(X)) -> U22(active(X))
            active(U22(tt())) -> mark(tt())
            active(U31(X1,X2)) -> U31(active(X1),X2)
            active(U31(tt(),V)) -> mark(U32(isNatList(V)))
            active(U32(X)) -> U32(active(X))
            active(U32(tt())) -> mark(tt())
            active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
            active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2))
            active(U42(X1,X2)) -> U42(active(X1),X2)
            active(U42(tt(),V2)) -> mark(U43(isNatIList(V2)))
            active(U43(X)) -> U43(active(X))
            active(U43(tt())) -> mark(tt())
            active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
            active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2))
            active(U52(X1,X2)) -> U52(active(X1),X2)
            active(U52(tt(),V2)) -> mark(U53(isNatList(V2)))
            active(U53(X)) -> U53(active(X))
            active(U53(tt())) -> mark(tt())
            active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3)
            active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2))
            active(U62(X1,X2)) -> U62(active(X1),X2)
            active(U62(tt(),V2)) -> mark(U63(isNatIList(V2)))
            active(U63(X)) -> U63(active(X))
            active(U63(tt())) -> mark(tt())
            active(U71(X1,X2)) -> U71(active(X1),X2)
            active(U71(tt(),L)) -> mark(s(length(L)))
            active(U81(X)) -> U81(active(X))
            active(U81(tt())) -> mark(nil())
            active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4)
            active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
            active(and(X1,X2)) -> and(active(X1),X2)
            active(and(tt(),X)) -> mark(X)
            active(cons(X1,X2)) -> cons(active(X1),X2)
            active(isNat(0())) -> mark(tt())
            active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
            active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
            active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
            active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2))
            active(isNatIList(zeros())) -> mark(tt())
            active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2)))
            active(isNatIListKind(nil())) -> mark(tt())
            active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2)))
            active(isNatIListKind(zeros())) -> mark(tt())
            active(isNatKind(0())) -> mark(tt())
            active(isNatKind(length(V1))) -> mark(isNatIListKind(V1))
            active(isNatKind(s(V1))) -> mark(isNatKind(V1))
            active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2))
            active(isNatList(nil())) -> mark(tt())
            active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2))
            active(length(X)) -> length(active(X))
            active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N)))
                                                 ,L))
            active(length(nil())) -> mark(0())
            active(s(X)) -> s(active(X))
            active(take(X1,X2)) -> take(X1,active(X2))
            active(take(X1,X2)) -> take(active(X1),X2)
            active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL))))
            active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL))
                                                         ,and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))))
                                                     ,IL
                                                     ,M
                                                     ,N))
            active(zeros()) -> mark(cons(0(),zeros()))
            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))
            isNat(ok(X)) -> ok(isNat(X))
            isNatIList(ok(X)) -> ok(isNatIList(X))
            isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
            isNatKind(ok(X)) -> ok(isNatKind(X))
            isNatList(ok(X)) -> ok(isNatList(X))
            length(mark(X)) -> mark(length(X))
            length(ok(X)) -> ok(length(X))
            proper(0()) -> ok(0())
            proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
            proper(U12(X)) -> U12(proper(X))
            proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
            proper(U22(X)) -> U22(proper(X))
            proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
            proper(U32(X)) -> U32(proper(X))
            proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
            proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
            proper(U43(X)) -> U43(proper(X))
            proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
            proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
            proper(U53(X)) -> U53(proper(X))
            proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3))
            proper(U62(X1,X2)) -> U62(proper(X1),proper(X2))
            proper(U63(X)) -> U63(proper(X))
            proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
            proper(U81(X)) -> U81(proper(X))
            proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4))
            proper(and(X1,X2)) -> and(proper(X1),proper(X2))
            proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
            proper(isNat(X)) -> isNat(proper(X))
            proper(isNatIList(X)) -> isNatIList(proper(X))
            proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
            proper(isNatKind(X)) -> isNatKind(proper(X))
            proper(isNatList(X)) -> isNatList(proper(X))
            proper(length(X)) -> length(proper(X))
            proper(nil()) -> ok(nil())
            proper(s(X)) -> s(proper(X))
            proper(take(X1,X2)) -> take(proper(X1),proper(X2))
            proper(tt()) -> ok(tt())
            proper(zeros()) -> ok(zeros())
            s(mark(X)) -> mark(s(X))
            s(ok(X)) -> ok(s(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:
            {U11/2,U12/1,U21/2,U22/1,U31/2,U32/1,U41/3,U42/2,U43/1,U51/3,U52/2,U53/1,U61/3,U62/2,U63/1,U71/2,U81/1,U91/4
            ,active/1,and/2,cons/2,isNat/1,isNatIList/1,isNatIListKind/1,isNatKind/1,isNatList/1,length/1,proper/1,s/1
            ,take/2,top/1} / {0/0,mark/1,nil/0,ok/1,tt/0,zeros/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {U11,U12,U21,U22,U31,U32,U41,U42,U43,U51,U52,U53,U61,U62
            ,U63,U71,U81,U91,active,and,cons,isNat,isNatIList,isNatIListKind,isNatKind,isNatList,length,proper,s,take
            ,top} and constructors {0,mark,nil,ok,tt,zeros}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          U11(x,y){x -> mark(x)} =
            U11(mark(x),y) ->^+ mark(U11(x,y))
              = C[U11(x,y) = U11(x,y){}]

WORST_CASE(Omega(n^1),?)