/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(NON_POLY, ?) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). (0) CpxTRS (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (2) TRS for Loop Detection (3) DecreasingLoopProof [LOWER BOUND(ID), 1225 ms] (4) BEST (5) proven lower bound (6) LowerBoundPropagationProof [FINISHED, 0 ms] (7) BOUNDS(n^1, INF) (8) TRS for Loop Detection (9) DecreasingLoopProof [FINISHED, 19.5 s] (10) BOUNDS(EXP, INF) ---------------------------------------- (0) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). The TRS R consists of the following rules: U101(tt, V1, V2) -> U102(isNaturalKind(activate(V1)), activate(V1), activate(V2)) U102(tt, V1, V2) -> U103(isLNatKind(activate(V2)), activate(V1), activate(V2)) U103(tt, V1, V2) -> U104(isLNatKind(activate(V2)), activate(V1), activate(V2)) U104(tt, V1, V2) -> U105(isNatural(activate(V1)), activate(V2)) U105(tt, V2) -> U106(isLNat(activate(V2))) U106(tt) -> tt U11(tt, N, XS) -> U12(isNaturalKind(activate(N)), activate(N), activate(XS)) U111(tt, V2) -> U112(isLNatKind(activate(V2))) U112(tt) -> tt U12(tt, N, XS) -> U13(isLNat(activate(XS)), activate(N), activate(XS)) U121(tt, V2) -> U122(isLNatKind(activate(V2))) U122(tt) -> tt U13(tt, N, XS) -> U14(isLNatKind(activate(XS)), activate(N), activate(XS)) U131(tt) -> tt U14(tt, N, XS) -> snd(splitAt(activate(N), activate(XS))) U141(tt) -> tt U151(tt) -> tt U161(tt) -> tt U171(tt, V2) -> U172(isLNatKind(activate(V2))) U172(tt) -> tt U181(tt, V1) -> U182(isLNatKind(activate(V1)), activate(V1)) U182(tt, V1) -> U183(isLNat(activate(V1))) U183(tt) -> tt U191(tt, V1) -> U192(isNaturalKind(activate(V1)), activate(V1)) U192(tt, V1) -> U193(isNatural(activate(V1))) U193(tt) -> tt U201(tt, V1, V2) -> U202(isNaturalKind(activate(V1)), activate(V1), activate(V2)) U202(tt, V1, V2) -> U203(isLNatKind(activate(V2)), activate(V1), activate(V2)) U203(tt, V1, V2) -> U204(isLNatKind(activate(V2)), activate(V1), activate(V2)) U204(tt, V1, V2) -> U205(isNatural(activate(V1)), activate(V2)) U205(tt, V2) -> U206(isLNat(activate(V2))) U206(tt) -> tt U21(tt, X, Y) -> U22(isLNatKind(activate(X)), activate(X), activate(Y)) U211(tt) -> tt U22(tt, X, Y) -> U23(isLNat(activate(Y)), activate(X), activate(Y)) U221(tt) -> tt U23(tt, X, Y) -> U24(isLNatKind(activate(Y)), activate(X)) U231(tt, V2) -> U232(isLNatKind(activate(V2))) U232(tt) -> tt U24(tt, X) -> activate(X) U241(tt, V1, V2) -> U242(isLNatKind(activate(V1)), activate(V1), activate(V2)) U242(tt, V1, V2) -> U243(isLNatKind(activate(V2)), activate(V1), activate(V2)) U243(tt, V1, V2) -> U244(isLNatKind(activate(V2)), activate(V1), activate(V2)) U244(tt, V1, V2) -> U245(isLNat(activate(V1)), activate(V2)) U245(tt, V2) -> U246(isLNat(activate(V2))) U246(tt) -> tt U251(tt, V1, V2) -> U252(isNaturalKind(activate(V1)), activate(V1), activate(V2)) U252(tt, V1, V2) -> U253(isLNatKind(activate(V2)), activate(V1), activate(V2)) U253(tt, V1, V2) -> U254(isLNatKind(activate(V2)), activate(V1), activate(V2)) U254(tt, V1, V2) -> U255(isNatural(activate(V1)), activate(V2)) U255(tt, V2) -> U256(isLNat(activate(V2))) U256(tt) -> tt U261(tt, V2) -> U262(isLNatKind(activate(V2))) U262(tt) -> tt U271(tt, V2) -> U272(isLNatKind(activate(V2))) U272(tt) -> tt U281(tt, N) -> U282(isNaturalKind(activate(N)), activate(N)) U282(tt, N) -> cons(activate(N), n__natsFrom(s(activate(N)))) U291(tt, N, XS) -> U292(isNaturalKind(activate(N)), activate(N), activate(XS)) U292(tt, N, XS) -> U293(isLNat(activate(XS)), activate(N), activate(XS)) U293(tt, N, XS) -> U294(isLNatKind(activate(XS)), activate(N), activate(XS)) U294(tt, N, XS) -> head(afterNth(activate(N), activate(XS))) U301(tt, X, Y) -> U302(isLNatKind(activate(X)), activate(Y)) U302(tt, Y) -> U303(isLNat(activate(Y)), activate(Y)) U303(tt, Y) -> U304(isLNatKind(activate(Y)), activate(Y)) U304(tt, Y) -> activate(Y) U31(tt, N, XS) -> U32(isNaturalKind(activate(N)), activate(N), activate(XS)) U311(tt, XS) -> U312(isLNatKind(activate(XS)), activate(XS)) U312(tt, XS) -> pair(nil, activate(XS)) U32(tt, N, XS) -> U33(isLNat(activate(XS)), activate(N), activate(XS)) U321(tt, N, X, XS) -> U322(isNaturalKind(activate(N)), activate(N), activate(X), activate(XS)) U322(tt, N, X, XS) -> U323(isNatural(activate(X)), activate(N), activate(X), activate(XS)) U323(tt, N, X, XS) -> U324(isNaturalKind(activate(X)), activate(N), activate(X), activate(XS)) U324(tt, N, X, XS) -> U325(isLNat(activate(XS)), activate(N), activate(X), activate(XS)) U325(tt, N, X, XS) -> U326(isLNatKind(activate(XS)), activate(N), activate(X), activate(XS)) U326(tt, N, X, XS) -> U327(splitAt(activate(N), activate(XS)), activate(X)) U327(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS) U33(tt, N, XS) -> U34(isLNatKind(activate(XS)), activate(N)) U331(tt, N, XS) -> U332(isNaturalKind(activate(N)), activate(XS)) U332(tt, XS) -> U333(isLNat(activate(XS)), activate(XS)) U333(tt, XS) -> U334(isLNatKind(activate(XS)), activate(XS)) U334(tt, XS) -> activate(XS) U34(tt, N) -> activate(N) U341(tt, N, XS) -> U342(isNaturalKind(activate(N)), activate(N), activate(XS)) U342(tt, N, XS) -> U343(isLNat(activate(XS)), activate(N), activate(XS)) U343(tt, N, XS) -> U344(isLNatKind(activate(XS)), activate(N), activate(XS)) U344(tt, N, XS) -> fst(splitAt(activate(N), activate(XS))) U41(tt, V1, V2) -> U42(isNaturalKind(activate(V1)), activate(V1), activate(V2)) U42(tt, V1, V2) -> U43(isLNatKind(activate(V2)), activate(V1), activate(V2)) U43(tt, V1, V2) -> U44(isLNatKind(activate(V2)), activate(V1), activate(V2)) U44(tt, V1, V2) -> U45(isNatural(activate(V1)), activate(V2)) U45(tt, V2) -> U46(isLNat(activate(V2))) U46(tt) -> tt U51(tt, V1, V2) -> U52(isNaturalKind(activate(V1)), activate(V1), activate(V2)) U52(tt, V1, V2) -> U53(isLNatKind(activate(V2)), activate(V1), activate(V2)) U53(tt, V1, V2) -> U54(isLNatKind(activate(V2)), activate(V1), activate(V2)) U54(tt, V1, V2) -> U55(isNatural(activate(V1)), activate(V2)) U55(tt, V2) -> U56(isLNat(activate(V2))) U56(tt) -> tt U61(tt, V1) -> U62(isPLNatKind(activate(V1)), activate(V1)) U62(tt, V1) -> U63(isPLNat(activate(V1))) U63(tt) -> tt U71(tt, V1) -> U72(isNaturalKind(activate(V1)), activate(V1)) U72(tt, V1) -> U73(isNatural(activate(V1))) U73(tt) -> tt U81(tt, V1) -> U82(isPLNatKind(activate(V1)), activate(V1)) U82(tt, V1) -> U83(isPLNat(activate(V1))) U83(tt) -> tt U91(tt, V1) -> U92(isLNatKind(activate(V1)), activate(V1)) U92(tt, V1) -> U93(isLNat(activate(V1))) U93(tt) -> tt afterNth(N, XS) -> U11(isNatural(N), N, XS) fst(pair(X, Y)) -> U21(isLNat(X), X, Y) head(cons(N, XS)) -> U31(isNatural(N), N, activate(XS)) isLNat(n__nil) -> tt isLNat(n__afterNth(V1, V2)) -> U41(isNaturalKind(activate(V1)), activate(V1), activate(V2)) isLNat(n__cons(V1, V2)) -> U51(isNaturalKind(activate(V1)), activate(V1), activate(V2)) isLNat(n__fst(V1)) -> U61(isPLNatKind(activate(V1)), activate(V1)) isLNat(n__natsFrom(V1)) -> U71(isNaturalKind(activate(V1)), activate(V1)) isLNat(n__snd(V1)) -> U81(isPLNatKind(activate(V1)), activate(V1)) isLNat(n__tail(V1)) -> U91(isLNatKind(activate(V1)), activate(V1)) isLNat(n__take(V1, V2)) -> U101(isNaturalKind(activate(V1)), activate(V1), activate(V2)) isLNatKind(n__nil) -> tt isLNatKind(n__afterNth(V1, V2)) -> U111(isNaturalKind(activate(V1)), activate(V2)) isLNatKind(n__cons(V1, V2)) -> U121(isNaturalKind(activate(V1)), activate(V2)) isLNatKind(n__fst(V1)) -> U131(isPLNatKind(activate(V1))) isLNatKind(n__natsFrom(V1)) -> U141(isNaturalKind(activate(V1))) isLNatKind(n__snd(V1)) -> U151(isPLNatKind(activate(V1))) isLNatKind(n__tail(V1)) -> U161(isLNatKind(activate(V1))) isLNatKind(n__take(V1, V2)) -> U171(isNaturalKind(activate(V1)), activate(V2)) isNatural(n__0) -> tt isNatural(n__head(V1)) -> U181(isLNatKind(activate(V1)), activate(V1)) isNatural(n__s(V1)) -> U191(isNaturalKind(activate(V1)), activate(V1)) isNatural(n__sel(V1, V2)) -> U201(isNaturalKind(activate(V1)), activate(V1), activate(V2)) isNaturalKind(n__0) -> tt isNaturalKind(n__head(V1)) -> U211(isLNatKind(activate(V1))) isNaturalKind(n__s(V1)) -> U221(isNaturalKind(activate(V1))) isNaturalKind(n__sel(V1, V2)) -> U231(isNaturalKind(activate(V1)), activate(V2)) isPLNat(n__pair(V1, V2)) -> U241(isLNatKind(activate(V1)), activate(V1), activate(V2)) isPLNat(n__splitAt(V1, V2)) -> U251(isNaturalKind(activate(V1)), activate(V1), activate(V2)) isPLNatKind(n__pair(V1, V2)) -> U261(isLNatKind(activate(V1)), activate(V2)) isPLNatKind(n__splitAt(V1, V2)) -> U271(isNaturalKind(activate(V1)), activate(V2)) natsFrom(N) -> U281(isNatural(N), N) sel(N, XS) -> U291(isNatural(N), N, XS) snd(pair(X, Y)) -> U301(isLNat(X), X, Y) splitAt(0, XS) -> U311(isLNat(XS), XS) splitAt(s(N), cons(X, XS)) -> U321(isNatural(N), N, X, activate(XS)) tail(cons(N, XS)) -> U331(isNatural(N), N, activate(XS)) take(N, XS) -> U341(isNatural(N), N, XS) natsFrom(X) -> n__natsFrom(X) nil -> n__nil afterNth(X1, X2) -> n__afterNth(X1, X2) cons(X1, X2) -> n__cons(X1, X2) fst(X) -> n__fst(X) snd(X) -> n__snd(X) tail(X) -> n__tail(X) take(X1, X2) -> n__take(X1, X2) 0 -> n__0 head(X) -> n__head(X) s(X) -> n__s(X) sel(X1, X2) -> n__sel(X1, X2) pair(X1, X2) -> n__pair(X1, X2) splitAt(X1, X2) -> n__splitAt(X1, X2) activate(n__natsFrom(X)) -> natsFrom(X) activate(n__nil) -> nil activate(n__afterNth(X1, X2)) -> afterNth(X1, X2) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__fst(X)) -> fst(X) activate(n__snd(X)) -> snd(X) activate(n__tail(X)) -> tail(X) activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__head(X)) -> head(X) activate(n__s(X)) -> s(X) activate(n__sel(X1, X2)) -> sel(X1, X2) activate(n__pair(X1, X2)) -> pair(X1, X2) activate(n__splitAt(X1, X2)) -> splitAt(X1, X2) activate(X) -> X S is empty. Rewrite Strategy: FULL ---------------------------------------- (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (2) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). The TRS R consists of the following rules: U101(tt, V1, V2) -> U102(isNaturalKind(activate(V1)), activate(V1), activate(V2)) U102(tt, V1, V2) -> U103(isLNatKind(activate(V2)), activate(V1), activate(V2)) U103(tt, V1, V2) -> U104(isLNatKind(activate(V2)), activate(V1), activate(V2)) U104(tt, V1, V2) -> U105(isNatural(activate(V1)), activate(V2)) U105(tt, V2) -> U106(isLNat(activate(V2))) U106(tt) -> tt U11(tt, N, XS) -> U12(isNaturalKind(activate(N)), activate(N), activate(XS)) U111(tt, V2) -> U112(isLNatKind(activate(V2))) U112(tt) -> tt U12(tt, N, XS) -> U13(isLNat(activate(XS)), activate(N), activate(XS)) U121(tt, V2) -> U122(isLNatKind(activate(V2))) U122(tt) -> tt U13(tt, N, XS) -> U14(isLNatKind(activate(XS)), activate(N), activate(XS)) U131(tt) -> tt U14(tt, N, XS) -> snd(splitAt(activate(N), activate(XS))) U141(tt) -> tt U151(tt) -> tt U161(tt) -> tt U171(tt, V2) -> U172(isLNatKind(activate(V2))) U172(tt) -> tt U181(tt, V1) -> U182(isLNatKind(activate(V1)), activate(V1)) U182(tt, V1) -> U183(isLNat(activate(V1))) U183(tt) -> tt U191(tt, V1) -> U192(isNaturalKind(activate(V1)), activate(V1)) U192(tt, V1) -> U193(isNatural(activate(V1))) U193(tt) -> tt U201(tt, V1, V2) -> U202(isNaturalKind(activate(V1)), activate(V1), activate(V2)) U202(tt, V1, V2) -> U203(isLNatKind(activate(V2)), activate(V1), activate(V2)) U203(tt, V1, V2) -> U204(isLNatKind(activate(V2)), activate(V1), activate(V2)) U204(tt, V1, V2) -> U205(isNatural(activate(V1)), activate(V2)) U205(tt, V2) -> U206(isLNat(activate(V2))) U206(tt) -> tt U21(tt, X, Y) -> U22(isLNatKind(activate(X)), activate(X), activate(Y)) U211(tt) -> tt U22(tt, X, Y) -> U23(isLNat(activate(Y)), activate(X), activate(Y)) U221(tt) -> tt U23(tt, X, Y) -> U24(isLNatKind(activate(Y)), activate(X)) U231(tt, V2) -> U232(isLNatKind(activate(V2))) U232(tt) -> tt U24(tt, X) -> activate(X) U241(tt, V1, V2) -> U242(isLNatKind(activate(V1)), activate(V1), activate(V2)) U242(tt, V1, V2) -> U243(isLNatKind(activate(V2)), activate(V1), activate(V2)) U243(tt, V1, V2) -> U244(isLNatKind(activate(V2)), activate(V1), activate(V2)) U244(tt, V1, V2) -> U245(isLNat(activate(V1)), activate(V2)) U245(tt, V2) -> U246(isLNat(activate(V2))) U246(tt) -> tt U251(tt, V1, V2) -> U252(isNaturalKind(activate(V1)), activate(V1), activate(V2)) U252(tt, V1, V2) -> U253(isLNatKind(activate(V2)), activate(V1), activate(V2)) U253(tt, V1, V2) -> U254(isLNatKind(activate(V2)), activate(V1), activate(V2)) U254(tt, V1, V2) -> U255(isNatural(activate(V1)), activate(V2)) U255(tt, V2) -> U256(isLNat(activate(V2))) U256(tt) -> tt U261(tt, V2) -> U262(isLNatKind(activate(V2))) U262(tt) -> tt U271(tt, V2) -> U272(isLNatKind(activate(V2))) U272(tt) -> tt U281(tt, N) -> U282(isNaturalKind(activate(N)), activate(N)) U282(tt, N) -> cons(activate(N), n__natsFrom(s(activate(N)))) U291(tt, N, XS) -> U292(isNaturalKind(activate(N)), activate(N), activate(XS)) U292(tt, N, XS) -> U293(isLNat(activate(XS)), activate(N), activate(XS)) U293(tt, N, XS) -> U294(isLNatKind(activate(XS)), activate(N), activate(XS)) U294(tt, N, XS) -> head(afterNth(activate(N), activate(XS))) U301(tt, X, Y) -> U302(isLNatKind(activate(X)), activate(Y)) U302(tt, Y) -> U303(isLNat(activate(Y)), activate(Y)) U303(tt, Y) -> U304(isLNatKind(activate(Y)), activate(Y)) U304(tt, Y) -> activate(Y) U31(tt, N, XS) -> U32(isNaturalKind(activate(N)), activate(N), activate(XS)) U311(tt, XS) -> U312(isLNatKind(activate(XS)), activate(XS)) U312(tt, XS) -> pair(nil, activate(XS)) U32(tt, N, XS) -> U33(isLNat(activate(XS)), activate(N), activate(XS)) U321(tt, N, X, XS) -> U322(isNaturalKind(activate(N)), activate(N), activate(X), activate(XS)) U322(tt, N, X, XS) -> U323(isNatural(activate(X)), activate(N), activate(X), activate(XS)) U323(tt, N, X, XS) -> U324(isNaturalKind(activate(X)), activate(N), activate(X), activate(XS)) U324(tt, N, X, XS) -> U325(isLNat(activate(XS)), activate(N), activate(X), activate(XS)) U325(tt, N, X, XS) -> U326(isLNatKind(activate(XS)), activate(N), activate(X), activate(XS)) U326(tt, N, X, XS) -> U327(splitAt(activate(N), activate(XS)), activate(X)) U327(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS) U33(tt, N, XS) -> U34(isLNatKind(activate(XS)), activate(N)) U331(tt, N, XS) -> U332(isNaturalKind(activate(N)), activate(XS)) U332(tt, XS) -> U333(isLNat(activate(XS)), activate(XS)) U333(tt, XS) -> U334(isLNatKind(activate(XS)), activate(XS)) U334(tt, XS) -> activate(XS) U34(tt, N) -> activate(N) U341(tt, N, XS) -> U342(isNaturalKind(activate(N)), activate(N), activate(XS)) U342(tt, N, XS) -> U343(isLNat(activate(XS)), activate(N), activate(XS)) U343(tt, N, XS) -> U344(isLNatKind(activate(XS)), activate(N), activate(XS)) U344(tt, N, XS) -> fst(splitAt(activate(N), activate(XS))) U41(tt, V1, V2) -> U42(isNaturalKind(activate(V1)), activate(V1), activate(V2)) U42(tt, V1, V2) -> U43(isLNatKind(activate(V2)), activate(V1), activate(V2)) U43(tt, V1, V2) -> U44(isLNatKind(activate(V2)), activate(V1), activate(V2)) U44(tt, V1, V2) -> U45(isNatural(activate(V1)), activate(V2)) U45(tt, V2) -> U46(isLNat(activate(V2))) U46(tt) -> tt U51(tt, V1, V2) -> U52(isNaturalKind(activate(V1)), activate(V1), activate(V2)) U52(tt, V1, V2) -> U53(isLNatKind(activate(V2)), activate(V1), activate(V2)) U53(tt, V1, V2) -> U54(isLNatKind(activate(V2)), activate(V1), activate(V2)) U54(tt, V1, V2) -> U55(isNatural(activate(V1)), activate(V2)) U55(tt, V2) -> U56(isLNat(activate(V2))) U56(tt) -> tt U61(tt, V1) -> U62(isPLNatKind(activate(V1)), activate(V1)) U62(tt, V1) -> U63(isPLNat(activate(V1))) U63(tt) -> tt U71(tt, V1) -> U72(isNaturalKind(activate(V1)), activate(V1)) U72(tt, V1) -> U73(isNatural(activate(V1))) U73(tt) -> tt U81(tt, V1) -> U82(isPLNatKind(activate(V1)), activate(V1)) U82(tt, V1) -> U83(isPLNat(activate(V1))) U83(tt) -> tt U91(tt, V1) -> U92(isLNatKind(activate(V1)), activate(V1)) U92(tt, V1) -> U93(isLNat(activate(V1))) U93(tt) -> tt afterNth(N, XS) -> U11(isNatural(N), N, XS) fst(pair(X, Y)) -> U21(isLNat(X), X, Y) head(cons(N, XS)) -> U31(isNatural(N), N, activate(XS)) isLNat(n__nil) -> tt isLNat(n__afterNth(V1, V2)) -> U41(isNaturalKind(activate(V1)), activate(V1), activate(V2)) isLNat(n__cons(V1, V2)) -> U51(isNaturalKind(activate(V1)), activate(V1), activate(V2)) isLNat(n__fst(V1)) -> U61(isPLNatKind(activate(V1)), activate(V1)) isLNat(n__natsFrom(V1)) -> U71(isNaturalKind(activate(V1)), activate(V1)) isLNat(n__snd(V1)) -> U81(isPLNatKind(activate(V1)), activate(V1)) isLNat(n__tail(V1)) -> U91(isLNatKind(activate(V1)), activate(V1)) isLNat(n__take(V1, V2)) -> U101(isNaturalKind(activate(V1)), activate(V1), activate(V2)) isLNatKind(n__nil) -> tt isLNatKind(n__afterNth(V1, V2)) -> U111(isNaturalKind(activate(V1)), activate(V2)) isLNatKind(n__cons(V1, V2)) -> U121(isNaturalKind(activate(V1)), activate(V2)) isLNatKind(n__fst(V1)) -> U131(isPLNatKind(activate(V1))) isLNatKind(n__natsFrom(V1)) -> U141(isNaturalKind(activate(V1))) isLNatKind(n__snd(V1)) -> U151(isPLNatKind(activate(V1))) isLNatKind(n__tail(V1)) -> U161(isLNatKind(activate(V1))) isLNatKind(n__take(V1, V2)) -> U171(isNaturalKind(activate(V1)), activate(V2)) isNatural(n__0) -> tt isNatural(n__head(V1)) -> U181(isLNatKind(activate(V1)), activate(V1)) isNatural(n__s(V1)) -> U191(isNaturalKind(activate(V1)), activate(V1)) isNatural(n__sel(V1, V2)) -> U201(isNaturalKind(activate(V1)), activate(V1), activate(V2)) isNaturalKind(n__0) -> tt isNaturalKind(n__head(V1)) -> U211(isLNatKind(activate(V1))) isNaturalKind(n__s(V1)) -> U221(isNaturalKind(activate(V1))) isNaturalKind(n__sel(V1, V2)) -> U231(isNaturalKind(activate(V1)), activate(V2)) isPLNat(n__pair(V1, V2)) -> U241(isLNatKind(activate(V1)), activate(V1), activate(V2)) isPLNat(n__splitAt(V1, V2)) -> U251(isNaturalKind(activate(V1)), activate(V1), activate(V2)) isPLNatKind(n__pair(V1, V2)) -> U261(isLNatKind(activate(V1)), activate(V2)) isPLNatKind(n__splitAt(V1, V2)) -> U271(isNaturalKind(activate(V1)), activate(V2)) natsFrom(N) -> U281(isNatural(N), N) sel(N, XS) -> U291(isNatural(N), N, XS) snd(pair(X, Y)) -> U301(isLNat(X), X, Y) splitAt(0, XS) -> U311(isLNat(XS), XS) splitAt(s(N), cons(X, XS)) -> U321(isNatural(N), N, X, activate(XS)) tail(cons(N, XS)) -> U331(isNatural(N), N, activate(XS)) take(N, XS) -> U341(isNatural(N), N, XS) natsFrom(X) -> n__natsFrom(X) nil -> n__nil afterNth(X1, X2) -> n__afterNth(X1, X2) cons(X1, X2) -> n__cons(X1, X2) fst(X) -> n__fst(X) snd(X) -> n__snd(X) tail(X) -> n__tail(X) take(X1, X2) -> n__take(X1, X2) 0 -> n__0 head(X) -> n__head(X) s(X) -> n__s(X) sel(X1, X2) -> n__sel(X1, X2) pair(X1, X2) -> n__pair(X1, X2) splitAt(X1, X2) -> n__splitAt(X1, X2) activate(n__natsFrom(X)) -> natsFrom(X) activate(n__nil) -> nil activate(n__afterNth(X1, X2)) -> afterNth(X1, X2) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__fst(X)) -> fst(X) activate(n__snd(X)) -> snd(X) activate(n__tail(X)) -> tail(X) activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__head(X)) -> head(X) activate(n__s(X)) -> s(X) activate(n__sel(X1, X2)) -> sel(X1, X2) activate(n__pair(X1, X2)) -> pair(X1, X2) activate(n__splitAt(X1, X2)) -> splitAt(X1, X2) activate(X) -> X S is empty. Rewrite Strategy: FULL ---------------------------------------- (3) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence isNaturalKind(n__sel(V1, V2)) ->^+ U231(isNaturalKind(V1), activate(V2)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [V1 / n__sel(V1, V2)]. The result substitution is [ ]. ---------------------------------------- (4) Complex Obligation (BEST) ---------------------------------------- (5) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). The TRS R consists of the following rules: U101(tt, V1, V2) -> U102(isNaturalKind(activate(V1)), activate(V1), activate(V2)) U102(tt, V1, V2) -> U103(isLNatKind(activate(V2)), activate(V1), activate(V2)) U103(tt, V1, V2) -> U104(isLNatKind(activate(V2)), activate(V1), activate(V2)) U104(tt, V1, V2) -> U105(isNatural(activate(V1)), activate(V2)) U105(tt, V2) -> U106(isLNat(activate(V2))) U106(tt) -> tt U11(tt, N, XS) -> U12(isNaturalKind(activate(N)), activate(N), activate(XS)) U111(tt, V2) -> U112(isLNatKind(activate(V2))) U112(tt) -> tt U12(tt, N, XS) -> U13(isLNat(activate(XS)), activate(N), activate(XS)) U121(tt, V2) -> U122(isLNatKind(activate(V2))) U122(tt) -> tt U13(tt, N, XS) -> U14(isLNatKind(activate(XS)), activate(N), activate(XS)) U131(tt) -> tt U14(tt, N, XS) -> snd(splitAt(activate(N), activate(XS))) U141(tt) -> tt U151(tt) -> tt U161(tt) -> tt U171(tt, V2) -> U172(isLNatKind(activate(V2))) U172(tt) -> tt U181(tt, V1) -> U182(isLNatKind(activate(V1)), activate(V1)) U182(tt, V1) -> U183(isLNat(activate(V1))) U183(tt) -> tt U191(tt, V1) -> U192(isNaturalKind(activate(V1)), activate(V1)) U192(tt, V1) -> U193(isNatural(activate(V1))) U193(tt) -> tt U201(tt, V1, V2) -> U202(isNaturalKind(activate(V1)), activate(V1), activate(V2)) U202(tt, V1, V2) -> U203(isLNatKind(activate(V2)), activate(V1), activate(V2)) U203(tt, V1, V2) -> U204(isLNatKind(activate(V2)), activate(V1), activate(V2)) U204(tt, V1, V2) -> U205(isNatural(activate(V1)), activate(V2)) U205(tt, V2) -> U206(isLNat(activate(V2))) U206(tt) -> tt U21(tt, X, Y) -> U22(isLNatKind(activate(X)), activate(X), activate(Y)) U211(tt) -> tt U22(tt, X, Y) -> U23(isLNat(activate(Y)), activate(X), activate(Y)) U221(tt) -> tt U23(tt, X, Y) -> U24(isLNatKind(activate(Y)), activate(X)) U231(tt, V2) -> U232(isLNatKind(activate(V2))) U232(tt) -> tt U24(tt, X) -> activate(X) U241(tt, V1, V2) -> U242(isLNatKind(activate(V1)), activate(V1), activate(V2)) U242(tt, V1, V2) -> U243(isLNatKind(activate(V2)), activate(V1), activate(V2)) U243(tt, V1, V2) -> U244(isLNatKind(activate(V2)), activate(V1), activate(V2)) U244(tt, V1, V2) -> U245(isLNat(activate(V1)), activate(V2)) U245(tt, V2) -> U246(isLNat(activate(V2))) U246(tt) -> tt U251(tt, V1, V2) -> U252(isNaturalKind(activate(V1)), activate(V1), activate(V2)) U252(tt, V1, V2) -> U253(isLNatKind(activate(V2)), activate(V1), activate(V2)) U253(tt, V1, V2) -> U254(isLNatKind(activate(V2)), activate(V1), activate(V2)) U254(tt, V1, V2) -> U255(isNatural(activate(V1)), activate(V2)) U255(tt, V2) -> U256(isLNat(activate(V2))) U256(tt) -> tt U261(tt, V2) -> U262(isLNatKind(activate(V2))) U262(tt) -> tt U271(tt, V2) -> U272(isLNatKind(activate(V2))) U272(tt) -> tt U281(tt, N) -> U282(isNaturalKind(activate(N)), activate(N)) U282(tt, N) -> cons(activate(N), n__natsFrom(s(activate(N)))) U291(tt, N, XS) -> U292(isNaturalKind(activate(N)), activate(N), activate(XS)) U292(tt, N, XS) -> U293(isLNat(activate(XS)), activate(N), activate(XS)) U293(tt, N, XS) -> U294(isLNatKind(activate(XS)), activate(N), activate(XS)) U294(tt, N, XS) -> head(afterNth(activate(N), activate(XS))) U301(tt, X, Y) -> U302(isLNatKind(activate(X)), activate(Y)) U302(tt, Y) -> U303(isLNat(activate(Y)), activate(Y)) U303(tt, Y) -> U304(isLNatKind(activate(Y)), activate(Y)) U304(tt, Y) -> activate(Y) U31(tt, N, XS) -> U32(isNaturalKind(activate(N)), activate(N), activate(XS)) U311(tt, XS) -> U312(isLNatKind(activate(XS)), activate(XS)) U312(tt, XS) -> pair(nil, activate(XS)) U32(tt, N, XS) -> U33(isLNat(activate(XS)), activate(N), activate(XS)) U321(tt, N, X, XS) -> U322(isNaturalKind(activate(N)), activate(N), activate(X), activate(XS)) U322(tt, N, X, XS) -> U323(isNatural(activate(X)), activate(N), activate(X), activate(XS)) U323(tt, N, X, XS) -> U324(isNaturalKind(activate(X)), activate(N), activate(X), activate(XS)) U324(tt, N, X, XS) -> U325(isLNat(activate(XS)), activate(N), activate(X), activate(XS)) U325(tt, N, X, XS) -> U326(isLNatKind(activate(XS)), activate(N), activate(X), activate(XS)) U326(tt, N, X, XS) -> U327(splitAt(activate(N), activate(XS)), activate(X)) U327(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS) U33(tt, N, XS) -> U34(isLNatKind(activate(XS)), activate(N)) U331(tt, N, XS) -> U332(isNaturalKind(activate(N)), activate(XS)) U332(tt, XS) -> U333(isLNat(activate(XS)), activate(XS)) U333(tt, XS) -> U334(isLNatKind(activate(XS)), activate(XS)) U334(tt, XS) -> activate(XS) U34(tt, N) -> activate(N) U341(tt, N, XS) -> U342(isNaturalKind(activate(N)), activate(N), activate(XS)) U342(tt, N, XS) -> U343(isLNat(activate(XS)), activate(N), activate(XS)) U343(tt, N, XS) -> U344(isLNatKind(activate(XS)), activate(N), activate(XS)) U344(tt, N, XS) -> fst(splitAt(activate(N), activate(XS))) U41(tt, V1, V2) -> U42(isNaturalKind(activate(V1)), activate(V1), activate(V2)) U42(tt, V1, V2) -> U43(isLNatKind(activate(V2)), activate(V1), activate(V2)) U43(tt, V1, V2) -> U44(isLNatKind(activate(V2)), activate(V1), activate(V2)) U44(tt, V1, V2) -> U45(isNatural(activate(V1)), activate(V2)) U45(tt, V2) -> U46(isLNat(activate(V2))) U46(tt) -> tt U51(tt, V1, V2) -> U52(isNaturalKind(activate(V1)), activate(V1), activate(V2)) U52(tt, V1, V2) -> U53(isLNatKind(activate(V2)), activate(V1), activate(V2)) U53(tt, V1, V2) -> U54(isLNatKind(activate(V2)), activate(V1), activate(V2)) U54(tt, V1, V2) -> U55(isNatural(activate(V1)), activate(V2)) U55(tt, V2) -> U56(isLNat(activate(V2))) U56(tt) -> tt U61(tt, V1) -> U62(isPLNatKind(activate(V1)), activate(V1)) U62(tt, V1) -> U63(isPLNat(activate(V1))) U63(tt) -> tt U71(tt, V1) -> U72(isNaturalKind(activate(V1)), activate(V1)) U72(tt, V1) -> U73(isNatural(activate(V1))) U73(tt) -> tt U81(tt, V1) -> U82(isPLNatKind(activate(V1)), activate(V1)) U82(tt, V1) -> U83(isPLNat(activate(V1))) U83(tt) -> tt U91(tt, V1) -> U92(isLNatKind(activate(V1)), activate(V1)) U92(tt, V1) -> U93(isLNat(activate(V1))) U93(tt) -> tt afterNth(N, XS) -> U11(isNatural(N), N, XS) fst(pair(X, Y)) -> U21(isLNat(X), X, Y) head(cons(N, XS)) -> U31(isNatural(N), N, activate(XS)) isLNat(n__nil) -> tt isLNat(n__afterNth(V1, V2)) -> U41(isNaturalKind(activate(V1)), activate(V1), activate(V2)) isLNat(n__cons(V1, V2)) -> U51(isNaturalKind(activate(V1)), activate(V1), activate(V2)) isLNat(n__fst(V1)) -> U61(isPLNatKind(activate(V1)), activate(V1)) isLNat(n__natsFrom(V1)) -> U71(isNaturalKind(activate(V1)), activate(V1)) isLNat(n__snd(V1)) -> U81(isPLNatKind(activate(V1)), activate(V1)) isLNat(n__tail(V1)) -> U91(isLNatKind(activate(V1)), activate(V1)) isLNat(n__take(V1, V2)) -> U101(isNaturalKind(activate(V1)), activate(V1), activate(V2)) isLNatKind(n__nil) -> tt isLNatKind(n__afterNth(V1, V2)) -> U111(isNaturalKind(activate(V1)), activate(V2)) isLNatKind(n__cons(V1, V2)) -> U121(isNaturalKind(activate(V1)), activate(V2)) isLNatKind(n__fst(V1)) -> U131(isPLNatKind(activate(V1))) isLNatKind(n__natsFrom(V1)) -> U141(isNaturalKind(activate(V1))) isLNatKind(n__snd(V1)) -> U151(isPLNatKind(activate(V1))) isLNatKind(n__tail(V1)) -> U161(isLNatKind(activate(V1))) isLNatKind(n__take(V1, V2)) -> U171(isNaturalKind(activate(V1)), activate(V2)) isNatural(n__0) -> tt isNatural(n__head(V1)) -> U181(isLNatKind(activate(V1)), activate(V1)) isNatural(n__s(V1)) -> U191(isNaturalKind(activate(V1)), activate(V1)) isNatural(n__sel(V1, V2)) -> U201(isNaturalKind(activate(V1)), activate(V1), activate(V2)) isNaturalKind(n__0) -> tt isNaturalKind(n__head(V1)) -> U211(isLNatKind(activate(V1))) isNaturalKind(n__s(V1)) -> U221(isNaturalKind(activate(V1))) isNaturalKind(n__sel(V1, V2)) -> U231(isNaturalKind(activate(V1)), activate(V2)) isPLNat(n__pair(V1, V2)) -> U241(isLNatKind(activate(V1)), activate(V1), activate(V2)) isPLNat(n__splitAt(V1, V2)) -> U251(isNaturalKind(activate(V1)), activate(V1), activate(V2)) isPLNatKind(n__pair(V1, V2)) -> U261(isLNatKind(activate(V1)), activate(V2)) isPLNatKind(n__splitAt(V1, V2)) -> U271(isNaturalKind(activate(V1)), activate(V2)) natsFrom(N) -> U281(isNatural(N), N) sel(N, XS) -> U291(isNatural(N), N, XS) snd(pair(X, Y)) -> U301(isLNat(X), X, Y) splitAt(0, XS) -> U311(isLNat(XS), XS) splitAt(s(N), cons(X, XS)) -> U321(isNatural(N), N, X, activate(XS)) tail(cons(N, XS)) -> U331(isNatural(N), N, activate(XS)) take(N, XS) -> U341(isNatural(N), N, XS) natsFrom(X) -> n__natsFrom(X) nil -> n__nil afterNth(X1, X2) -> n__afterNth(X1, X2) cons(X1, X2) -> n__cons(X1, X2) fst(X) -> n__fst(X) snd(X) -> n__snd(X) tail(X) -> n__tail(X) take(X1, X2) -> n__take(X1, X2) 0 -> n__0 head(X) -> n__head(X) s(X) -> n__s(X) sel(X1, X2) -> n__sel(X1, X2) pair(X1, X2) -> n__pair(X1, X2) splitAt(X1, X2) -> n__splitAt(X1, X2) activate(n__natsFrom(X)) -> natsFrom(X) activate(n__nil) -> nil activate(n__afterNth(X1, X2)) -> afterNth(X1, X2) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__fst(X)) -> fst(X) activate(n__snd(X)) -> snd(X) activate(n__tail(X)) -> tail(X) activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__head(X)) -> head(X) activate(n__s(X)) -> s(X) activate(n__sel(X1, X2)) -> sel(X1, X2) activate(n__pair(X1, X2)) -> pair(X1, X2) activate(n__splitAt(X1, X2)) -> splitAt(X1, X2) activate(X) -> X S is empty. Rewrite Strategy: FULL ---------------------------------------- (6) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (7) BOUNDS(n^1, INF) ---------------------------------------- (8) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). The TRS R consists of the following rules: U101(tt, V1, V2) -> U102(isNaturalKind(activate(V1)), activate(V1), activate(V2)) U102(tt, V1, V2) -> U103(isLNatKind(activate(V2)), activate(V1), activate(V2)) U103(tt, V1, V2) -> U104(isLNatKind(activate(V2)), activate(V1), activate(V2)) U104(tt, V1, V2) -> U105(isNatural(activate(V1)), activate(V2)) U105(tt, V2) -> U106(isLNat(activate(V2))) U106(tt) -> tt U11(tt, N, XS) -> U12(isNaturalKind(activate(N)), activate(N), activate(XS)) U111(tt, V2) -> U112(isLNatKind(activate(V2))) U112(tt) -> tt U12(tt, N, XS) -> U13(isLNat(activate(XS)), activate(N), activate(XS)) U121(tt, V2) -> U122(isLNatKind(activate(V2))) U122(tt) -> tt U13(tt, N, XS) -> U14(isLNatKind(activate(XS)), activate(N), activate(XS)) U131(tt) -> tt U14(tt, N, XS) -> snd(splitAt(activate(N), activate(XS))) U141(tt) -> tt U151(tt) -> tt U161(tt) -> tt U171(tt, V2) -> U172(isLNatKind(activate(V2))) U172(tt) -> tt U181(tt, V1) -> U182(isLNatKind(activate(V1)), activate(V1)) U182(tt, V1) -> U183(isLNat(activate(V1))) U183(tt) -> tt U191(tt, V1) -> U192(isNaturalKind(activate(V1)), activate(V1)) U192(tt, V1) -> U193(isNatural(activate(V1))) U193(tt) -> tt U201(tt, V1, V2) -> U202(isNaturalKind(activate(V1)), activate(V1), activate(V2)) U202(tt, V1, V2) -> U203(isLNatKind(activate(V2)), activate(V1), activate(V2)) U203(tt, V1, V2) -> U204(isLNatKind(activate(V2)), activate(V1), activate(V2)) U204(tt, V1, V2) -> U205(isNatural(activate(V1)), activate(V2)) U205(tt, V2) -> U206(isLNat(activate(V2))) U206(tt) -> tt U21(tt, X, Y) -> U22(isLNatKind(activate(X)), activate(X), activate(Y)) U211(tt) -> tt U22(tt, X, Y) -> U23(isLNat(activate(Y)), activate(X), activate(Y)) U221(tt) -> tt U23(tt, X, Y) -> U24(isLNatKind(activate(Y)), activate(X)) U231(tt, V2) -> U232(isLNatKind(activate(V2))) U232(tt) -> tt U24(tt, X) -> activate(X) U241(tt, V1, V2) -> U242(isLNatKind(activate(V1)), activate(V1), activate(V2)) U242(tt, V1, V2) -> U243(isLNatKind(activate(V2)), activate(V1), activate(V2)) U243(tt, V1, V2) -> U244(isLNatKind(activate(V2)), activate(V1), activate(V2)) U244(tt, V1, V2) -> U245(isLNat(activate(V1)), activate(V2)) U245(tt, V2) -> U246(isLNat(activate(V2))) U246(tt) -> tt U251(tt, V1, V2) -> U252(isNaturalKind(activate(V1)), activate(V1), activate(V2)) U252(tt, V1, V2) -> U253(isLNatKind(activate(V2)), activate(V1), activate(V2)) U253(tt, V1, V2) -> U254(isLNatKind(activate(V2)), activate(V1), activate(V2)) U254(tt, V1, V2) -> U255(isNatural(activate(V1)), activate(V2)) U255(tt, V2) -> U256(isLNat(activate(V2))) U256(tt) -> tt U261(tt, V2) -> U262(isLNatKind(activate(V2))) U262(tt) -> tt U271(tt, V2) -> U272(isLNatKind(activate(V2))) U272(tt) -> tt U281(tt, N) -> U282(isNaturalKind(activate(N)), activate(N)) U282(tt, N) -> cons(activate(N), n__natsFrom(s(activate(N)))) U291(tt, N, XS) -> U292(isNaturalKind(activate(N)), activate(N), activate(XS)) U292(tt, N, XS) -> U293(isLNat(activate(XS)), activate(N), activate(XS)) U293(tt, N, XS) -> U294(isLNatKind(activate(XS)), activate(N), activate(XS)) U294(tt, N, XS) -> head(afterNth(activate(N), activate(XS))) U301(tt, X, Y) -> U302(isLNatKind(activate(X)), activate(Y)) U302(tt, Y) -> U303(isLNat(activate(Y)), activate(Y)) U303(tt, Y) -> U304(isLNatKind(activate(Y)), activate(Y)) U304(tt, Y) -> activate(Y) U31(tt, N, XS) -> U32(isNaturalKind(activate(N)), activate(N), activate(XS)) U311(tt, XS) -> U312(isLNatKind(activate(XS)), activate(XS)) U312(tt, XS) -> pair(nil, activate(XS)) U32(tt, N, XS) -> U33(isLNat(activate(XS)), activate(N), activate(XS)) U321(tt, N, X, XS) -> U322(isNaturalKind(activate(N)), activate(N), activate(X), activate(XS)) U322(tt, N, X, XS) -> U323(isNatural(activate(X)), activate(N), activate(X), activate(XS)) U323(tt, N, X, XS) -> U324(isNaturalKind(activate(X)), activate(N), activate(X), activate(XS)) U324(tt, N, X, XS) -> U325(isLNat(activate(XS)), activate(N), activate(X), activate(XS)) U325(tt, N, X, XS) -> U326(isLNatKind(activate(XS)), activate(N), activate(X), activate(XS)) U326(tt, N, X, XS) -> U327(splitAt(activate(N), activate(XS)), activate(X)) U327(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS) U33(tt, N, XS) -> U34(isLNatKind(activate(XS)), activate(N)) U331(tt, N, XS) -> U332(isNaturalKind(activate(N)), activate(XS)) U332(tt, XS) -> U333(isLNat(activate(XS)), activate(XS)) U333(tt, XS) -> U334(isLNatKind(activate(XS)), activate(XS)) U334(tt, XS) -> activate(XS) U34(tt, N) -> activate(N) U341(tt, N, XS) -> U342(isNaturalKind(activate(N)), activate(N), activate(XS)) U342(tt, N, XS) -> U343(isLNat(activate(XS)), activate(N), activate(XS)) U343(tt, N, XS) -> U344(isLNatKind(activate(XS)), activate(N), activate(XS)) U344(tt, N, XS) -> fst(splitAt(activate(N), activate(XS))) U41(tt, V1, V2) -> U42(isNaturalKind(activate(V1)), activate(V1), activate(V2)) U42(tt, V1, V2) -> U43(isLNatKind(activate(V2)), activate(V1), activate(V2)) U43(tt, V1, V2) -> U44(isLNatKind(activate(V2)), activate(V1), activate(V2)) U44(tt, V1, V2) -> U45(isNatural(activate(V1)), activate(V2)) U45(tt, V2) -> U46(isLNat(activate(V2))) U46(tt) -> tt U51(tt, V1, V2) -> U52(isNaturalKind(activate(V1)), activate(V1), activate(V2)) U52(tt, V1, V2) -> U53(isLNatKind(activate(V2)), activate(V1), activate(V2)) U53(tt, V1, V2) -> U54(isLNatKind(activate(V2)), activate(V1), activate(V2)) U54(tt, V1, V2) -> U55(isNatural(activate(V1)), activate(V2)) U55(tt, V2) -> U56(isLNat(activate(V2))) U56(tt) -> tt U61(tt, V1) -> U62(isPLNatKind(activate(V1)), activate(V1)) U62(tt, V1) -> U63(isPLNat(activate(V1))) U63(tt) -> tt U71(tt, V1) -> U72(isNaturalKind(activate(V1)), activate(V1)) U72(tt, V1) -> U73(isNatural(activate(V1))) U73(tt) -> tt U81(tt, V1) -> U82(isPLNatKind(activate(V1)), activate(V1)) U82(tt, V1) -> U83(isPLNat(activate(V1))) U83(tt) -> tt U91(tt, V1) -> U92(isLNatKind(activate(V1)), activate(V1)) U92(tt, V1) -> U93(isLNat(activate(V1))) U93(tt) -> tt afterNth(N, XS) -> U11(isNatural(N), N, XS) fst(pair(X, Y)) -> U21(isLNat(X), X, Y) head(cons(N, XS)) -> U31(isNatural(N), N, activate(XS)) isLNat(n__nil) -> tt isLNat(n__afterNth(V1, V2)) -> U41(isNaturalKind(activate(V1)), activate(V1), activate(V2)) isLNat(n__cons(V1, V2)) -> U51(isNaturalKind(activate(V1)), activate(V1), activate(V2)) isLNat(n__fst(V1)) -> U61(isPLNatKind(activate(V1)), activate(V1)) isLNat(n__natsFrom(V1)) -> U71(isNaturalKind(activate(V1)), activate(V1)) isLNat(n__snd(V1)) -> U81(isPLNatKind(activate(V1)), activate(V1)) isLNat(n__tail(V1)) -> U91(isLNatKind(activate(V1)), activate(V1)) isLNat(n__take(V1, V2)) -> U101(isNaturalKind(activate(V1)), activate(V1), activate(V2)) isLNatKind(n__nil) -> tt isLNatKind(n__afterNth(V1, V2)) -> U111(isNaturalKind(activate(V1)), activate(V2)) isLNatKind(n__cons(V1, V2)) -> U121(isNaturalKind(activate(V1)), activate(V2)) isLNatKind(n__fst(V1)) -> U131(isPLNatKind(activate(V1))) isLNatKind(n__natsFrom(V1)) -> U141(isNaturalKind(activate(V1))) isLNatKind(n__snd(V1)) -> U151(isPLNatKind(activate(V1))) isLNatKind(n__tail(V1)) -> U161(isLNatKind(activate(V1))) isLNatKind(n__take(V1, V2)) -> U171(isNaturalKind(activate(V1)), activate(V2)) isNatural(n__0) -> tt isNatural(n__head(V1)) -> U181(isLNatKind(activate(V1)), activate(V1)) isNatural(n__s(V1)) -> U191(isNaturalKind(activate(V1)), activate(V1)) isNatural(n__sel(V1, V2)) -> U201(isNaturalKind(activate(V1)), activate(V1), activate(V2)) isNaturalKind(n__0) -> tt isNaturalKind(n__head(V1)) -> U211(isLNatKind(activate(V1))) isNaturalKind(n__s(V1)) -> U221(isNaturalKind(activate(V1))) isNaturalKind(n__sel(V1, V2)) -> U231(isNaturalKind(activate(V1)), activate(V2)) isPLNat(n__pair(V1, V2)) -> U241(isLNatKind(activate(V1)), activate(V1), activate(V2)) isPLNat(n__splitAt(V1, V2)) -> U251(isNaturalKind(activate(V1)), activate(V1), activate(V2)) isPLNatKind(n__pair(V1, V2)) -> U261(isLNatKind(activate(V1)), activate(V2)) isPLNatKind(n__splitAt(V1, V2)) -> U271(isNaturalKind(activate(V1)), activate(V2)) natsFrom(N) -> U281(isNatural(N), N) sel(N, XS) -> U291(isNatural(N), N, XS) snd(pair(X, Y)) -> U301(isLNat(X), X, Y) splitAt(0, XS) -> U311(isLNat(XS), XS) splitAt(s(N), cons(X, XS)) -> U321(isNatural(N), N, X, activate(XS)) tail(cons(N, XS)) -> U331(isNatural(N), N, activate(XS)) take(N, XS) -> U341(isNatural(N), N, XS) natsFrom(X) -> n__natsFrom(X) nil -> n__nil afterNth(X1, X2) -> n__afterNth(X1, X2) cons(X1, X2) -> n__cons(X1, X2) fst(X) -> n__fst(X) snd(X) -> n__snd(X) tail(X) -> n__tail(X) take(X1, X2) -> n__take(X1, X2) 0 -> n__0 head(X) -> n__head(X) s(X) -> n__s(X) sel(X1, X2) -> n__sel(X1, X2) pair(X1, X2) -> n__pair(X1, X2) splitAt(X1, X2) -> n__splitAt(X1, X2) activate(n__natsFrom(X)) -> natsFrom(X) activate(n__nil) -> nil activate(n__afterNth(X1, X2)) -> afterNth(X1, X2) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__fst(X)) -> fst(X) activate(n__snd(X)) -> snd(X) activate(n__tail(X)) -> tail(X) activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__head(X)) -> head(X) activate(n__s(X)) -> s(X) activate(n__sel(X1, X2)) -> sel(X1, X2) activate(n__pair(X1, X2)) -> pair(X1, X2) activate(n__splitAt(X1, X2)) -> splitAt(X1, X2) activate(X) -> X S is empty. Rewrite Strategy: FULL ---------------------------------------- (9) DecreasingLoopProof (FINISHED) The following loop(s) give(s) rise to the lower bound EXP: The rewrite sequence activate(n__take(n__head(V11_0), X2)) ->^+ U341(U181(isLNatKind(activate(V11_0)), activate(V11_0)), n__head(V11_0), X2) gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0]. The pumping substitution is [V11_0 / n__take(n__head(V11_0), X2)]. The result substitution is [ ]. The rewrite sequence activate(n__take(n__head(V11_0), X2)) ->^+ U341(U181(isLNatKind(activate(V11_0)), activate(V11_0)), n__head(V11_0), X2) gives rise to a decreasing loop by considering the right hand sides subterm at position [0,1]. The pumping substitution is [V11_0 / n__take(n__head(V11_0), X2)]. The result substitution is [ ]. ---------------------------------------- (10) BOUNDS(EXP, INF)