MAYBE 1071.42/297.23 MAYBE 1071.42/297.23 1071.42/297.23 We are left with following problem, upon which TcT provides the 1071.42/297.23 certificate MAYBE. 1071.42/297.23 1071.42/297.23 Strict Trs: 1071.42/297.23 { U101(tt(), V1, V2) -> 1071.42/297.23 U102(isNaturalKind(activate(V1)), activate(V1), activate(V2)) 1071.42/297.23 , U102(tt(), V1, V2) -> 1071.42/297.23 U103(isLNatKind(activate(V2)), activate(V1), activate(V2)) 1071.42/297.23 , isNaturalKind(n__s(V1)) -> U221(isNaturalKind(activate(V1))) 1071.42/297.23 , isNaturalKind(n__0()) -> tt() 1071.42/297.23 , isNaturalKind(n__head(V1)) -> U211(isLNatKind(activate(V1))) 1071.42/297.23 , isNaturalKind(n__sel(V1, V2)) -> 1071.42/297.23 U231(isNaturalKind(activate(V1)), activate(V2)) 1071.42/297.23 , activate(X) -> X 1071.42/297.23 , activate(n__natsFrom(X)) -> natsFrom(activate(X)) 1071.42/297.23 , activate(n__s(X)) -> s(activate(X)) 1071.42/297.23 , activate(n__nil()) -> nil() 1071.42/297.23 , activate(n__afterNth(X1, X2)) -> 1071.42/297.23 afterNth(activate(X1), activate(X2)) 1071.42/297.23 , activate(n__cons(X1, X2)) -> cons(activate(X1), X2) 1071.42/297.23 , activate(n__fst(X)) -> fst(activate(X)) 1071.42/297.23 , activate(n__snd(X)) -> snd(activate(X)) 1071.42/297.23 , activate(n__tail(X)) -> tail(activate(X)) 1071.42/297.23 , activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) 1071.42/297.23 , activate(n__0()) -> 0() 1071.42/297.23 , activate(n__head(X)) -> head(activate(X)) 1071.42/297.23 , activate(n__sel(X1, X2)) -> sel(activate(X1), activate(X2)) 1071.42/297.23 , activate(n__pair(X1, X2)) -> pair(activate(X1), activate(X2)) 1071.42/297.23 , activate(n__splitAt(X1, X2)) -> 1071.42/297.23 splitAt(activate(X1), activate(X2)) 1071.42/297.23 , U103(tt(), V1, V2) -> 1071.42/297.23 U104(isLNatKind(activate(V2)), activate(V1), activate(V2)) 1071.42/297.23 , isLNatKind(n__natsFrom(V1)) -> U141(isNaturalKind(activate(V1))) 1071.42/297.23 , isLNatKind(n__nil()) -> tt() 1071.42/297.23 , isLNatKind(n__afterNth(V1, V2)) -> 1071.42/297.23 U111(isNaturalKind(activate(V1)), activate(V2)) 1071.42/297.23 , isLNatKind(n__cons(V1, V2)) -> 1071.42/297.23 U121(isNaturalKind(activate(V1)), activate(V2)) 1071.42/297.23 , isLNatKind(n__fst(V1)) -> U131(isPLNatKind(activate(V1))) 1071.42/297.23 , isLNatKind(n__snd(V1)) -> U151(isPLNatKind(activate(V1))) 1071.42/297.23 , isLNatKind(n__tail(V1)) -> U161(isLNatKind(activate(V1))) 1071.42/297.23 , isLNatKind(n__take(V1, V2)) -> 1071.42/297.23 U171(isNaturalKind(activate(V1)), activate(V2)) 1071.42/297.23 , U104(tt(), V1, V2) -> U105(isNatural(activate(V1)), activate(V2)) 1071.42/297.23 , U105(tt(), V2) -> U106(isLNat(activate(V2))) 1071.42/297.23 , isNatural(n__s(V1)) -> 1071.42/297.23 U191(isNaturalKind(activate(V1)), activate(V1)) 1071.42/297.23 , isNatural(n__0()) -> tt() 1071.42/297.23 , isNatural(n__head(V1)) -> 1071.42/297.23 U181(isLNatKind(activate(V1)), activate(V1)) 1071.42/297.23 , isNatural(n__sel(V1, V2)) -> 1071.42/297.23 U201(isNaturalKind(activate(V1)), activate(V1), activate(V2)) 1071.42/297.23 , U106(tt()) -> tt() 1071.42/297.23 , isLNat(n__natsFrom(V1)) -> 1071.42/297.23 U71(isNaturalKind(activate(V1)), activate(V1)) 1071.42/297.23 , isLNat(n__nil()) -> tt() 1071.42/297.23 , isLNat(n__afterNth(V1, V2)) -> 1071.42/297.23 U41(isNaturalKind(activate(V1)), activate(V1), activate(V2)) 1071.42/297.23 , isLNat(n__cons(V1, V2)) -> 1071.42/297.23 U51(isNaturalKind(activate(V1)), activate(V1), activate(V2)) 1071.42/297.23 , isLNat(n__fst(V1)) -> 1071.42/297.23 U61(isPLNatKind(activate(V1)), activate(V1)) 1071.42/297.23 , isLNat(n__snd(V1)) -> 1071.42/297.23 U81(isPLNatKind(activate(V1)), activate(V1)) 1071.42/297.23 , isLNat(n__tail(V1)) -> 1071.42/297.23 U91(isLNatKind(activate(V1)), activate(V1)) 1071.42/297.23 , isLNat(n__take(V1, V2)) -> 1071.42/297.23 U101(isNaturalKind(activate(V1)), activate(V1), activate(V2)) 1071.42/297.23 , U11(tt(), N, XS) -> 1071.42/297.23 U12(isNaturalKind(activate(N)), activate(N), activate(XS)) 1071.42/297.23 , U12(tt(), N, XS) -> 1071.42/297.23 U13(isLNat(activate(XS)), activate(N), activate(XS)) 1071.42/297.23 , U111(tt(), V2) -> U112(isLNatKind(activate(V2))) 1071.42/297.23 , U112(tt()) -> tt() 1071.42/297.23 , U13(tt(), N, XS) -> 1071.42/297.23 U14(isLNatKind(activate(XS)), activate(N), activate(XS)) 1071.42/297.23 , U121(tt(), V2) -> U122(isLNatKind(activate(V2))) 1071.42/297.23 , U122(tt()) -> tt() 1071.42/297.23 , U14(tt(), N, XS) -> snd(splitAt(activate(N), activate(XS))) 1071.42/297.23 , U131(tt()) -> tt() 1071.42/297.23 , snd(X) -> n__snd(X) 1071.42/297.23 , snd(pair(X, Y)) -> U301(isLNat(X), X, Y) 1071.42/297.23 , splitAt(X1, X2) -> n__splitAt(X1, X2) 1071.42/297.23 , splitAt(0(), XS) -> U311(isLNat(XS), XS) 1071.42/297.23 , splitAt(s(N), cons(X, XS)) -> 1071.42/297.23 U321(isNatural(N), N, X, activate(XS)) 1071.42/297.23 , U141(tt()) -> tt() 1071.42/297.23 , U151(tt()) -> tt() 1071.42/297.23 , U161(tt()) -> tt() 1071.42/297.23 , U171(tt(), V2) -> U172(isLNatKind(activate(V2))) 1071.42/297.23 , U172(tt()) -> tt() 1071.42/297.23 , U181(tt(), V1) -> U182(isLNatKind(activate(V1)), activate(V1)) 1071.42/297.23 , U182(tt(), V1) -> U183(isLNat(activate(V1))) 1071.42/297.23 , U183(tt()) -> tt() 1071.42/297.23 , U191(tt(), V1) -> U192(isNaturalKind(activate(V1)), activate(V1)) 1071.42/297.23 , U192(tt(), V1) -> U193(isNatural(activate(V1))) 1071.42/297.23 , U193(tt()) -> tt() 1071.42/297.23 , U201(tt(), V1, V2) -> 1071.42/297.23 U202(isNaturalKind(activate(V1)), activate(V1), activate(V2)) 1071.42/297.23 , U202(tt(), V1, V2) -> 1071.42/297.23 U203(isLNatKind(activate(V2)), activate(V1), activate(V2)) 1071.42/297.23 , U203(tt(), V1, V2) -> 1071.42/297.23 U204(isLNatKind(activate(V2)), activate(V1), activate(V2)) 1071.42/297.23 , U204(tt(), V1, V2) -> U205(isNatural(activate(V1)), activate(V2)) 1071.42/297.23 , U205(tt(), V2) -> U206(isLNat(activate(V2))) 1071.42/297.23 , U206(tt()) -> tt() 1071.42/297.23 , U21(tt(), X, Y) -> 1071.42/297.23 U22(isLNatKind(activate(X)), activate(X), activate(Y)) 1071.42/297.23 , U22(tt(), X, Y) -> 1071.42/297.23 U23(isLNat(activate(Y)), activate(X), activate(Y)) 1071.42/297.23 , U211(tt()) -> tt() 1071.42/297.23 , U23(tt(), X, Y) -> U24(isLNatKind(activate(Y)), activate(X)) 1071.42/297.23 , U221(tt()) -> tt() 1071.42/297.23 , U24(tt(), X) -> activate(X) 1071.42/297.23 , U231(tt(), V2) -> U232(isLNatKind(activate(V2))) 1071.42/297.23 , U232(tt()) -> tt() 1071.42/297.23 , U241(tt(), V1, V2) -> 1071.42/297.23 U242(isLNatKind(activate(V1)), activate(V1), activate(V2)) 1071.42/297.23 , U242(tt(), V1, V2) -> 1071.42/297.23 U243(isLNatKind(activate(V2)), activate(V1), activate(V2)) 1071.42/297.23 , U243(tt(), V1, V2) -> 1071.42/297.23 U244(isLNatKind(activate(V2)), activate(V1), activate(V2)) 1071.42/297.23 , U244(tt(), V1, V2) -> U245(isLNat(activate(V1)), activate(V2)) 1071.42/297.23 , U245(tt(), V2) -> U246(isLNat(activate(V2))) 1071.42/297.23 , U246(tt()) -> tt() 1071.42/297.23 , U251(tt(), V1, V2) -> 1071.42/297.23 U252(isNaturalKind(activate(V1)), activate(V1), activate(V2)) 1071.42/297.23 , U252(tt(), V1, V2) -> 1071.42/297.23 U253(isLNatKind(activate(V2)), activate(V1), activate(V2)) 1071.42/297.23 , U253(tt(), V1, V2) -> 1071.42/297.23 U254(isLNatKind(activate(V2)), activate(V1), activate(V2)) 1071.42/297.23 , U254(tt(), V1, V2) -> U255(isNatural(activate(V1)), activate(V2)) 1071.42/297.23 , U255(tt(), V2) -> U256(isLNat(activate(V2))) 1071.42/297.23 , U256(tt()) -> tt() 1071.42/297.23 , U261(tt(), V2) -> U262(isLNatKind(activate(V2))) 1071.42/297.23 , U262(tt()) -> tt() 1071.42/297.23 , U271(tt(), V2) -> U272(isLNatKind(activate(V2))) 1071.42/297.23 , U272(tt()) -> tt() 1071.42/297.23 , U281(tt(), N) -> U282(isNaturalKind(activate(N)), activate(N)) 1071.42/297.23 , U282(tt(), N) -> 1071.42/297.23 cons(activate(N), n__natsFrom(n__s(activate(N)))) 1071.42/297.23 , cons(X1, X2) -> n__cons(X1, X2) 1071.42/297.23 , U291(tt(), N, XS) -> 1071.42/297.23 U292(isNaturalKind(activate(N)), activate(N), activate(XS)) 1071.42/297.23 , U292(tt(), N, XS) -> 1071.42/297.23 U293(isLNat(activate(XS)), activate(N), activate(XS)) 1071.42/297.23 , U293(tt(), N, XS) -> 1071.42/297.23 U294(isLNatKind(activate(XS)), activate(N), activate(XS)) 1071.42/297.23 , U294(tt(), N, XS) -> head(afterNth(activate(N), activate(XS))) 1071.42/297.23 , head(X) -> n__head(X) 1071.42/297.23 , head(cons(N, XS)) -> U31(isNatural(N), N, activate(XS)) 1071.42/297.23 , afterNth(N, XS) -> U11(isNatural(N), N, XS) 1071.42/297.23 , afterNth(X1, X2) -> n__afterNth(X1, X2) 1071.42/297.23 , U301(tt(), X, Y) -> U302(isLNatKind(activate(X)), activate(Y)) 1071.42/297.23 , U302(tt(), Y) -> U303(isLNat(activate(Y)), activate(Y)) 1071.42/297.23 , U303(tt(), Y) -> U304(isLNatKind(activate(Y)), activate(Y)) 1071.42/297.23 , U304(tt(), Y) -> activate(Y) 1071.42/297.23 , U31(tt(), N, XS) -> 1071.42/297.23 U32(isNaturalKind(activate(N)), activate(N), activate(XS)) 1071.42/297.23 , U32(tt(), N, XS) -> 1071.42/297.23 U33(isLNat(activate(XS)), activate(N), activate(XS)) 1071.42/297.23 , U311(tt(), XS) -> U312(isLNatKind(activate(XS)), activate(XS)) 1071.42/297.23 , U312(tt(), XS) -> pair(nil(), activate(XS)) 1071.42/297.23 , pair(X1, X2) -> n__pair(X1, X2) 1071.42/297.23 , nil() -> n__nil() 1071.42/297.23 , U33(tt(), N, XS) -> U34(isLNatKind(activate(XS)), activate(N)) 1071.42/297.23 , U321(tt(), N, X, XS) -> 1071.42/297.23 U322(isNaturalKind(activate(N)), 1071.42/297.23 activate(N), 1071.42/297.23 activate(X), 1071.42/297.23 activate(XS)) 1071.42/297.23 , U322(tt(), N, X, XS) -> 1071.42/297.23 U323(isNatural(activate(X)), 1071.42/297.23 activate(N), 1071.42/297.23 activate(X), 1071.42/297.23 activate(XS)) 1071.42/297.23 , U323(tt(), N, X, XS) -> 1071.42/297.23 U324(isNaturalKind(activate(X)), 1071.42/297.23 activate(N), 1071.42/297.23 activate(X), 1071.42/297.23 activate(XS)) 1071.42/297.23 , U324(tt(), N, X, XS) -> 1071.42/297.23 U325(isLNat(activate(XS)), activate(N), activate(X), activate(XS)) 1071.42/297.23 , U325(tt(), N, X, XS) -> 1071.42/297.23 U326(isLNatKind(activate(XS)), 1071.42/297.23 activate(N), 1071.42/297.23 activate(X), 1071.42/297.23 activate(XS)) 1071.42/297.23 , U326(tt(), N, X, XS) -> 1071.42/297.23 U327(splitAt(activate(N), activate(XS)), activate(X)) 1071.42/297.23 , U327(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS) 1071.42/297.23 , U34(tt(), N) -> activate(N) 1071.42/297.23 , U331(tt(), N, XS) -> 1071.42/297.23 U332(isNaturalKind(activate(N)), activate(XS)) 1071.42/297.23 , U332(tt(), XS) -> U333(isLNat(activate(XS)), activate(XS)) 1071.42/297.23 , U333(tt(), XS) -> U334(isLNatKind(activate(XS)), activate(XS)) 1071.42/297.23 , U334(tt(), XS) -> activate(XS) 1071.42/297.23 , U341(tt(), N, XS) -> 1071.42/297.23 U342(isNaturalKind(activate(N)), activate(N), activate(XS)) 1071.42/297.23 , U342(tt(), N, XS) -> 1071.42/297.23 U343(isLNat(activate(XS)), activate(N), activate(XS)) 1071.42/297.23 , U343(tt(), N, XS) -> 1071.42/297.23 U344(isLNatKind(activate(XS)), activate(N), activate(XS)) 1071.42/297.23 , U344(tt(), N, XS) -> fst(splitAt(activate(N), activate(XS))) 1071.42/297.23 , fst(X) -> n__fst(X) 1071.42/297.23 , fst(pair(X, Y)) -> U21(isLNat(X), X, Y) 1071.42/297.23 , U41(tt(), V1, V2) -> 1071.42/297.23 U42(isNaturalKind(activate(V1)), activate(V1), activate(V2)) 1071.42/297.23 , U42(tt(), V1, V2) -> 1071.42/297.23 U43(isLNatKind(activate(V2)), activate(V1), activate(V2)) 1071.42/297.23 , U43(tt(), V1, V2) -> 1071.42/297.23 U44(isLNatKind(activate(V2)), activate(V1), activate(V2)) 1071.42/297.23 , U44(tt(), V1, V2) -> U45(isNatural(activate(V1)), activate(V2)) 1071.42/297.23 , U45(tt(), V2) -> U46(isLNat(activate(V2))) 1071.42/297.23 , U46(tt()) -> tt() 1071.42/297.23 , U51(tt(), V1, V2) -> 1071.42/297.23 U52(isNaturalKind(activate(V1)), activate(V1), activate(V2)) 1071.42/297.23 , U52(tt(), V1, V2) -> 1071.42/297.23 U53(isLNatKind(activate(V2)), activate(V1), activate(V2)) 1071.42/297.23 , U53(tt(), V1, V2) -> 1071.42/297.23 U54(isLNatKind(activate(V2)), activate(V1), activate(V2)) 1071.42/297.23 , U54(tt(), V1, V2) -> U55(isNatural(activate(V1)), activate(V2)) 1071.42/297.23 , U55(tt(), V2) -> U56(isLNat(activate(V2))) 1071.42/297.23 , U56(tt()) -> tt() 1071.42/297.23 , U61(tt(), V1) -> U62(isPLNatKind(activate(V1)), activate(V1)) 1071.42/297.23 , U62(tt(), V1) -> U63(isPLNat(activate(V1))) 1071.42/297.23 , isPLNatKind(n__pair(V1, V2)) -> 1071.42/297.23 U261(isLNatKind(activate(V1)), activate(V2)) 1071.42/297.23 , isPLNatKind(n__splitAt(V1, V2)) -> 1071.42/297.23 U271(isNaturalKind(activate(V1)), activate(V2)) 1071.42/297.23 , U63(tt()) -> tt() 1071.42/297.23 , isPLNat(n__pair(V1, V2)) -> 1071.42/297.23 U241(isLNatKind(activate(V1)), activate(V1), activate(V2)) 1071.42/297.23 , isPLNat(n__splitAt(V1, V2)) -> 1071.42/297.23 U251(isNaturalKind(activate(V1)), activate(V1), activate(V2)) 1071.42/297.23 , U71(tt(), V1) -> U72(isNaturalKind(activate(V1)), activate(V1)) 1071.42/297.23 , U72(tt(), V1) -> U73(isNatural(activate(V1))) 1071.42/297.23 , U73(tt()) -> tt() 1071.42/297.23 , U81(tt(), V1) -> U82(isPLNatKind(activate(V1)), activate(V1)) 1071.42/297.23 , U82(tt(), V1) -> U83(isPLNat(activate(V1))) 1071.42/297.23 , U83(tt()) -> tt() 1071.42/297.23 , U91(tt(), V1) -> U92(isLNatKind(activate(V1)), activate(V1)) 1071.42/297.23 , U92(tt(), V1) -> U93(isLNat(activate(V1))) 1071.42/297.23 , U93(tt()) -> tt() 1071.42/297.23 , natsFrom(N) -> U281(isNatural(N), N) 1071.42/297.23 , natsFrom(X) -> n__natsFrom(X) 1071.42/297.23 , sel(N, XS) -> U291(isNatural(N), N, XS) 1071.42/297.23 , sel(X1, X2) -> n__sel(X1, X2) 1071.42/297.23 , 0() -> n__0() 1071.42/297.23 , s(X) -> n__s(X) 1071.42/297.23 , tail(X) -> n__tail(X) 1071.42/297.23 , tail(cons(N, XS)) -> U331(isNatural(N), N, activate(XS)) 1071.42/297.23 , take(N, XS) -> U341(isNatural(N), N, XS) 1071.42/297.23 , take(X1, X2) -> n__take(X1, X2) } 1071.42/297.23 Obligation: 1071.42/297.23 innermost runtime complexity 1071.42/297.23 Answer: 1071.42/297.23 MAYBE 1071.42/297.23 1071.42/297.23 Arguments of following rules are not normal-forms: 1071.42/297.23 1071.42/297.23 { snd(pair(X, Y)) -> U301(isLNat(X), X, Y) 1071.42/297.23 , splitAt(0(), XS) -> U311(isLNat(XS), XS) 1071.42/297.23 , splitAt(s(N), cons(X, XS)) -> 1071.42/297.23 U321(isNatural(N), N, X, activate(XS)) 1071.42/297.23 , head(cons(N, XS)) -> U31(isNatural(N), N, activate(XS)) 1071.42/297.23 , U327(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS) 1071.42/297.23 , fst(pair(X, Y)) -> U21(isLNat(X), X, Y) 1071.42/297.23 , tail(cons(N, XS)) -> U331(isNatural(N), N, activate(XS)) } 1071.42/297.23 1071.42/297.23 All above mentioned rules can be savely removed. 1071.42/297.23 1071.42/297.23 We are left with following problem, upon which TcT provides the 1071.42/297.23 certificate MAYBE. 1071.42/297.23 1071.42/297.23 Strict Trs: 1071.42/297.23 { U101(tt(), V1, V2) -> 1071.42/297.23 U102(isNaturalKind(activate(V1)), activate(V1), activate(V2)) 1071.42/297.23 , U102(tt(), V1, V2) -> 1071.42/297.23 U103(isLNatKind(activate(V2)), activate(V1), activate(V2)) 1071.42/297.23 , isNaturalKind(n__s(V1)) -> U221(isNaturalKind(activate(V1))) 1071.42/297.23 , isNaturalKind(n__0()) -> tt() 1071.42/297.23 , isNaturalKind(n__head(V1)) -> U211(isLNatKind(activate(V1))) 1071.42/297.23 , isNaturalKind(n__sel(V1, V2)) -> 1071.42/297.23 U231(isNaturalKind(activate(V1)), activate(V2)) 1071.42/297.23 , activate(X) -> X 1071.42/297.23 , activate(n__natsFrom(X)) -> natsFrom(activate(X)) 1071.42/297.23 , activate(n__s(X)) -> s(activate(X)) 1071.42/297.23 , activate(n__nil()) -> nil() 1071.42/297.23 , activate(n__afterNth(X1, X2)) -> 1071.42/297.23 afterNth(activate(X1), activate(X2)) 1071.42/297.23 , activate(n__cons(X1, X2)) -> cons(activate(X1), X2) 1071.42/297.23 , activate(n__fst(X)) -> fst(activate(X)) 1071.42/297.23 , activate(n__snd(X)) -> snd(activate(X)) 1071.42/297.23 , activate(n__tail(X)) -> tail(activate(X)) 1071.42/297.23 , activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) 1071.42/297.23 , activate(n__0()) -> 0() 1071.42/297.23 , activate(n__head(X)) -> head(activate(X)) 1071.42/297.23 , activate(n__sel(X1, X2)) -> sel(activate(X1), activate(X2)) 1071.42/297.23 , activate(n__pair(X1, X2)) -> pair(activate(X1), activate(X2)) 1071.42/297.23 , activate(n__splitAt(X1, X2)) -> 1071.42/297.23 splitAt(activate(X1), activate(X2)) 1071.42/297.23 , U103(tt(), V1, V2) -> 1071.42/297.23 U104(isLNatKind(activate(V2)), activate(V1), activate(V2)) 1071.42/297.23 , isLNatKind(n__natsFrom(V1)) -> U141(isNaturalKind(activate(V1))) 1071.42/297.23 , isLNatKind(n__nil()) -> tt() 1071.42/297.23 , isLNatKind(n__afterNth(V1, V2)) -> 1071.42/297.23 U111(isNaturalKind(activate(V1)), activate(V2)) 1071.42/297.23 , isLNatKind(n__cons(V1, V2)) -> 1071.42/297.23 U121(isNaturalKind(activate(V1)), activate(V2)) 1071.42/297.23 , isLNatKind(n__fst(V1)) -> U131(isPLNatKind(activate(V1))) 1071.42/297.23 , isLNatKind(n__snd(V1)) -> U151(isPLNatKind(activate(V1))) 1071.42/297.23 , isLNatKind(n__tail(V1)) -> U161(isLNatKind(activate(V1))) 1071.42/297.23 , isLNatKind(n__take(V1, V2)) -> 1071.42/297.23 U171(isNaturalKind(activate(V1)), activate(V2)) 1071.42/297.23 , U104(tt(), V1, V2) -> U105(isNatural(activate(V1)), activate(V2)) 1071.42/297.23 , U105(tt(), V2) -> U106(isLNat(activate(V2))) 1071.42/297.23 , isNatural(n__s(V1)) -> 1071.42/297.23 U191(isNaturalKind(activate(V1)), activate(V1)) 1071.42/297.23 , isNatural(n__0()) -> tt() 1071.42/297.23 , isNatural(n__head(V1)) -> 1071.42/297.23 U181(isLNatKind(activate(V1)), activate(V1)) 1071.42/297.23 , isNatural(n__sel(V1, V2)) -> 1071.42/297.23 U201(isNaturalKind(activate(V1)), activate(V1), activate(V2)) 1071.42/297.23 , U106(tt()) -> tt() 1071.42/297.23 , isLNat(n__natsFrom(V1)) -> 1071.42/297.23 U71(isNaturalKind(activate(V1)), activate(V1)) 1071.42/297.23 , isLNat(n__nil()) -> tt() 1071.42/297.23 , isLNat(n__afterNth(V1, V2)) -> 1071.42/297.23 U41(isNaturalKind(activate(V1)), activate(V1), activate(V2)) 1071.42/297.23 , isLNat(n__cons(V1, V2)) -> 1071.42/297.23 U51(isNaturalKind(activate(V1)), activate(V1), activate(V2)) 1071.42/297.23 , isLNat(n__fst(V1)) -> 1071.42/297.23 U61(isPLNatKind(activate(V1)), activate(V1)) 1071.42/297.23 , isLNat(n__snd(V1)) -> 1071.42/297.23 U81(isPLNatKind(activate(V1)), activate(V1)) 1071.42/297.23 , isLNat(n__tail(V1)) -> 1071.42/297.23 U91(isLNatKind(activate(V1)), activate(V1)) 1071.42/297.23 , isLNat(n__take(V1, V2)) -> 1071.42/297.23 U101(isNaturalKind(activate(V1)), activate(V1), activate(V2)) 1071.42/297.23 , U11(tt(), N, XS) -> 1071.42/297.23 U12(isNaturalKind(activate(N)), activate(N), activate(XS)) 1071.42/297.23 , U12(tt(), N, XS) -> 1071.42/297.23 U13(isLNat(activate(XS)), activate(N), activate(XS)) 1071.42/297.23 , U111(tt(), V2) -> U112(isLNatKind(activate(V2))) 1071.42/297.23 , U112(tt()) -> tt() 1071.42/297.23 , U13(tt(), N, XS) -> 1071.42/297.23 U14(isLNatKind(activate(XS)), activate(N), activate(XS)) 1071.42/297.23 , U121(tt(), V2) -> U122(isLNatKind(activate(V2))) 1071.42/297.23 , U122(tt()) -> tt() 1071.42/297.23 , U14(tt(), N, XS) -> snd(splitAt(activate(N), activate(XS))) 1071.42/297.23 , U131(tt()) -> tt() 1071.42/297.23 , snd(X) -> n__snd(X) 1071.42/297.23 , splitAt(X1, X2) -> n__splitAt(X1, X2) 1071.42/297.23 , U141(tt()) -> tt() 1071.42/297.23 , U151(tt()) -> tt() 1071.42/297.23 , U161(tt()) -> tt() 1071.42/297.23 , U171(tt(), V2) -> U172(isLNatKind(activate(V2))) 1071.42/297.23 , U172(tt()) -> tt() 1071.42/297.23 , U181(tt(), V1) -> U182(isLNatKind(activate(V1)), activate(V1)) 1071.42/297.23 , U182(tt(), V1) -> U183(isLNat(activate(V1))) 1071.42/297.23 , U183(tt()) -> tt() 1071.42/297.23 , U191(tt(), V1) -> U192(isNaturalKind(activate(V1)), activate(V1)) 1071.42/297.23 , U192(tt(), V1) -> U193(isNatural(activate(V1))) 1071.42/297.23 , U193(tt()) -> tt() 1071.42/297.23 , U201(tt(), V1, V2) -> 1071.42/297.23 U202(isNaturalKind(activate(V1)), activate(V1), activate(V2)) 1071.42/297.23 , U202(tt(), V1, V2) -> 1071.42/297.23 U203(isLNatKind(activate(V2)), activate(V1), activate(V2)) 1071.42/297.23 , U203(tt(), V1, V2) -> 1071.42/297.23 U204(isLNatKind(activate(V2)), activate(V1), activate(V2)) 1071.42/297.23 , U204(tt(), V1, V2) -> U205(isNatural(activate(V1)), activate(V2)) 1071.42/297.23 , U205(tt(), V2) -> U206(isLNat(activate(V2))) 1071.42/297.23 , U206(tt()) -> tt() 1071.42/297.23 , U21(tt(), X, Y) -> 1071.42/297.23 U22(isLNatKind(activate(X)), activate(X), activate(Y)) 1071.42/297.23 , U22(tt(), X, Y) -> 1071.42/297.23 U23(isLNat(activate(Y)), activate(X), activate(Y)) 1071.42/297.23 , U211(tt()) -> tt() 1071.42/297.23 , U23(tt(), X, Y) -> U24(isLNatKind(activate(Y)), activate(X)) 1071.42/297.23 , U221(tt()) -> tt() 1071.42/297.23 , U24(tt(), X) -> activate(X) 1071.42/297.23 , U231(tt(), V2) -> U232(isLNatKind(activate(V2))) 1071.42/297.23 , U232(tt()) -> tt() 1071.42/297.23 , U241(tt(), V1, V2) -> 1071.42/297.23 U242(isLNatKind(activate(V1)), activate(V1), activate(V2)) 1071.42/297.23 , U242(tt(), V1, V2) -> 1071.42/297.23 U243(isLNatKind(activate(V2)), activate(V1), activate(V2)) 1071.42/297.23 , U243(tt(), V1, V2) -> 1071.42/297.23 U244(isLNatKind(activate(V2)), activate(V1), activate(V2)) 1071.42/297.23 , U244(tt(), V1, V2) -> U245(isLNat(activate(V1)), activate(V2)) 1071.42/297.23 , U245(tt(), V2) -> U246(isLNat(activate(V2))) 1071.42/297.23 , U246(tt()) -> tt() 1071.42/297.23 , U251(tt(), V1, V2) -> 1071.42/297.23 U252(isNaturalKind(activate(V1)), activate(V1), activate(V2)) 1071.42/297.23 , U252(tt(), V1, V2) -> 1071.42/297.23 U253(isLNatKind(activate(V2)), activate(V1), activate(V2)) 1071.42/297.23 , U253(tt(), V1, V2) -> 1071.42/297.23 U254(isLNatKind(activate(V2)), activate(V1), activate(V2)) 1071.42/297.23 , U254(tt(), V1, V2) -> U255(isNatural(activate(V1)), activate(V2)) 1071.42/297.23 , U255(tt(), V2) -> U256(isLNat(activate(V2))) 1071.42/297.23 , U256(tt()) -> tt() 1071.42/297.23 , U261(tt(), V2) -> U262(isLNatKind(activate(V2))) 1071.42/297.23 , U262(tt()) -> tt() 1071.42/297.23 , U271(tt(), V2) -> U272(isLNatKind(activate(V2))) 1071.42/297.23 , U272(tt()) -> tt() 1071.42/297.23 , U281(tt(), N) -> U282(isNaturalKind(activate(N)), activate(N)) 1071.42/297.23 , U282(tt(), N) -> 1071.42/297.23 cons(activate(N), n__natsFrom(n__s(activate(N)))) 1071.42/297.23 , cons(X1, X2) -> n__cons(X1, X2) 1071.42/297.23 , U291(tt(), N, XS) -> 1071.42/297.23 U292(isNaturalKind(activate(N)), activate(N), activate(XS)) 1071.42/297.23 , U292(tt(), N, XS) -> 1071.42/297.23 U293(isLNat(activate(XS)), activate(N), activate(XS)) 1071.42/297.23 , U293(tt(), N, XS) -> 1071.42/297.23 U294(isLNatKind(activate(XS)), activate(N), activate(XS)) 1071.42/297.23 , U294(tt(), N, XS) -> head(afterNth(activate(N), activate(XS))) 1071.42/297.23 , head(X) -> n__head(X) 1071.42/297.23 , afterNth(N, XS) -> U11(isNatural(N), N, XS) 1071.42/297.23 , afterNth(X1, X2) -> n__afterNth(X1, X2) 1071.42/297.23 , U301(tt(), X, Y) -> U302(isLNatKind(activate(X)), activate(Y)) 1071.42/297.23 , U302(tt(), Y) -> U303(isLNat(activate(Y)), activate(Y)) 1071.42/297.23 , U303(tt(), Y) -> U304(isLNatKind(activate(Y)), activate(Y)) 1071.42/297.23 , U304(tt(), Y) -> activate(Y) 1071.42/297.23 , U31(tt(), N, XS) -> 1071.42/297.23 U32(isNaturalKind(activate(N)), activate(N), activate(XS)) 1071.42/297.23 , U32(tt(), N, XS) -> 1071.42/297.23 U33(isLNat(activate(XS)), activate(N), activate(XS)) 1071.42/297.23 , U311(tt(), XS) -> U312(isLNatKind(activate(XS)), activate(XS)) 1071.42/297.23 , U312(tt(), XS) -> pair(nil(), activate(XS)) 1071.42/297.23 , pair(X1, X2) -> n__pair(X1, X2) 1071.42/297.23 , nil() -> n__nil() 1071.42/297.23 , U33(tt(), N, XS) -> U34(isLNatKind(activate(XS)), activate(N)) 1071.42/297.23 , U321(tt(), N, X, XS) -> 1071.42/297.23 U322(isNaturalKind(activate(N)), 1071.42/297.23 activate(N), 1071.42/297.23 activate(X), 1071.42/297.23 activate(XS)) 1071.42/297.23 , U322(tt(), N, X, XS) -> 1071.42/297.23 U323(isNatural(activate(X)), 1071.42/297.23 activate(N), 1071.42/297.23 activate(X), 1071.42/297.23 activate(XS)) 1071.42/297.23 , U323(tt(), N, X, XS) -> 1071.42/297.23 U324(isNaturalKind(activate(X)), 1071.42/297.23 activate(N), 1071.42/297.23 activate(X), 1071.42/297.23 activate(XS)) 1071.42/297.23 , U324(tt(), N, X, XS) -> 1071.42/297.23 U325(isLNat(activate(XS)), activate(N), activate(X), activate(XS)) 1071.42/297.23 , U325(tt(), N, X, XS) -> 1071.42/297.23 U326(isLNatKind(activate(XS)), 1071.42/297.23 activate(N), 1071.42/297.23 activate(X), 1071.42/297.23 activate(XS)) 1071.42/297.23 , U326(tt(), N, X, XS) -> 1071.42/297.23 U327(splitAt(activate(N), activate(XS)), activate(X)) 1071.42/297.23 , U34(tt(), N) -> activate(N) 1071.42/297.23 , U331(tt(), N, XS) -> 1071.42/297.23 U332(isNaturalKind(activate(N)), activate(XS)) 1071.42/297.23 , U332(tt(), XS) -> U333(isLNat(activate(XS)), activate(XS)) 1071.42/297.23 , U333(tt(), XS) -> U334(isLNatKind(activate(XS)), activate(XS)) 1071.42/297.23 , U334(tt(), XS) -> activate(XS) 1071.42/297.23 , U341(tt(), N, XS) -> 1071.42/297.23 U342(isNaturalKind(activate(N)), activate(N), activate(XS)) 1071.42/297.23 , U342(tt(), N, XS) -> 1071.42/297.23 U343(isLNat(activate(XS)), activate(N), activate(XS)) 1071.42/297.23 , U343(tt(), N, XS) -> 1071.42/297.23 U344(isLNatKind(activate(XS)), activate(N), activate(XS)) 1071.42/297.23 , U344(tt(), N, XS) -> fst(splitAt(activate(N), activate(XS))) 1071.42/297.23 , fst(X) -> n__fst(X) 1071.42/297.23 , U41(tt(), V1, V2) -> 1071.42/297.23 U42(isNaturalKind(activate(V1)), activate(V1), activate(V2)) 1071.42/297.23 , U42(tt(), V1, V2) -> 1071.42/297.23 U43(isLNatKind(activate(V2)), activate(V1), activate(V2)) 1071.42/297.23 , U43(tt(), V1, V2) -> 1071.42/297.23 U44(isLNatKind(activate(V2)), activate(V1), activate(V2)) 1071.42/297.23 , U44(tt(), V1, V2) -> U45(isNatural(activate(V1)), activate(V2)) 1071.42/297.23 , U45(tt(), V2) -> U46(isLNat(activate(V2))) 1071.42/297.23 , U46(tt()) -> tt() 1071.42/297.23 , U51(tt(), V1, V2) -> 1071.42/297.23 U52(isNaturalKind(activate(V1)), activate(V1), activate(V2)) 1071.42/297.23 , U52(tt(), V1, V2) -> 1071.42/297.23 U53(isLNatKind(activate(V2)), activate(V1), activate(V2)) 1071.42/297.23 , U53(tt(), V1, V2) -> 1071.42/297.23 U54(isLNatKind(activate(V2)), activate(V1), activate(V2)) 1071.42/297.23 , U54(tt(), V1, V2) -> U55(isNatural(activate(V1)), activate(V2)) 1071.42/297.23 , U55(tt(), V2) -> U56(isLNat(activate(V2))) 1071.42/297.23 , U56(tt()) -> tt() 1071.42/297.23 , U61(tt(), V1) -> U62(isPLNatKind(activate(V1)), activate(V1)) 1071.42/297.23 , U62(tt(), V1) -> U63(isPLNat(activate(V1))) 1071.42/297.23 , isPLNatKind(n__pair(V1, V2)) -> 1071.42/297.23 U261(isLNatKind(activate(V1)), activate(V2)) 1071.42/297.23 , isPLNatKind(n__splitAt(V1, V2)) -> 1071.42/297.23 U271(isNaturalKind(activate(V1)), activate(V2)) 1071.42/297.23 , U63(tt()) -> tt() 1071.42/297.23 , isPLNat(n__pair(V1, V2)) -> 1071.42/297.23 U241(isLNatKind(activate(V1)), activate(V1), activate(V2)) 1071.42/297.23 , isPLNat(n__splitAt(V1, V2)) -> 1071.42/297.23 U251(isNaturalKind(activate(V1)), activate(V1), activate(V2)) 1071.42/297.23 , U71(tt(), V1) -> U72(isNaturalKind(activate(V1)), activate(V1)) 1071.42/297.23 , U72(tt(), V1) -> U73(isNatural(activate(V1))) 1071.42/297.23 , U73(tt()) -> tt() 1071.42/297.23 , U81(tt(), V1) -> U82(isPLNatKind(activate(V1)), activate(V1)) 1071.42/297.23 , U82(tt(), V1) -> U83(isPLNat(activate(V1))) 1071.42/297.23 , U83(tt()) -> tt() 1071.42/297.23 , U91(tt(), V1) -> U92(isLNatKind(activate(V1)), activate(V1)) 1071.42/297.23 , U92(tt(), V1) -> U93(isLNat(activate(V1))) 1071.42/297.23 , U93(tt()) -> tt() 1071.42/297.23 , natsFrom(N) -> U281(isNatural(N), N) 1071.42/297.23 , natsFrom(X) -> n__natsFrom(X) 1071.42/297.23 , sel(N, XS) -> U291(isNatural(N), N, XS) 1071.42/297.23 , sel(X1, X2) -> n__sel(X1, X2) 1071.42/297.23 , 0() -> n__0() 1071.42/297.23 , s(X) -> n__s(X) 1071.42/297.23 , tail(X) -> n__tail(X) 1071.42/297.23 , take(N, XS) -> U341(isNatural(N), N, XS) 1071.42/297.23 , take(X1, X2) -> n__take(X1, X2) } 1071.42/297.23 Obligation: 1071.42/297.23 innermost runtime complexity 1071.42/297.23 Answer: 1071.42/297.23 MAYBE 1071.42/297.23 1071.42/297.23 None of the processors succeeded. 1071.42/297.23 1071.42/297.23 Details of failed attempt(s): 1071.42/297.23 ----------------------------- 1071.42/297.23 1) 'empty' failed due to the following reason: 1071.42/297.23 1071.42/297.23 Empty strict component of the problem is NOT empty. 1071.42/297.23 1071.42/297.23 2) 'Best' failed due to the following reason: 1071.42/297.23 1071.42/297.23 None of the processors succeeded. 1071.42/297.23 1071.42/297.23 Details of failed attempt(s): 1071.42/297.23 ----------------------------- 1071.42/297.23 1) 'With Problem ... (timeout of 297 seconds)' failed due to the 1071.42/297.23 following reason: 1071.42/297.23 1071.42/297.23 Computation stopped due to timeout after 297.0 seconds. 1071.42/297.23 1071.42/297.23 2) 'Best' failed due to the following reason: 1071.42/297.23 1071.42/297.23 None of the processors succeeded. 1071.42/297.23 1071.42/297.23 Details of failed attempt(s): 1071.42/297.23 ----------------------------- 1071.42/297.23 1) 'With Problem ... (timeout of 148 seconds) (timeout of 297 1071.42/297.23 seconds)' failed due to the following reason: 1071.42/297.23 1071.42/297.23 Computation stopped due to timeout after 148.0 seconds. 1071.42/297.23 1071.42/297.23 2) 'Best' failed due to the following reason: 1071.42/297.23 1071.42/297.23 None of the processors succeeded. 1071.42/297.23 1071.42/297.23 Details of failed attempt(s): 1071.42/297.23 ----------------------------- 1071.42/297.23 1) 'bsearch-popstar (timeout of 297 seconds)' failed due to the 1071.42/297.23 following reason: 1071.42/297.23 1071.42/297.23 The input cannot be shown compatible 1071.42/297.23 1071.42/297.23 2) 'Polynomial Path Order (PS) (timeout of 297 seconds)' failed due 1071.42/297.23 to the following reason: 1071.42/297.23 1071.42/297.23 The input cannot be shown compatible 1071.42/297.23 1071.42/297.23 1071.42/297.23 3) 'Fastest (timeout of 24 seconds) (timeout of 297 seconds)' 1071.42/297.23 failed due to the following reason: 1071.42/297.23 1071.42/297.23 None of the processors succeeded. 1071.42/297.23 1071.42/297.23 Details of failed attempt(s): 1071.42/297.23 ----------------------------- 1071.42/297.23 1) 'Bounds with minimal-enrichment and initial automaton 'match'' 1071.42/297.23 failed due to the following reason: 1071.42/297.23 1071.42/297.23 match-boundness of the problem could not be verified. 1071.42/297.23 1071.42/297.23 2) 'Bounds with perSymbol-enrichment and initial automaton 'match'' 1071.42/297.23 failed due to the following reason: 1071.42/297.23 1071.42/297.23 match-boundness of the problem could not be verified. 1071.42/297.23 1071.42/297.23 1071.42/297.23 1071.42/297.23 1071.42/297.23 1071.42/297.23 Arrrr.. 1071.42/297.24 EOF