MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { U101(tt(), V2) -> U102(isLNat(activate(V2))) , U102(tt()) -> tt() , isLNat(n__natsFrom(V1)) -> U71(isNatural(activate(V1))) , isLNat(n__nil()) -> tt() , isLNat(n__afterNth(V1, V2)) -> U41(isNatural(activate(V1)), activate(V2)) , isLNat(n__cons(V1, V2)) -> U51(isNatural(activate(V1)), activate(V2)) , isLNat(n__fst(V1)) -> U61(isPLNat(activate(V1))) , isLNat(n__snd(V1)) -> U81(isPLNat(activate(V1))) , isLNat(n__tail(V1)) -> U91(isLNat(activate(V1))) , isLNat(n__take(V1, V2)) -> U101(isNatural(activate(V1)), activate(V2)) , activate(X) -> X , 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) , U11(tt(), N, XS) -> U12(isLNat(activate(XS)), activate(N), activate(XS)) , U12(tt(), N, XS) -> snd(splitAt(activate(N), activate(XS))) , U111(tt()) -> tt() , snd(X) -> n__snd(X) , snd(pair(X, Y)) -> U181(isLNat(X), Y) , splitAt(X1, X2) -> n__splitAt(X1, X2) , splitAt(s(N), cons(X, XS)) -> U201(isNatural(N), N, X, activate(XS)) , splitAt(0(), XS) -> U191(isLNat(XS), XS) , U121(tt()) -> tt() , U131(tt(), V2) -> U132(isLNat(activate(V2))) , U132(tt()) -> tt() , U141(tt(), V2) -> U142(isLNat(activate(V2))) , U142(tt()) -> tt() , U151(tt(), V2) -> U152(isLNat(activate(V2))) , U152(tt()) -> tt() , U161(tt(), N) -> cons(activate(N), n__natsFrom(s(activate(N)))) , cons(X1, X2) -> n__cons(X1, X2) , s(X) -> n__s(X) , U171(tt(), N, XS) -> U172(isLNat(activate(XS)), activate(N), activate(XS)) , U172(tt(), N, XS) -> head(afterNth(activate(N), activate(XS))) , head(X) -> n__head(X) , head(cons(N, XS)) -> U31(isNatural(N), N, activate(XS)) , afterNth(N, XS) -> U11(isNatural(N), N, XS) , afterNth(X1, X2) -> n__afterNth(X1, X2) , U181(tt(), Y) -> U182(isLNat(activate(Y)), activate(Y)) , U182(tt(), Y) -> activate(Y) , U191(tt(), XS) -> pair(nil(), activate(XS)) , pair(X1, X2) -> n__pair(X1, X2) , nil() -> n__nil() , U201(tt(), N, X, XS) -> U202(isNatural(activate(X)), activate(N), activate(X), activate(XS)) , U202(tt(), N, X, XS) -> U203(isLNat(activate(XS)), activate(N), activate(X), activate(XS)) , isNatural(n__0()) -> tt() , isNatural(n__head(V1)) -> U111(isLNat(activate(V1))) , isNatural(n__s(V1)) -> U121(isNatural(activate(V1))) , isNatural(n__sel(V1, V2)) -> U131(isNatural(activate(V1)), activate(V2)) , U203(tt(), N, X, XS) -> U204(splitAt(activate(N), activate(XS)), activate(X)) , U204(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS) , U21(tt(), X, Y) -> U22(isLNat(activate(Y)), activate(X)) , U22(tt(), X) -> activate(X) , U211(tt(), XS) -> U212(isLNat(activate(XS)), activate(XS)) , U212(tt(), XS) -> activate(XS) , U221(tt(), N, XS) -> U222(isLNat(activate(XS)), activate(N), activate(XS)) , U222(tt(), N, XS) -> fst(splitAt(activate(N), activate(XS))) , fst(X) -> n__fst(X) , fst(pair(X, Y)) -> U21(isLNat(X), X, Y) , U31(tt(), N, XS) -> U32(isLNat(activate(XS)), activate(N)) , U32(tt(), N) -> activate(N) , U41(tt(), V2) -> U42(isLNat(activate(V2))) , U42(tt()) -> tt() , U51(tt(), V2) -> U52(isLNat(activate(V2))) , U52(tt()) -> tt() , U61(tt()) -> tt() , U71(tt()) -> tt() , U81(tt()) -> tt() , U91(tt()) -> tt() , isPLNat(n__pair(V1, V2)) -> U141(isLNat(activate(V1)), activate(V2)) , isPLNat(n__splitAt(V1, V2)) -> U151(isNatural(activate(V1)), activate(V2)) , natsFrom(N) -> U161(isNatural(N), N) , natsFrom(X) -> n__natsFrom(X) , sel(N, XS) -> U171(isNatural(N), N, XS) , sel(X1, X2) -> n__sel(X1, X2) , 0() -> n__0() , tail(X) -> n__tail(X) , tail(cons(N, XS)) -> U211(isNatural(N), activate(XS)) , take(N, XS) -> U221(isNatural(N), N, XS) , take(X1, X2) -> n__take(X1, X2) } Obligation: innermost runtime complexity Answer: MAYBE Arguments of following rules are not normal-forms: { snd(pair(X, Y)) -> U181(isLNat(X), Y) , splitAt(s(N), cons(X, XS)) -> U201(isNatural(N), N, X, activate(XS)) , splitAt(0(), XS) -> U191(isLNat(XS), XS) , head(cons(N, XS)) -> U31(isNatural(N), N, activate(XS)) , U204(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS) , fst(pair(X, Y)) -> U21(isLNat(X), X, Y) , tail(cons(N, XS)) -> U211(isNatural(N), activate(XS)) } All above mentioned rules can be savely removed. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { U101(tt(), V2) -> U102(isLNat(activate(V2))) , U102(tt()) -> tt() , isLNat(n__natsFrom(V1)) -> U71(isNatural(activate(V1))) , isLNat(n__nil()) -> tt() , isLNat(n__afterNth(V1, V2)) -> U41(isNatural(activate(V1)), activate(V2)) , isLNat(n__cons(V1, V2)) -> U51(isNatural(activate(V1)), activate(V2)) , isLNat(n__fst(V1)) -> U61(isPLNat(activate(V1))) , isLNat(n__snd(V1)) -> U81(isPLNat(activate(V1))) , isLNat(n__tail(V1)) -> U91(isLNat(activate(V1))) , isLNat(n__take(V1, V2)) -> U101(isNatural(activate(V1)), activate(V2)) , activate(X) -> X , 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) , U11(tt(), N, XS) -> U12(isLNat(activate(XS)), activate(N), activate(XS)) , U12(tt(), N, XS) -> snd(splitAt(activate(N), activate(XS))) , U111(tt()) -> tt() , snd(X) -> n__snd(X) , splitAt(X1, X2) -> n__splitAt(X1, X2) , U121(tt()) -> tt() , U131(tt(), V2) -> U132(isLNat(activate(V2))) , U132(tt()) -> tt() , U141(tt(), V2) -> U142(isLNat(activate(V2))) , U142(tt()) -> tt() , U151(tt(), V2) -> U152(isLNat(activate(V2))) , U152(tt()) -> tt() , U161(tt(), N) -> cons(activate(N), n__natsFrom(s(activate(N)))) , cons(X1, X2) -> n__cons(X1, X2) , s(X) -> n__s(X) , U171(tt(), N, XS) -> U172(isLNat(activate(XS)), activate(N), activate(XS)) , U172(tt(), N, XS) -> head(afterNth(activate(N), activate(XS))) , head(X) -> n__head(X) , afterNth(N, XS) -> U11(isNatural(N), N, XS) , afterNth(X1, X2) -> n__afterNth(X1, X2) , U181(tt(), Y) -> U182(isLNat(activate(Y)), activate(Y)) , U182(tt(), Y) -> activate(Y) , U191(tt(), XS) -> pair(nil(), activate(XS)) , pair(X1, X2) -> n__pair(X1, X2) , nil() -> n__nil() , U201(tt(), N, X, XS) -> U202(isNatural(activate(X)), activate(N), activate(X), activate(XS)) , U202(tt(), N, X, XS) -> U203(isLNat(activate(XS)), activate(N), activate(X), activate(XS)) , isNatural(n__0()) -> tt() , isNatural(n__head(V1)) -> U111(isLNat(activate(V1))) , isNatural(n__s(V1)) -> U121(isNatural(activate(V1))) , isNatural(n__sel(V1, V2)) -> U131(isNatural(activate(V1)), activate(V2)) , U203(tt(), N, X, XS) -> U204(splitAt(activate(N), activate(XS)), activate(X)) , U21(tt(), X, Y) -> U22(isLNat(activate(Y)), activate(X)) , U22(tt(), X) -> activate(X) , U211(tt(), XS) -> U212(isLNat(activate(XS)), activate(XS)) , U212(tt(), XS) -> activate(XS) , U221(tt(), N, XS) -> U222(isLNat(activate(XS)), activate(N), activate(XS)) , U222(tt(), N, XS) -> fst(splitAt(activate(N), activate(XS))) , fst(X) -> n__fst(X) , U31(tt(), N, XS) -> U32(isLNat(activate(XS)), activate(N)) , U32(tt(), N) -> activate(N) , U41(tt(), V2) -> U42(isLNat(activate(V2))) , U42(tt()) -> tt() , U51(tt(), V2) -> U52(isLNat(activate(V2))) , U52(tt()) -> tt() , U61(tt()) -> tt() , U71(tt()) -> tt() , U81(tt()) -> tt() , U91(tt()) -> tt() , isPLNat(n__pair(V1, V2)) -> U141(isLNat(activate(V1)), activate(V2)) , isPLNat(n__splitAt(V1, V2)) -> U151(isNatural(activate(V1)), activate(V2)) , natsFrom(N) -> U161(isNatural(N), N) , natsFrom(X) -> n__natsFrom(X) , sel(N, XS) -> U171(isNatural(N), N, XS) , sel(X1, X2) -> n__sel(X1, X2) , 0() -> n__0() , tail(X) -> n__tail(X) , take(N, XS) -> U221(isNatural(N), N, XS) , take(X1, X2) -> n__take(X1, X2) } Obligation: innermost runtime complexity Answer: MAYBE None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'Best' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'WithProblem (timeout of 60 seconds)' failed due to the following reason: Computation stopped due to timeout after 60.0 seconds. 2) 'Best' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'WithProblem (timeout of 30 seconds) (timeout of 60 seconds)' failed due to the following reason: Computation stopped due to timeout after 30.0 seconds. 2) 'Best' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'Polynomial Path Order (PS) (timeout of 60 seconds)' failed due to the following reason: The input cannot be shown compatible 2) 'bsearch-popstar (timeout of 60 seconds)' failed due to the following reason: The input cannot be shown compatible 3) 'Fastest (timeout of 5 seconds) (timeout of 60 seconds)' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. 2) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. Arrrr..