MAYBE MAYBE TRS: { U102(mark(X1), X2) -> mark(U102(X1, X2)), U102(ok(X1), ok(X2)) -> ok(U102(X1, X2)), isNatural(ok(X)) -> ok(isNatural(X)), active(U102(X1, X2)) -> U102(active(X1), X2), active(U102(tt(), V2)) -> mark(U103(isLNat(V2))), active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1), V1)), active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1), V1)), active(isNatural(0())) -> mark(tt()), active(isNatural(sel(V1, V2))) -> mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2)), active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3), active(U101(tt(), V1, V2)) -> mark(U102(isNatural(V1), V2)), active(U103(X)) -> U103(active(X)), active(U103(tt())) -> mark(tt()), active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1), V1)), active(isLNat(cons(V1, V2))) -> mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2)), active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1), V1)), active(isLNat(afterNth(V1, V2))) -> mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2)), active(isLNat(nil())) -> mark(tt()), active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1), V1)), active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1), V1)), active(isLNat(take(V1, V2))) -> mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2)), active(snd(X)) -> snd(active(X)), active(snd(pair(X, Y))) -> mark( U181( and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y ) ), active(splitAt(X1, X2)) -> splitAt(X1, active(X2)), active(splitAt(X1, X2)) -> splitAt(active(X1), X2), active(splitAt(s(N), cons(X, XS))) -> mark( U201( and( and(isNatural(N), isNaturalKind(N)), and( and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)) ) ), N, X, XS ) ), active(splitAt(0(), XS)) -> mark(U191(and(isLNat(XS), isLNatKind(XS)), XS)), active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3), active(U11(tt(), N, XS)) -> mark(snd(splitAt(N, XS))), active(U112(X)) -> U112(active(X)), active(U112(tt())) -> mark(tt()), active(U111(X1, X2)) -> U111(active(X1), X2), active(U111(tt(), V1)) -> mark(U112(isLNat(V1))), active(U122(X)) -> U122(active(X)), active(U122(tt())) -> mark(tt()), active(U121(X1, X2)) -> U121(active(X1), X2), active(U121(tt(), V1)) -> mark(U122(isNatural(V1))), active(U132(X1, X2)) -> U132(active(X1), X2), active(U132(tt(), V2)) -> mark(U133(isLNat(V2))), active(U131(X1, X2, X3)) -> U131(active(X1), X2, X3), active(U131(tt(), V1, V2)) -> mark(U132(isNatural(V1), V2)), active(U133(X)) -> U133(active(X)), active(U133(tt())) -> mark(tt()), active(U142(X1, X2)) -> U142(active(X1), X2), active(U142(tt(), V2)) -> mark(U143(isLNat(V2))), active(U141(X1, X2, X3)) -> U141(active(X1), X2, X3), active(U141(tt(), V1, V2)) -> mark(U142(isLNat(V1), V2)), active(U143(X)) -> U143(active(X)), active(U143(tt())) -> mark(tt()), active(U152(X1, X2)) -> U152(active(X1), X2), active(U152(tt(), V2)) -> mark(U153(isLNat(V2))), active(U151(X1, X2, X3)) -> U151(active(X1), X2, X3), active(U151(tt(), V1, V2)) -> mark(U152(isNatural(V1), V2)), active(U153(X)) -> U153(active(X)), active(U153(tt())) -> mark(tt()), active(cons(X1, X2)) -> cons(active(X1), X2), active(natsFrom(N)) -> mark(U161(and(isNatural(N), isNaturalKind(N)), N)), active(natsFrom(X)) -> natsFrom(active(X)), active(s(X)) -> s(active(X)), active(U161(X1, X2)) -> U161(active(X1), X2), active(U161(tt(), N)) -> mark(cons(N, natsFrom(s(N)))), active(head(X)) -> head(active(X)), active(head(cons(N, XS))) -> mark( U31( and( and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS)) ), N ) ), active(afterNth(N, XS)) -> mark( U11( and( and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS)) ), N, XS ) ), active(afterNth(X1, X2)) -> afterNth(X1, active(X2)), active(afterNth(X1, X2)) -> afterNth(active(X1), X2), active(U171(X1, X2, X3)) -> U171(active(X1), X2, X3), active(U171(tt(), N, XS)) -> mark(head(afterNth(N, XS))), active(U181(X1, X2)) -> U181(active(X1), X2), active(U181(tt(), Y)) -> mark(Y), active(pair(X1, X2)) -> pair(X1, active(X2)), active(pair(X1, X2)) -> pair(active(X1), X2), active(U191(X1, X2)) -> U191(active(X1), X2), active(U191(tt(), XS)) -> mark(pair(nil(), XS)), active(U202(X1, X2)) -> U202(active(X1), X2), active(U202(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)), active(U201(X1, X2, X3, X4)) -> U201(active(X1), X2, X3, X4), active(U201(tt(), N, X, XS)) -> mark(U202(splitAt(N, XS), X)), active(U21(X1, X2)) -> U21(active(X1), X2), active(U21(tt(), X)) -> mark(X), active(U211(X1, X2)) -> U211(active(X1), X2), active(U211(tt(), XS)) -> mark(XS), active(fst(X)) -> fst(active(X)), active(fst(pair(X, Y))) -> mark( U21( and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X ) ), active(U221(X1, X2, X3)) -> U221(active(X1), X2, X3), active(U221(tt(), N, XS)) -> mark(fst(splitAt(N, XS))), active(U31(X1, X2)) -> U31(active(X1), X2), active(U31(tt(), N)) -> mark(N), active(U42(X1, X2)) -> U42(active(X1), X2), active(U42(tt(), V2)) -> mark(U43(isLNat(V2))), active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3), active(U41(tt(), V1, V2)) -> mark(U42(isNatural(V1), V2)), active(U43(X)) -> U43(active(X)), active(U43(tt())) -> mark(tt()), active(U52(X1, X2)) -> U52(active(X1), X2), active(U52(tt(), V2)) -> mark(U53(isLNat(V2))), active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3), active(U51(tt(), V1, V2)) -> mark(U52(isNatural(V1), V2)), active(U53(X)) -> U53(active(X)), active(U53(tt())) -> mark(tt()), active(U62(X)) -> U62(active(X)), active(U62(tt())) -> mark(tt()), active(isPLNat(splitAt(V1, V2))) -> mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2)), active(isPLNat(pair(V1, V2))) -> mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2)), active(U61(X1, X2)) -> U61(active(X1), X2), active(U61(tt(), V1)) -> mark(U62(isPLNat(V1))), active(U72(X)) -> U72(active(X)), active(U72(tt())) -> mark(tt()), active(U71(X1, X2)) -> U71(active(X1), X2), active(U71(tt(), V1)) -> mark(U72(isNatural(V1))), active(U82(X)) -> U82(active(X)), active(U82(tt())) -> mark(tt()), active(U81(X1, X2)) -> U81(active(X1), X2), active(U81(tt(), V1)) -> mark(U82(isPLNat(V1))), active(U92(X)) -> U92(active(X)), active(U92(tt())) -> mark(tt()), active(U91(X1, X2)) -> U91(active(X1), X2), active(U91(tt(), V1)) -> mark(U92(isLNat(V1))), active(and(X1, X2)) -> and(active(X1), X2), active(and(tt(), X)) -> mark(X), active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1)), active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1)), active(isNaturalKind(0())) -> mark(tt()), active(isNaturalKind(sel(V1, V2))) -> mark(and(isNaturalKind(V1), isLNatKind(V2))), active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1)), active(isLNatKind(cons(V1, V2))) -> mark(and(isNaturalKind(V1), isLNatKind(V2))), active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1)), active(isLNatKind(afterNth(V1, V2))) -> mark(and(isNaturalKind(V1), isLNatKind(V2))), active(isLNatKind(nil())) -> mark(tt()), active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1)), active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1)), active(isLNatKind(take(V1, V2))) -> mark(and(isNaturalKind(V1), isLNatKind(V2))), active(isPLNatKind(splitAt(V1, V2))) -> mark(and(isNaturalKind(V1), isLNatKind(V2))), active(isPLNatKind(pair(V1, V2))) -> mark(and(isLNatKind(V1), isLNatKind(V2))), active(tail(X)) -> tail(active(X)), active(tail(cons(N, XS))) -> mark( U211( and( and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS)) ), XS ) ), active(take(N, XS)) -> mark( U221( and( and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS)) ), N, XS ) ), active(take(X1, X2)) -> take(X1, active(X2)), active(take(X1, X2)) -> take(active(X1), X2), active(sel(N, XS)) -> mark( U171( and( and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS)) ), N, XS ) ), active(sel(X1, X2)) -> sel(X1, active(X2)), active(sel(X1, X2)) -> sel(active(X1), X2), U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)), U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)), U103(mark(X)) -> mark(U103(X)), U103(ok(X)) -> ok(U103(X)), isLNat(ok(X)) -> ok(isLNat(X)), snd(mark(X)) -> mark(snd(X)), snd(ok(X)) -> ok(snd(X)), splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)), splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)), splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)), U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)), U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)), U112(mark(X)) -> mark(U112(X)), U112(ok(X)) -> ok(U112(X)), U111(mark(X1), X2) -> mark(U111(X1, X2)), U111(ok(X1), ok(X2)) -> ok(U111(X1, X2)), U122(mark(X)) -> mark(U122(X)), U122(ok(X)) -> ok(U122(X)), U121(mark(X1), X2) -> mark(U121(X1, X2)), U121(ok(X1), ok(X2)) -> ok(U121(X1, X2)), U132(mark(X1), X2) -> mark(U132(X1, X2)), U132(ok(X1), ok(X2)) -> ok(U132(X1, X2)), U131(mark(X1), X2, X3) -> mark(U131(X1, X2, X3)), U131(ok(X1), ok(X2), ok(X3)) -> ok(U131(X1, X2, X3)), U133(mark(X)) -> mark(U133(X)), U133(ok(X)) -> ok(U133(X)), U142(mark(X1), X2) -> mark(U142(X1, X2)), U142(ok(X1), ok(X2)) -> ok(U142(X1, X2)), U141(mark(X1), X2, X3) -> mark(U141(X1, X2, X3)), U141(ok(X1), ok(X2), ok(X3)) -> ok(U141(X1, X2, X3)), U143(mark(X)) -> mark(U143(X)), U143(ok(X)) -> ok(U143(X)), U152(mark(X1), X2) -> mark(U152(X1, X2)), U152(ok(X1), ok(X2)) -> ok(U152(X1, X2)), U151(mark(X1), X2, X3) -> mark(U151(X1, X2, X3)), U151(ok(X1), ok(X2), ok(X3)) -> ok(U151(X1, X2, X3)), U153(mark(X)) -> mark(U153(X)), U153(ok(X)) -> ok(U153(X)), cons(mark(X1), X2) -> mark(cons(X1, X2)), cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)), natsFrom(mark(X)) -> mark(natsFrom(X)), natsFrom(ok(X)) -> ok(natsFrom(X)), s(mark(X)) -> mark(s(X)), s(ok(X)) -> ok(s(X)), U161(mark(X1), X2) -> mark(U161(X1, X2)), U161(ok(X1), ok(X2)) -> ok(U161(X1, X2)), head(mark(X)) -> mark(head(X)), head(ok(X)) -> ok(head(X)), afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)), afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)), afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)), U171(mark(X1), X2, X3) -> mark(U171(X1, X2, X3)), U171(ok(X1), ok(X2), ok(X3)) -> ok(U171(X1, X2, X3)), U181(mark(X1), X2) -> mark(U181(X1, X2)), U181(ok(X1), ok(X2)) -> ok(U181(X1, X2)), pair(X1, mark(X2)) -> mark(pair(X1, X2)), pair(mark(X1), X2) -> mark(pair(X1, X2)), pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)), U191(mark(X1), X2) -> mark(U191(X1, X2)), U191(ok(X1), ok(X2)) -> ok(U191(X1, X2)), U202(mark(X1), X2) -> mark(U202(X1, X2)), U202(ok(X1), ok(X2)) -> ok(U202(X1, X2)), U201(mark(X1), X2, X3, X4) -> mark(U201(X1, X2, X3, X4)), U201(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U201(X1, X2, X3, X4)), U21(mark(X1), X2) -> mark(U21(X1, X2)), U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)), U211(mark(X1), X2) -> mark(U211(X1, X2)), U211(ok(X1), ok(X2)) -> ok(U211(X1, X2)), fst(mark(X)) -> mark(fst(X)), fst(ok(X)) -> ok(fst(X)), U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3)), U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3)), U31(mark(X1), X2) -> mark(U31(X1, X2)), U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)), U42(mark(X1), X2) -> mark(U42(X1, X2)), U42(ok(X1), ok(X2)) -> ok(U42(X1, X2)), U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)), U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)), U43(mark(X)) -> mark(U43(X)), U43(ok(X)) -> ok(U43(X)), U52(mark(X1), X2) -> mark(U52(X1, X2)), U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)), U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)), U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)), U53(mark(X)) -> mark(U53(X)), U53(ok(X)) -> ok(U53(X)), U62(mark(X)) -> mark(U62(X)), U62(ok(X)) -> ok(U62(X)), isPLNat(ok(X)) -> ok(isPLNat(X)), U61(mark(X1), X2) -> mark(U61(X1, X2)), U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)), U72(mark(X)) -> mark(U72(X)), U72(ok(X)) -> ok(U72(X)), U71(mark(X1), X2) -> mark(U71(X1, X2)), U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)), U82(mark(X)) -> mark(U82(X)), U82(ok(X)) -> ok(U82(X)), U81(mark(X1), X2) -> mark(U81(X1, X2)), U81(ok(X1), ok(X2)) -> ok(U81(X1, X2)), U92(mark(X)) -> mark(U92(X)), U92(ok(X)) -> ok(U92(X)), U91(mark(X1), X2) -> mark(U91(X1, X2)), U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)), and(mark(X1), X2) -> mark(and(X1, X2)), and(ok(X1), ok(X2)) -> ok(and(X1, X2)), isNaturalKind(ok(X)) -> ok(isNaturalKind(X)), isLNatKind(ok(X)) -> ok(isLNatKind(X)), isPLNatKind(ok(X)) -> ok(isPLNatKind(X)), tail(mark(X)) -> mark(tail(X)), tail(ok(X)) -> ok(tail(X)), take(X1, mark(X2)) -> mark(take(X1, X2)), take(mark(X1), X2) -> mark(take(X1, X2)), take(ok(X1), ok(X2)) -> ok(take(X1, X2)), sel(X1, mark(X2)) -> mark(sel(X1, X2)), sel(mark(X1), X2) -> mark(sel(X1, X2)), sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)), proper(U102(X1, X2)) -> U102(proper(X1), proper(X2)), proper(isNatural(X)) -> isNatural(proper(X)), proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)), proper(tt()) -> ok(tt()), proper(U103(X)) -> U103(proper(X)), proper(isLNat(X)) -> isLNat(proper(X)), proper(snd(X)) -> snd(proper(X)), proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)), proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)), proper(U112(X)) -> U112(proper(X)), proper(U111(X1, X2)) -> U111(proper(X1), proper(X2)), proper(U122(X)) -> U122(proper(X)), proper(U121(X1, X2)) -> U121(proper(X1), proper(X2)), proper(U132(X1, X2)) -> U132(proper(X1), proper(X2)), proper(U131(X1, X2, X3)) -> U131(proper(X1), proper(X2), proper(X3)), proper(U133(X)) -> U133(proper(X)), proper(U142(X1, X2)) -> U142(proper(X1), proper(X2)), proper(U141(X1, X2, X3)) -> U141(proper(X1), proper(X2), proper(X3)), proper(U143(X)) -> U143(proper(X)), proper(U152(X1, X2)) -> U152(proper(X1), proper(X2)), proper(U151(X1, X2, X3)) -> U151(proper(X1), proper(X2), proper(X3)), proper(U153(X)) -> U153(proper(X)), proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)), proper(natsFrom(X)) -> natsFrom(proper(X)), proper(s(X)) -> s(proper(X)), proper(U161(X1, X2)) -> U161(proper(X1), proper(X2)), proper(head(X)) -> head(proper(X)), proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)), proper(U171(X1, X2, X3)) -> U171(proper(X1), proper(X2), proper(X3)), proper(U181(X1, X2)) -> U181(proper(X1), proper(X2)), proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)), proper(nil()) -> ok(nil()), proper(U191(X1, X2)) -> U191(proper(X1), proper(X2)), proper(U202(X1, X2)) -> U202(proper(X1), proper(X2)), proper(U201(X1, X2, X3, X4)) -> U201(proper(X1), proper(X2), proper(X3), proper(X4)), proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)), proper(U211(X1, X2)) -> U211(proper(X1), proper(X2)), proper(fst(X)) -> fst(proper(X)), proper(U221(X1, X2, X3)) -> U221(proper(X1), proper(X2), proper(X3)), proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)), proper(U42(X1, X2)) -> U42(proper(X1), proper(X2)), proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)), proper(U43(X)) -> U43(proper(X)), proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)), proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)), proper(U53(X)) -> U53(proper(X)), proper(U62(X)) -> U62(proper(X)), proper(isPLNat(X)) -> isPLNat(proper(X)), proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)), proper(U72(X)) -> U72(proper(X)), proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)), proper(U82(X)) -> U82(proper(X)), proper(U81(X1, X2)) -> U81(proper(X1), proper(X2)), proper(U92(X)) -> U92(proper(X)), proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)), proper(and(X1, X2)) -> and(proper(X1), proper(X2)), proper(isNaturalKind(X)) -> isNaturalKind(proper(X)), proper(isLNatKind(X)) -> isLNatKind(proper(X)), proper(isPLNatKind(X)) -> isPLNatKind(proper(X)), proper(tail(X)) -> tail(proper(X)), proper(take(X1, X2)) -> take(proper(X1), proper(X2)), proper(0()) -> ok(0()), proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)), top(mark(X)) -> top(proper(X)), top(ok(X)) -> top(active(X)) } DUP: We consider a duplicating system. Trs: { U102(mark(X1), X2) -> mark(U102(X1, X2)), U102(ok(X1), ok(X2)) -> ok(U102(X1, X2)), isNatural(ok(X)) -> ok(isNatural(X)), active(U102(X1, X2)) -> U102(active(X1), X2), active(U102(tt(), V2)) -> mark(U103(isLNat(V2))), active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1), V1)), active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1), V1)), active(isNatural(0())) -> mark(tt()), active(isNatural(sel(V1, V2))) -> mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2)), active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3), active(U101(tt(), V1, V2)) -> mark(U102(isNatural(V1), V2)), active(U103(X)) -> U103(active(X)), active(U103(tt())) -> mark(tt()), active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1), V1)), active(isLNat(cons(V1, V2))) -> mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2)), active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1), V1)), active(isLNat(afterNth(V1, V2))) -> mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2)), active(isLNat(nil())) -> mark(tt()), active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1), V1)), active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1), V1)), active(isLNat(take(V1, V2))) -> mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2)), active(snd(X)) -> snd(active(X)), active(snd(pair(X, Y))) -> mark( U181( and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y ) ), active(splitAt(X1, X2)) -> splitAt(X1, active(X2)), active(splitAt(X1, X2)) -> splitAt(active(X1), X2), active(splitAt(s(N), cons(X, XS))) -> mark( U201( and( and(isNatural(N), isNaturalKind(N)), and( and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)) ) ), N, X, XS ) ), active(splitAt(0(), XS)) -> mark(U191(and(isLNat(XS), isLNatKind(XS)), XS)), active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3), active(U11(tt(), N, XS)) -> mark(snd(splitAt(N, XS))), active(U112(X)) -> U112(active(X)), active(U112(tt())) -> mark(tt()), active(U111(X1, X2)) -> U111(active(X1), X2), active(U111(tt(), V1)) -> mark(U112(isLNat(V1))), active(U122(X)) -> U122(active(X)), active(U122(tt())) -> mark(tt()), active(U121(X1, X2)) -> U121(active(X1), X2), active(U121(tt(), V1)) -> mark(U122(isNatural(V1))), active(U132(X1, X2)) -> U132(active(X1), X2), active(U132(tt(), V2)) -> mark(U133(isLNat(V2))), active(U131(X1, X2, X3)) -> U131(active(X1), X2, X3), active(U131(tt(), V1, V2)) -> mark(U132(isNatural(V1), V2)), active(U133(X)) -> U133(active(X)), active(U133(tt())) -> mark(tt()), active(U142(X1, X2)) -> U142(active(X1), X2), active(U142(tt(), V2)) -> mark(U143(isLNat(V2))), active(U141(X1, X2, X3)) -> U141(active(X1), X2, X3), active(U141(tt(), V1, V2)) -> mark(U142(isLNat(V1), V2)), active(U143(X)) -> U143(active(X)), active(U143(tt())) -> mark(tt()), active(U152(X1, X2)) -> U152(active(X1), X2), active(U152(tt(), V2)) -> mark(U153(isLNat(V2))), active(U151(X1, X2, X3)) -> U151(active(X1), X2, X3), active(U151(tt(), V1, V2)) -> mark(U152(isNatural(V1), V2)), active(U153(X)) -> U153(active(X)), active(U153(tt())) -> mark(tt()), active(cons(X1, X2)) -> cons(active(X1), X2), active(natsFrom(N)) -> mark(U161(and(isNatural(N), isNaturalKind(N)), N)), active(natsFrom(X)) -> natsFrom(active(X)), active(s(X)) -> s(active(X)), active(U161(X1, X2)) -> U161(active(X1), X2), active(U161(tt(), N)) -> mark(cons(N, natsFrom(s(N)))), active(head(X)) -> head(active(X)), active(head(cons(N, XS))) -> mark( U31( and( and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS)) ), N ) ), active(afterNth(N, XS)) -> mark( U11( and( and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS)) ), N, XS ) ), active(afterNth(X1, X2)) -> afterNth(X1, active(X2)), active(afterNth(X1, X2)) -> afterNth(active(X1), X2), active(U171(X1, X2, X3)) -> U171(active(X1), X2, X3), active(U171(tt(), N, XS)) -> mark(head(afterNth(N, XS))), active(U181(X1, X2)) -> U181(active(X1), X2), active(U181(tt(), Y)) -> mark(Y), active(pair(X1, X2)) -> pair(X1, active(X2)), active(pair(X1, X2)) -> pair(active(X1), X2), active(U191(X1, X2)) -> U191(active(X1), X2), active(U191(tt(), XS)) -> mark(pair(nil(), XS)), active(U202(X1, X2)) -> U202(active(X1), X2), active(U202(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)), active(U201(X1, X2, X3, X4)) -> U201(active(X1), X2, X3, X4), active(U201(tt(), N, X, XS)) -> mark(U202(splitAt(N, XS), X)), active(U21(X1, X2)) -> U21(active(X1), X2), active(U21(tt(), X)) -> mark(X), active(U211(X1, X2)) -> U211(active(X1), X2), active(U211(tt(), XS)) -> mark(XS), active(fst(X)) -> fst(active(X)), active(fst(pair(X, Y))) -> mark( U21( and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X ) ), active(U221(X1, X2, X3)) -> U221(active(X1), X2, X3), active(U221(tt(), N, XS)) -> mark(fst(splitAt(N, XS))), active(U31(X1, X2)) -> U31(active(X1), X2), active(U31(tt(), N)) -> mark(N), active(U42(X1, X2)) -> U42(active(X1), X2), active(U42(tt(), V2)) -> mark(U43(isLNat(V2))), active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3), active(U41(tt(), V1, V2)) -> mark(U42(isNatural(V1), V2)), active(U43(X)) -> U43(active(X)), active(U43(tt())) -> mark(tt()), active(U52(X1, X2)) -> U52(active(X1), X2), active(U52(tt(), V2)) -> mark(U53(isLNat(V2))), active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3), active(U51(tt(), V1, V2)) -> mark(U52(isNatural(V1), V2)), active(U53(X)) -> U53(active(X)), active(U53(tt())) -> mark(tt()), active(U62(X)) -> U62(active(X)), active(U62(tt())) -> mark(tt()), active(isPLNat(splitAt(V1, V2))) -> mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2)), active(isPLNat(pair(V1, V2))) -> mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2)), active(U61(X1, X2)) -> U61(active(X1), X2), active(U61(tt(), V1)) -> mark(U62(isPLNat(V1))), active(U72(X)) -> U72(active(X)), active(U72(tt())) -> mark(tt()), active(U71(X1, X2)) -> U71(active(X1), X2), active(U71(tt(), V1)) -> mark(U72(isNatural(V1))), active(U82(X)) -> U82(active(X)), active(U82(tt())) -> mark(tt()), active(U81(X1, X2)) -> U81(active(X1), X2), active(U81(tt(), V1)) -> mark(U82(isPLNat(V1))), active(U92(X)) -> U92(active(X)), active(U92(tt())) -> mark(tt()), active(U91(X1, X2)) -> U91(active(X1), X2), active(U91(tt(), V1)) -> mark(U92(isLNat(V1))), active(and(X1, X2)) -> and(active(X1), X2), active(and(tt(), X)) -> mark(X), active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1)), active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1)), active(isNaturalKind(0())) -> mark(tt()), active(isNaturalKind(sel(V1, V2))) -> mark(and(isNaturalKind(V1), isLNatKind(V2))), active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1)), active(isLNatKind(cons(V1, V2))) -> mark(and(isNaturalKind(V1), isLNatKind(V2))), active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1)), active(isLNatKind(afterNth(V1, V2))) -> mark(and(isNaturalKind(V1), isLNatKind(V2))), active(isLNatKind(nil())) -> mark(tt()), active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1)), active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1)), active(isLNatKind(take(V1, V2))) -> mark(and(isNaturalKind(V1), isLNatKind(V2))), active(isPLNatKind(splitAt(V1, V2))) -> mark(and(isNaturalKind(V1), isLNatKind(V2))), active(isPLNatKind(pair(V1, V2))) -> mark(and(isLNatKind(V1), isLNatKind(V2))), active(tail(X)) -> tail(active(X)), active(tail(cons(N, XS))) -> mark( U211( and( and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS)) ), XS ) ), active(take(N, XS)) -> mark( U221( and( and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS)) ), N, XS ) ), active(take(X1, X2)) -> take(X1, active(X2)), active(take(X1, X2)) -> take(active(X1), X2), active(sel(N, XS)) -> mark( U171( and( and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS)) ), N, XS ) ), active(sel(X1, X2)) -> sel(X1, active(X2)), active(sel(X1, X2)) -> sel(active(X1), X2), U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)), U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)), U103(mark(X)) -> mark(U103(X)), U103(ok(X)) -> ok(U103(X)), isLNat(ok(X)) -> ok(isLNat(X)), snd(mark(X)) -> mark(snd(X)), snd(ok(X)) -> ok(snd(X)), splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)), splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)), splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)), U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)), U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)), U112(mark(X)) -> mark(U112(X)), U112(ok(X)) -> ok(U112(X)), U111(mark(X1), X2) -> mark(U111(X1, X2)), U111(ok(X1), ok(X2)) -> ok(U111(X1, X2)), U122(mark(X)) -> mark(U122(X)), U122(ok(X)) -> ok(U122(X)), U121(mark(X1), X2) -> mark(U121(X1, X2)), U121(ok(X1), ok(X2)) -> ok(U121(X1, X2)), U132(mark(X1), X2) -> mark(U132(X1, X2)), U132(ok(X1), ok(X2)) -> ok(U132(X1, X2)), U131(mark(X1), X2, X3) -> mark(U131(X1, X2, X3)), U131(ok(X1), ok(X2), ok(X3)) -> ok(U131(X1, X2, X3)), U133(mark(X)) -> mark(U133(X)), U133(ok(X)) -> ok(U133(X)), U142(mark(X1), X2) -> mark(U142(X1, X2)), U142(ok(X1), ok(X2)) -> ok(U142(X1, X2)), U141(mark(X1), X2, X3) -> mark(U141(X1, X2, X3)), U141(ok(X1), ok(X2), ok(X3)) -> ok(U141(X1, X2, X3)), U143(mark(X)) -> mark(U143(X)), U143(ok(X)) -> ok(U143(X)), U152(mark(X1), X2) -> mark(U152(X1, X2)), U152(ok(X1), ok(X2)) -> ok(U152(X1, X2)), U151(mark(X1), X2, X3) -> mark(U151(X1, X2, X3)), U151(ok(X1), ok(X2), ok(X3)) -> ok(U151(X1, X2, X3)), U153(mark(X)) -> mark(U153(X)), U153(ok(X)) -> ok(U153(X)), cons(mark(X1), X2) -> mark(cons(X1, X2)), cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)), natsFrom(mark(X)) -> mark(natsFrom(X)), natsFrom(ok(X)) -> ok(natsFrom(X)), s(mark(X)) -> mark(s(X)), s(ok(X)) -> ok(s(X)), U161(mark(X1), X2) -> mark(U161(X1, X2)), U161(ok(X1), ok(X2)) -> ok(U161(X1, X2)), head(mark(X)) -> mark(head(X)), head(ok(X)) -> ok(head(X)), afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)), afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)), afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)), U171(mark(X1), X2, X3) -> mark(U171(X1, X2, X3)), U171(ok(X1), ok(X2), ok(X3)) -> ok(U171(X1, X2, X3)), U181(mark(X1), X2) -> mark(U181(X1, X2)), U181(ok(X1), ok(X2)) -> ok(U181(X1, X2)), pair(X1, mark(X2)) -> mark(pair(X1, X2)), pair(mark(X1), X2) -> mark(pair(X1, X2)), pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)), U191(mark(X1), X2) -> mark(U191(X1, X2)), U191(ok(X1), ok(X2)) -> ok(U191(X1, X2)), U202(mark(X1), X2) -> mark(U202(X1, X2)), U202(ok(X1), ok(X2)) -> ok(U202(X1, X2)), U201(mark(X1), X2, X3, X4) -> mark(U201(X1, X2, X3, X4)), U201(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U201(X1, X2, X3, X4)), U21(mark(X1), X2) -> mark(U21(X1, X2)), U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)), U211(mark(X1), X2) -> mark(U211(X1, X2)), U211(ok(X1), ok(X2)) -> ok(U211(X1, X2)), fst(mark(X)) -> mark(fst(X)), fst(ok(X)) -> ok(fst(X)), U221(mark(X1), X2, X3) -> mark(U221(X1, X2, X3)), U221(ok(X1), ok(X2), ok(X3)) -> ok(U221(X1, X2, X3)), U31(mark(X1), X2) -> mark(U31(X1, X2)), U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)), U42(mark(X1), X2) -> mark(U42(X1, X2)), U42(ok(X1), ok(X2)) -> ok(U42(X1, X2)), U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)), U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)), U43(mark(X)) -> mark(U43(X)), U43(ok(X)) -> ok(U43(X)), U52(mark(X1), X2) -> mark(U52(X1, X2)), U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)), U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)), U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)), U53(mark(X)) -> mark(U53(X)), U53(ok(X)) -> ok(U53(X)), U62(mark(X)) -> mark(U62(X)), U62(ok(X)) -> ok(U62(X)), isPLNat(ok(X)) -> ok(isPLNat(X)), U61(mark(X1), X2) -> mark(U61(X1, X2)), U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)), U72(mark(X)) -> mark(U72(X)), U72(ok(X)) -> ok(U72(X)), U71(mark(X1), X2) -> mark(U71(X1, X2)), U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)), U82(mark(X)) -> mark(U82(X)), U82(ok(X)) -> ok(U82(X)), U81(mark(X1), X2) -> mark(U81(X1, X2)), U81(ok(X1), ok(X2)) -> ok(U81(X1, X2)), U92(mark(X)) -> mark(U92(X)), U92(ok(X)) -> ok(U92(X)), U91(mark(X1), X2) -> mark(U91(X1, X2)), U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)), and(mark(X1), X2) -> mark(and(X1, X2)), and(ok(X1), ok(X2)) -> ok(and(X1, X2)), isNaturalKind(ok(X)) -> ok(isNaturalKind(X)), isLNatKind(ok(X)) -> ok(isLNatKind(X)), isPLNatKind(ok(X)) -> ok(isPLNatKind(X)), tail(mark(X)) -> mark(tail(X)), tail(ok(X)) -> ok(tail(X)), take(X1, mark(X2)) -> mark(take(X1, X2)), take(mark(X1), X2) -> mark(take(X1, X2)), take(ok(X1), ok(X2)) -> ok(take(X1, X2)), sel(X1, mark(X2)) -> mark(sel(X1, X2)), sel(mark(X1), X2) -> mark(sel(X1, X2)), sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)), proper(U102(X1, X2)) -> U102(proper(X1), proper(X2)), proper(isNatural(X)) -> isNatural(proper(X)), proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)), proper(tt()) -> ok(tt()), proper(U103(X)) -> U103(proper(X)), proper(isLNat(X)) -> isLNat(proper(X)), proper(snd(X)) -> snd(proper(X)), proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)), proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)), proper(U112(X)) -> U112(proper(X)), proper(U111(X1, X2)) -> U111(proper(X1), proper(X2)), proper(U122(X)) -> U122(proper(X)), proper(U121(X1, X2)) -> U121(proper(X1), proper(X2)), proper(U132(X1, X2)) -> U132(proper(X1), proper(X2)), proper(U131(X1, X2, X3)) -> U131(proper(X1), proper(X2), proper(X3)), proper(U133(X)) -> U133(proper(X)), proper(U142(X1, X2)) -> U142(proper(X1), proper(X2)), proper(U141(X1, X2, X3)) -> U141(proper(X1), proper(X2), proper(X3)), proper(U143(X)) -> U143(proper(X)), proper(U152(X1, X2)) -> U152(proper(X1), proper(X2)), proper(U151(X1, X2, X3)) -> U151(proper(X1), proper(X2), proper(X3)), proper(U153(X)) -> U153(proper(X)), proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)), proper(natsFrom(X)) -> natsFrom(proper(X)), proper(s(X)) -> s(proper(X)), proper(U161(X1, X2)) -> U161(proper(X1), proper(X2)), proper(head(X)) -> head(proper(X)), proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)), proper(U171(X1, X2, X3)) -> U171(proper(X1), proper(X2), proper(X3)), proper(U181(X1, X2)) -> U181(proper(X1), proper(X2)), proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)), proper(nil()) -> ok(nil()), proper(U191(X1, X2)) -> U191(proper(X1), proper(X2)), proper(U202(X1, X2)) -> U202(proper(X1), proper(X2)), proper(U201(X1, X2, X3, X4)) -> U201(proper(X1), proper(X2), proper(X3), proper(X4)), proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)), proper(U211(X1, X2)) -> U211(proper(X1), proper(X2)), proper(fst(X)) -> fst(proper(X)), proper(U221(X1, X2, X3)) -> U221(proper(X1), proper(X2), proper(X3)), proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)), proper(U42(X1, X2)) -> U42(proper(X1), proper(X2)), proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)), proper(U43(X)) -> U43(proper(X)), proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)), proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)), proper(U53(X)) -> U53(proper(X)), proper(U62(X)) -> U62(proper(X)), proper(isPLNat(X)) -> isPLNat(proper(X)), proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)), proper(U72(X)) -> U72(proper(X)), proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)), proper(U82(X)) -> U82(proper(X)), proper(U81(X1, X2)) -> U81(proper(X1), proper(X2)), proper(U92(X)) -> U92(proper(X)), proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)), proper(and(X1, X2)) -> and(proper(X1), proper(X2)), proper(isNaturalKind(X)) -> isNaturalKind(proper(X)), proper(isLNatKind(X)) -> isLNatKind(proper(X)), proper(isPLNatKind(X)) -> isPLNatKind(proper(X)), proper(tail(X)) -> tail(proper(X)), proper(take(X1, X2)) -> take(proper(X1), proper(X2)), proper(0()) -> ok(0()), proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)), top(mark(X)) -> top(proper(X)), top(ok(X)) -> top(active(X)) } Fail