MAYBE MAYBE TRS: { U12(mark(X1), X2) -> mark(U12(X1, X2)), U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)), 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)), active(U12(X1, X2)) -> U12(active(X1), X2), active(U12(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)), active(splitAt(X1, X2)) -> splitAt(X1, active(X2)), active(splitAt(X1, X2)) -> splitAt(active(X1), X2), active(splitAt(s(N), cons(X, XS))) -> mark(U11(tt(), N, X, XS)), active(splitAt(0(), XS)) -> mark(pair(nil(), XS)), active(U11(X1, X2, X3, X4)) -> U11(active(X1), X2, X3, X4), active(U11(tt(), N, X, XS)) -> mark(U12(splitAt(N, XS), X)), active(pair(X1, X2)) -> pair(X1, active(X2)), active(pair(X1, X2)) -> pair(active(X1), X2), active(cons(X1, X2)) -> cons(active(X1), X2), active(snd(X)) -> snd(active(X)), active(snd(pair(X, Y))) -> mark(Y), active(afterNth(N, XS)) -> mark(snd(splitAt(N, XS))), active(afterNth(X1, X2)) -> afterNth(X1, active(X2)), active(afterNth(X1, X2)) -> afterNth(active(X1), X2), active(and(X1, X2)) -> and(active(X1), X2), active(and(tt(), X)) -> mark(X), active(fst(X)) -> fst(active(X)), active(fst(pair(X, Y))) -> mark(X), active(head(X)) -> head(active(X)), active(head(cons(N, XS))) -> mark(N), active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N)))), active(natsFrom(X)) -> natsFrom(active(X)), active(s(X)) -> s(active(X)), active(sel(N, XS)) -> mark(head(afterNth(N, XS))), active(sel(X1, X2)) -> sel(X1, active(X2)), active(sel(X1, X2)) -> sel(active(X1), X2), active(tail(X)) -> tail(active(X)), active(tail(cons(N, XS))) -> mark(XS), active(take(N, XS)) -> mark(fst(splitAt(N, XS))), active(take(X1, X2)) -> take(X1, active(X2)), active(take(X1, X2)) -> take(active(X1), X2), U11(mark(X1), X2, X3, X4) -> mark(U11(X1, X2, X3, X4)), U11(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U11(X1, X2, X3, X4)), 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)), cons(mark(X1), X2) -> mark(cons(X1, X2)), cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)), snd(mark(X)) -> mark(snd(X)), snd(ok(X)) -> ok(snd(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)), and(mark(X1), X2) -> mark(and(X1, X2)), and(ok(X1), ok(X2)) -> ok(and(X1, X2)), fst(mark(X)) -> mark(fst(X)), fst(ok(X)) -> ok(fst(X)), head(mark(X)) -> mark(head(X)), head(ok(X)) -> ok(head(X)), natsFrom(mark(X)) -> mark(natsFrom(X)), natsFrom(ok(X)) -> ok(natsFrom(X)), s(mark(X)) -> mark(s(X)), s(ok(X)) -> ok(s(X)), 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)), 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)), proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)), proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)), proper(U11(X1, X2, X3, X4)) -> U11(proper(X1), proper(X2), proper(X3), proper(X4)), proper(tt()) -> ok(tt()), proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)), proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)), proper(snd(X)) -> snd(proper(X)), proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)), proper(and(X1, X2)) -> and(proper(X1), proper(X2)), proper(fst(X)) -> fst(proper(X)), proper(head(X)) -> head(proper(X)), proper(natsFrom(X)) -> natsFrom(proper(X)), proper(s(X)) -> s(proper(X)), proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)), proper(nil()) -> ok(nil()), proper(0()) -> ok(0()), proper(tail(X)) -> tail(proper(X)), proper(take(X1, X2)) -> take(proper(X1), proper(X2)), top(mark(X)) -> top(proper(X)), top(ok(X)) -> top(active(X)) } DUP: We consider a duplicating system. Trs: { U12(mark(X1), X2) -> mark(U12(X1, X2)), U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)), 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)), active(U12(X1, X2)) -> U12(active(X1), X2), active(U12(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)), active(splitAt(X1, X2)) -> splitAt(X1, active(X2)), active(splitAt(X1, X2)) -> splitAt(active(X1), X2), active(splitAt(s(N), cons(X, XS))) -> mark(U11(tt(), N, X, XS)), active(splitAt(0(), XS)) -> mark(pair(nil(), XS)), active(U11(X1, X2, X3, X4)) -> U11(active(X1), X2, X3, X4), active(U11(tt(), N, X, XS)) -> mark(U12(splitAt(N, XS), X)), active(pair(X1, X2)) -> pair(X1, active(X2)), active(pair(X1, X2)) -> pair(active(X1), X2), active(cons(X1, X2)) -> cons(active(X1), X2), active(snd(X)) -> snd(active(X)), active(snd(pair(X, Y))) -> mark(Y), active(afterNth(N, XS)) -> mark(snd(splitAt(N, XS))), active(afterNth(X1, X2)) -> afterNth(X1, active(X2)), active(afterNth(X1, X2)) -> afterNth(active(X1), X2), active(and(X1, X2)) -> and(active(X1), X2), active(and(tt(), X)) -> mark(X), active(fst(X)) -> fst(active(X)), active(fst(pair(X, Y))) -> mark(X), active(head(X)) -> head(active(X)), active(head(cons(N, XS))) -> mark(N), active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N)))), active(natsFrom(X)) -> natsFrom(active(X)), active(s(X)) -> s(active(X)), active(sel(N, XS)) -> mark(head(afterNth(N, XS))), active(sel(X1, X2)) -> sel(X1, active(X2)), active(sel(X1, X2)) -> sel(active(X1), X2), active(tail(X)) -> tail(active(X)), active(tail(cons(N, XS))) -> mark(XS), active(take(N, XS)) -> mark(fst(splitAt(N, XS))), active(take(X1, X2)) -> take(X1, active(X2)), active(take(X1, X2)) -> take(active(X1), X2), U11(mark(X1), X2, X3, X4) -> mark(U11(X1, X2, X3, X4)), U11(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U11(X1, X2, X3, X4)), 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)), cons(mark(X1), X2) -> mark(cons(X1, X2)), cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)), snd(mark(X)) -> mark(snd(X)), snd(ok(X)) -> ok(snd(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)), and(mark(X1), X2) -> mark(and(X1, X2)), and(ok(X1), ok(X2)) -> ok(and(X1, X2)), fst(mark(X)) -> mark(fst(X)), fst(ok(X)) -> ok(fst(X)), head(mark(X)) -> mark(head(X)), head(ok(X)) -> ok(head(X)), natsFrom(mark(X)) -> mark(natsFrom(X)), natsFrom(ok(X)) -> ok(natsFrom(X)), s(mark(X)) -> mark(s(X)), s(ok(X)) -> ok(s(X)), 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)), 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)), proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)), proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)), proper(U11(X1, X2, X3, X4)) -> U11(proper(X1), proper(X2), proper(X3), proper(X4)), proper(tt()) -> ok(tt()), proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)), proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)), proper(snd(X)) -> snd(proper(X)), proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)), proper(and(X1, X2)) -> and(proper(X1), proper(X2)), proper(fst(X)) -> fst(proper(X)), proper(head(X)) -> head(proper(X)), proper(natsFrom(X)) -> natsFrom(proper(X)), proper(s(X)) -> s(proper(X)), proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)), proper(nil()) -> ok(nil()), proper(0()) -> ok(0()), proper(tail(X)) -> tail(proper(X)), proper(take(X1, X2)) -> take(proper(X1), proper(X2)), top(mark(X)) -> top(proper(X)), top(ok(X)) -> top(active(X)) } Fail