MAYBE MAYBE TRS: { active(and(X1, X2)) -> and(X1, active(X2)), active(and(X1, X2)) -> and(active(X1), X2), active(and(tt(), T)) -> mark(T), active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))), active(isNatList(nil())) -> mark(tt()), active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))), active(isNatIList(IL)) -> mark(isNatList(IL)), active(isNatIList(zeros())) -> mark(tt()), active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))), active(isNat(0())) -> mark(tt()), active(isNat(s(N))) -> mark(isNat(N)), active(isNat(length(L))) -> mark(isNatList(L)), active(s(X)) -> s(active(X)), active(length(X)) -> length(active(X)), active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)), active(zeros()) -> mark(cons(0(), zeros())), active(cons(X1, X2)) -> cons(active(X1), X2), active(take(X1, X2)) -> take(X1, active(X2)), active(take(X1, X2)) -> take(active(X1), X2), active(take(0(), IL)) -> mark(uTake1(isNatIList(IL))), active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)), active(uTake1(X)) -> uTake1(active(X)), active(uTake1(tt())) -> mark(nil()), active(uTake2(X1, X2, X3, X4)) -> uTake2(active(X1), X2, X3, X4), active(uTake2(tt(), M, N, IL)) -> mark(cons(N, take(M, IL))), active(uLength(X1, X2)) -> uLength(active(X1), X2), active(uLength(tt(), L)) -> mark(s(length(L))), and(X1, mark(X2)) -> mark(and(X1, X2)), and(mark(X1), X2) -> mark(and(X1, X2)), and(ok(X1), ok(X2)) -> ok(and(X1, X2)), isNatList(ok(X)) -> ok(isNatList(X)), isNatIList(ok(X)) -> ok(isNatIList(X)), isNat(ok(X)) -> ok(isNat(X)), s(mark(X)) -> mark(s(X)), s(ok(X)) -> ok(s(X)), length(mark(X)) -> mark(length(X)), length(ok(X)) -> ok(length(X)), cons(mark(X1), X2) -> mark(cons(X1, X2)), cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)), 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)), uTake1(mark(X)) -> mark(uTake1(X)), uTake1(ok(X)) -> ok(uTake1(X)), uTake2(mark(X1), X2, X3, X4) -> mark(uTake2(X1, X2, X3, X4)), uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(uTake2(X1, X2, X3, X4)), uLength(mark(X1), X2) -> mark(uLength(X1, X2)), uLength(ok(X1), ok(X2)) -> ok(uLength(X1, X2)), proper(and(X1, X2)) -> and(proper(X1), proper(X2)), proper(tt()) -> ok(tt()), proper(isNatList(X)) -> isNatList(proper(X)), proper(isNatIList(X)) -> isNatIList(proper(X)), proper(isNat(X)) -> isNat(proper(X)), proper(0()) -> ok(0()), proper(s(X)) -> s(proper(X)), proper(length(X)) -> length(proper(X)), proper(zeros()) -> ok(zeros()), proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)), proper(nil()) -> ok(nil()), proper(take(X1, X2)) -> take(proper(X1), proper(X2)), proper(uTake1(X)) -> uTake1(proper(X)), proper(uTake2(X1, X2, X3, X4)) -> uTake2(proper(X1), proper(X2), proper(X3), proper(X4)), proper(uLength(X1, X2)) -> uLength(proper(X1), proper(X2)), top(mark(X)) -> top(proper(X)), top(ok(X)) -> top(active(X)) } DUP: We consider a duplicating system. Trs: { active(and(X1, X2)) -> and(X1, active(X2)), active(and(X1, X2)) -> and(active(X1), X2), active(and(tt(), T)) -> mark(T), active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))), active(isNatList(nil())) -> mark(tt()), active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))), active(isNatIList(IL)) -> mark(isNatList(IL)), active(isNatIList(zeros())) -> mark(tt()), active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))), active(isNat(0())) -> mark(tt()), active(isNat(s(N))) -> mark(isNat(N)), active(isNat(length(L))) -> mark(isNatList(L)), active(s(X)) -> s(active(X)), active(length(X)) -> length(active(X)), active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)), active(zeros()) -> mark(cons(0(), zeros())), active(cons(X1, X2)) -> cons(active(X1), X2), active(take(X1, X2)) -> take(X1, active(X2)), active(take(X1, X2)) -> take(active(X1), X2), active(take(0(), IL)) -> mark(uTake1(isNatIList(IL))), active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)), active(uTake1(X)) -> uTake1(active(X)), active(uTake1(tt())) -> mark(nil()), active(uTake2(X1, X2, X3, X4)) -> uTake2(active(X1), X2, X3, X4), active(uTake2(tt(), M, N, IL)) -> mark(cons(N, take(M, IL))), active(uLength(X1, X2)) -> uLength(active(X1), X2), active(uLength(tt(), L)) -> mark(s(length(L))), and(X1, mark(X2)) -> mark(and(X1, X2)), and(mark(X1), X2) -> mark(and(X1, X2)), and(ok(X1), ok(X2)) -> ok(and(X1, X2)), isNatList(ok(X)) -> ok(isNatList(X)), isNatIList(ok(X)) -> ok(isNatIList(X)), isNat(ok(X)) -> ok(isNat(X)), s(mark(X)) -> mark(s(X)), s(ok(X)) -> ok(s(X)), length(mark(X)) -> mark(length(X)), length(ok(X)) -> ok(length(X)), cons(mark(X1), X2) -> mark(cons(X1, X2)), cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)), 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)), uTake1(mark(X)) -> mark(uTake1(X)), uTake1(ok(X)) -> ok(uTake1(X)), uTake2(mark(X1), X2, X3, X4) -> mark(uTake2(X1, X2, X3, X4)), uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(uTake2(X1, X2, X3, X4)), uLength(mark(X1), X2) -> mark(uLength(X1, X2)), uLength(ok(X1), ok(X2)) -> ok(uLength(X1, X2)), proper(and(X1, X2)) -> and(proper(X1), proper(X2)), proper(tt()) -> ok(tt()), proper(isNatList(X)) -> isNatList(proper(X)), proper(isNatIList(X)) -> isNatIList(proper(X)), proper(isNat(X)) -> isNat(proper(X)), proper(0()) -> ok(0()), proper(s(X)) -> s(proper(X)), proper(length(X)) -> length(proper(X)), proper(zeros()) -> ok(zeros()), proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)), proper(nil()) -> ok(nil()), proper(take(X1, X2)) -> take(proper(X1), proper(X2)), proper(uTake1(X)) -> uTake1(proper(X)), proper(uTake2(X1, X2, X3, X4)) -> uTake2(proper(X1), proper(X2), proper(X3), proper(X4)), proper(uLength(X1, X2)) -> uLength(proper(X1), proper(X2)), top(mark(X)) -> top(proper(X)), top(ok(X)) -> top(active(X)) } Fail