MAYBE MAYBE TRS: { cons(mark(X1), X2) -> mark(cons(X1, X2)), cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)), active(cons(X1, X2)) -> cons(active(X1), X2), active(zeros()) -> mark(cons(0(), zeros())), active(s(X)) -> s(active(X)), active(length(X)) -> length(active(X)), active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L)), active(length(nil())) -> mark(0()), active(U11(X1, X2)) -> U11(active(X1), X2), active(U11(tt(), L)) -> mark(s(length(L))), active(U21(X)) -> U21(active(X)), active(U21(tt())) -> mark(nil()), active(take(X1, X2)) -> take(X1, active(X2)), active(take(X1, X2)) -> take(active(X1), X2), active(take(0(), IL)) -> mark(U21(isNatIList(IL))), active(take(s(M), cons(N, IL))) -> mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)), active(U31(X1, X2, X3, X4)) -> U31(active(X1), X2, X3, X4), active(U31(tt(), IL, M, N)) -> mark(cons(N, take(M, IL))), active(and(X1, X2)) -> and(active(X1), X2), active(and(tt(), X)) -> mark(X), active(isNat(0())) -> mark(tt()), active(isNat(s(V1))) -> mark(isNat(V1)), active(isNat(length(V1))) -> mark(isNatList(V1)), active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2))), active(isNatList(nil())) -> mark(tt()), active(isNatList(take(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2))), active(isNatIList(V)) -> mark(isNatList(V)), active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2))), active(isNatIList(zeros())) -> mark(tt()), s(mark(X)) -> mark(s(X)), s(ok(X)) -> ok(s(X)), length(mark(X)) -> mark(length(X)), length(ok(X)) -> ok(length(X)), U11(mark(X1), X2) -> mark(U11(X1, X2)), U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)), U21(mark(X)) -> mark(U21(X)), U21(ok(X)) -> ok(U21(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)), U31(mark(X1), X2, X3, X4) -> mark(U31(X1, X2, X3, X4)), U31(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U31(X1, X2, X3, X4)), and(mark(X1), X2) -> mark(and(X1, X2)), and(ok(X1), ok(X2)) -> ok(and(X1, X2)), isNat(ok(X)) -> ok(isNat(X)), isNatList(ok(X)) -> ok(isNatList(X)), isNatIList(ok(X)) -> ok(isNatIList(X)), proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)), proper(0()) -> ok(0()), proper(zeros()) -> ok(zeros()), proper(s(X)) -> s(proper(X)), proper(length(X)) -> length(proper(X)), proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)), proper(tt()) -> ok(tt()), proper(nil()) -> ok(nil()), proper(U21(X)) -> U21(proper(X)), proper(take(X1, X2)) -> take(proper(X1), proper(X2)), proper(U31(X1, X2, X3, X4)) -> U31(proper(X1), proper(X2), proper(X3), proper(X4)), proper(and(X1, X2)) -> and(proper(X1), proper(X2)), proper(isNat(X)) -> isNat(proper(X)), proper(isNatList(X)) -> isNatList(proper(X)), proper(isNatIList(X)) -> isNatIList(proper(X)), top(mark(X)) -> top(proper(X)), top(ok(X)) -> top(active(X)) } DUP: We consider a duplicating system. Trs: { cons(mark(X1), X2) -> mark(cons(X1, X2)), cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)), active(cons(X1, X2)) -> cons(active(X1), X2), active(zeros()) -> mark(cons(0(), zeros())), active(s(X)) -> s(active(X)), active(length(X)) -> length(active(X)), active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L)), active(length(nil())) -> mark(0()), active(U11(X1, X2)) -> U11(active(X1), X2), active(U11(tt(), L)) -> mark(s(length(L))), active(U21(X)) -> U21(active(X)), active(U21(tt())) -> mark(nil()), active(take(X1, X2)) -> take(X1, active(X2)), active(take(X1, X2)) -> take(active(X1), X2), active(take(0(), IL)) -> mark(U21(isNatIList(IL))), active(take(s(M), cons(N, IL))) -> mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)), active(U31(X1, X2, X3, X4)) -> U31(active(X1), X2, X3, X4), active(U31(tt(), IL, M, N)) -> mark(cons(N, take(M, IL))), active(and(X1, X2)) -> and(active(X1), X2), active(and(tt(), X)) -> mark(X), active(isNat(0())) -> mark(tt()), active(isNat(s(V1))) -> mark(isNat(V1)), active(isNat(length(V1))) -> mark(isNatList(V1)), active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2))), active(isNatList(nil())) -> mark(tt()), active(isNatList(take(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2))), active(isNatIList(V)) -> mark(isNatList(V)), active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2))), active(isNatIList(zeros())) -> mark(tt()), s(mark(X)) -> mark(s(X)), s(ok(X)) -> ok(s(X)), length(mark(X)) -> mark(length(X)), length(ok(X)) -> ok(length(X)), U11(mark(X1), X2) -> mark(U11(X1, X2)), U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)), U21(mark(X)) -> mark(U21(X)), U21(ok(X)) -> ok(U21(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)), U31(mark(X1), X2, X3, X4) -> mark(U31(X1, X2, X3, X4)), U31(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U31(X1, X2, X3, X4)), and(mark(X1), X2) -> mark(and(X1, X2)), and(ok(X1), ok(X2)) -> ok(and(X1, X2)), isNat(ok(X)) -> ok(isNat(X)), isNatList(ok(X)) -> ok(isNatList(X)), isNatIList(ok(X)) -> ok(isNatIList(X)), proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)), proper(0()) -> ok(0()), proper(zeros()) -> ok(zeros()), proper(s(X)) -> s(proper(X)), proper(length(X)) -> length(proper(X)), proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)), proper(tt()) -> ok(tt()), proper(nil()) -> ok(nil()), proper(U21(X)) -> U21(proper(X)), proper(take(X1, X2)) -> take(proper(X1), proper(X2)), proper(U31(X1, X2, X3, X4)) -> U31(proper(X1), proper(X2), proper(X3), proper(X4)), proper(and(X1, X2)) -> and(proper(X1), proper(X2)), proper(isNat(X)) -> isNat(proper(X)), proper(isNatList(X)) -> isNatList(proper(X)), proper(isNatIList(X)) -> isNatIList(proper(X)), top(mark(X)) -> top(proper(X)), top(ok(X)) -> top(active(X)) } Fail