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(U12(X)) -> U12(active(X)), active(U12(tt())) -> mark(tt()), active(isNatList(cons(V1, V2))) -> mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)), active(isNatList(nil())) -> mark(tt()), active(isNatList(take(V1, V2))) -> mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)), active(U11(X1, X2)) -> U11(active(X1), X2), active(U11(tt(), V1)) -> mark(U12(isNatList(V1))), active(U22(X)) -> U22(active(X)), active(U22(tt())) -> mark(tt()), active(isNat(0())) -> mark(tt()), active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)), active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1)), active(U21(X1, X2)) -> U21(active(X1), X2), active(U21(tt(), V1)) -> mark(U22(isNat(V1))), active(U32(X)) -> U32(active(X)), active(U32(tt())) -> mark(tt()), active(U31(X1, X2)) -> U31(active(X1), X2), active(U31(tt(), V)) -> mark(U32(isNatList(V))), active(U42(X1, X2)) -> U42(active(X1), X2), active(U42(tt(), V2)) -> mark(U43(isNatIList(V2))), active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3), active(U41(tt(), V1, V2)) -> mark(U42(isNat(V1), V2)), active(U43(X)) -> U43(active(X)), active(U43(tt())) -> mark(tt()), active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V)), active(isNatIList(cons(V1, V2))) -> mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)), active(isNatIList(zeros())) -> mark(tt()), active(U52(X1, X2)) -> U52(active(X1), X2), active(U52(tt(), V2)) -> mark(U53(isNatList(V2))), active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3), active(U51(tt(), V1, V2)) -> mark(U52(isNat(V1), V2)), active(U53(X)) -> U53(active(X)), active(U53(tt())) -> mark(tt()), active(U62(X1, X2)) -> U62(active(X1), X2), active(U62(tt(), V2)) -> mark(U63(isNatIList(V2))), active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3), active(U61(tt(), V1, V2)) -> mark(U62(isNat(V1), V2)), active(U63(X)) -> U63(active(X)), active(U63(tt())) -> mark(tt()), active(s(X)) -> s(active(X)), active(length(X)) -> length(active(X)), active(length(cons(N, L))) -> mark( U71( and( and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N)) ), L ) ), active(length(nil())) -> mark(0()), active(U71(X1, X2)) -> U71(active(X1), X2), active(U71(tt(), L)) -> mark(s(length(L))), active(U81(X)) -> U81(active(X)), active(U81(tt())) -> mark(nil()), active(take(X1, X2)) -> take(X1, active(X2)), active(take(X1, X2)) -> take(active(X1), X2), active(take(0(), IL)) -> mark(U81(and(isNatIList(IL), isNatIListKind(IL)))), active(take(s(M), cons(N, IL))) -> mark( U91( and( and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))) ), IL, M, N ) ), active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4), active(U91(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(isNatIListKind(cons(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2))), active(isNatIListKind(zeros())) -> mark(tt()), active(isNatIListKind(nil())) -> mark(tt()), active(isNatIListKind(take(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2))), active(isNatKind(0())) -> mark(tt()), active(isNatKind(s(V1))) -> mark(isNatKind(V1)), active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)), U12(mark(X)) -> mark(U12(X)), U12(ok(X)) -> ok(U12(X)), isNatList(ok(X)) -> ok(isNatList(X)), U11(mark(X1), X2) -> mark(U11(X1, X2)), U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)), U22(mark(X)) -> mark(U22(X)), U22(ok(X)) -> ok(U22(X)), isNat(ok(X)) -> ok(isNat(X)), U21(mark(X1), X2) -> mark(U21(X1, X2)), U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)), U32(mark(X)) -> mark(U32(X)), U32(ok(X)) -> ok(U32(X)), 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)), isNatIList(ok(X)) -> ok(isNatIList(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(X1), X2) -> mark(U62(X1, X2)), U62(ok(X1), ok(X2)) -> ok(U62(X1, X2)), U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3)), U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3)), U63(mark(X)) -> mark(U63(X)), U63(ok(X)) -> ok(U63(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)), U71(mark(X1), X2) -> mark(U71(X1, X2)), U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)), U81(mark(X)) -> mark(U81(X)), U81(ok(X)) -> ok(U81(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)), U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4)), U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4)), and(mark(X1), X2) -> mark(and(X1, X2)), and(ok(X1), ok(X2)) -> ok(and(X1, X2)), isNatIListKind(ok(X)) -> ok(isNatIListKind(X)), isNatKind(ok(X)) -> ok(isNatKind(X)), proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)), proper(0()) -> ok(0()), proper(zeros()) -> ok(zeros()), proper(U12(X)) -> U12(proper(X)), proper(isNatList(X)) -> isNatList(proper(X)), proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)), proper(tt()) -> ok(tt()), proper(U22(X)) -> U22(proper(X)), proper(isNat(X)) -> isNat(proper(X)), proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)), proper(U32(X)) -> U32(proper(X)), 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(isNatIList(X)) -> isNatIList(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(X1, X2)) -> U62(proper(X1), proper(X2)), proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3)), proper(U63(X)) -> U63(proper(X)), proper(s(X)) -> s(proper(X)), proper(length(X)) -> length(proper(X)), proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)), proper(nil()) -> ok(nil()), proper(U81(X)) -> U81(proper(X)), proper(take(X1, X2)) -> take(proper(X1), proper(X2)), proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4)), proper(and(X1, X2)) -> and(proper(X1), proper(X2)), proper(isNatIListKind(X)) -> isNatIListKind(proper(X)), proper(isNatKind(X)) -> isNatKind(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(U12(X)) -> U12(active(X)), active(U12(tt())) -> mark(tt()), active(isNatList(cons(V1, V2))) -> mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)), active(isNatList(nil())) -> mark(tt()), active(isNatList(take(V1, V2))) -> mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)), active(U11(X1, X2)) -> U11(active(X1), X2), active(U11(tt(), V1)) -> mark(U12(isNatList(V1))), active(U22(X)) -> U22(active(X)), active(U22(tt())) -> mark(tt()), active(isNat(0())) -> mark(tt()), active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)), active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1)), active(U21(X1, X2)) -> U21(active(X1), X2), active(U21(tt(), V1)) -> mark(U22(isNat(V1))), active(U32(X)) -> U32(active(X)), active(U32(tt())) -> mark(tt()), active(U31(X1, X2)) -> U31(active(X1), X2), active(U31(tt(), V)) -> mark(U32(isNatList(V))), active(U42(X1, X2)) -> U42(active(X1), X2), active(U42(tt(), V2)) -> mark(U43(isNatIList(V2))), active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3), active(U41(tt(), V1, V2)) -> mark(U42(isNat(V1), V2)), active(U43(X)) -> U43(active(X)), active(U43(tt())) -> mark(tt()), active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V)), active(isNatIList(cons(V1, V2))) -> mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)), active(isNatIList(zeros())) -> mark(tt()), active(U52(X1, X2)) -> U52(active(X1), X2), active(U52(tt(), V2)) -> mark(U53(isNatList(V2))), active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3), active(U51(tt(), V1, V2)) -> mark(U52(isNat(V1), V2)), active(U53(X)) -> U53(active(X)), active(U53(tt())) -> mark(tt()), active(U62(X1, X2)) -> U62(active(X1), X2), active(U62(tt(), V2)) -> mark(U63(isNatIList(V2))), active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3), active(U61(tt(), V1, V2)) -> mark(U62(isNat(V1), V2)), active(U63(X)) -> U63(active(X)), active(U63(tt())) -> mark(tt()), active(s(X)) -> s(active(X)), active(length(X)) -> length(active(X)), active(length(cons(N, L))) -> mark( U71( and( and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N)) ), L ) ), active(length(nil())) -> mark(0()), active(U71(X1, X2)) -> U71(active(X1), X2), active(U71(tt(), L)) -> mark(s(length(L))), active(U81(X)) -> U81(active(X)), active(U81(tt())) -> mark(nil()), active(take(X1, X2)) -> take(X1, active(X2)), active(take(X1, X2)) -> take(active(X1), X2), active(take(0(), IL)) -> mark(U81(and(isNatIList(IL), isNatIListKind(IL)))), active(take(s(M), cons(N, IL))) -> mark( U91( and( and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))) ), IL, M, N ) ), active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4), active(U91(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(isNatIListKind(cons(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2))), active(isNatIListKind(zeros())) -> mark(tt()), active(isNatIListKind(nil())) -> mark(tt()), active(isNatIListKind(take(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2))), active(isNatKind(0())) -> mark(tt()), active(isNatKind(s(V1))) -> mark(isNatKind(V1)), active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)), U12(mark(X)) -> mark(U12(X)), U12(ok(X)) -> ok(U12(X)), isNatList(ok(X)) -> ok(isNatList(X)), U11(mark(X1), X2) -> mark(U11(X1, X2)), U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)), U22(mark(X)) -> mark(U22(X)), U22(ok(X)) -> ok(U22(X)), isNat(ok(X)) -> ok(isNat(X)), U21(mark(X1), X2) -> mark(U21(X1, X2)), U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)), U32(mark(X)) -> mark(U32(X)), U32(ok(X)) -> ok(U32(X)), 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)), isNatIList(ok(X)) -> ok(isNatIList(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(X1), X2) -> mark(U62(X1, X2)), U62(ok(X1), ok(X2)) -> ok(U62(X1, X2)), U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3)), U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3)), U63(mark(X)) -> mark(U63(X)), U63(ok(X)) -> ok(U63(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)), U71(mark(X1), X2) -> mark(U71(X1, X2)), U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)), U81(mark(X)) -> mark(U81(X)), U81(ok(X)) -> ok(U81(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)), U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4)), U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4)), and(mark(X1), X2) -> mark(and(X1, X2)), and(ok(X1), ok(X2)) -> ok(and(X1, X2)), isNatIListKind(ok(X)) -> ok(isNatIListKind(X)), isNatKind(ok(X)) -> ok(isNatKind(X)), proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)), proper(0()) -> ok(0()), proper(zeros()) -> ok(zeros()), proper(U12(X)) -> U12(proper(X)), proper(isNatList(X)) -> isNatList(proper(X)), proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)), proper(tt()) -> ok(tt()), proper(U22(X)) -> U22(proper(X)), proper(isNat(X)) -> isNat(proper(X)), proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)), proper(U32(X)) -> U32(proper(X)), 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(isNatIList(X)) -> isNatIList(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(X1, X2)) -> U62(proper(X1), proper(X2)), proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3)), proper(U63(X)) -> U63(proper(X)), proper(s(X)) -> s(proper(X)), proper(length(X)) -> length(proper(X)), proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)), proper(nil()) -> ok(nil()), proper(U81(X)) -> U81(proper(X)), proper(take(X1, X2)) -> take(proper(X1), proper(X2)), proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4)), proper(and(X1, X2)) -> and(proper(X1), proper(X2)), proper(isNatIListKind(X)) -> isNatIListKind(proper(X)), proper(isNatKind(X)) -> isNatKind(proper(X)), top(mark(X)) -> top(proper(X)), top(ok(X)) -> top(active(X)) } Fail