MAYBE MAYBE TRS: { cons(X1, X2) -> n__cons(X1, X2), 0() -> n__0(), zeros() -> cons(0(), n__zeros()), zeros() -> n__zeros(), U12(tt()) -> tt(), isNatList(n__take(V1, V2)) -> U61( and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2) ), isNatList(n__cons(V1, V2)) -> U51( and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2) ), isNatList(n__nil()) -> tt(), activate(X) -> X, activate(n__zeros()) -> zeros(), activate(n__take(X1, X2)) -> take(X1, X2), activate(n__0()) -> 0(), activate(n__length(X)) -> length(X), activate(n__s(X)) -> s(X), activate(n__isNatIListKind(X)) -> isNatIListKind(X), activate(n__cons(X1, X2)) -> cons(X1, X2), activate(n__nil()) -> nil(), activate(n__and(X1, X2)) -> and(X1, X2), activate(n__isNatKind(X)) -> isNatKind(X), U11(tt(), V1) -> U12(isNatList(activate(V1))), U22(tt()) -> tt(), isNat(n__0()) -> tt(), isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)), isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)), U21(tt(), V1) -> U22(isNat(activate(V1))), U32(tt()) -> tt(), U31(tt(), V) -> U32(isNatList(activate(V))), U42(tt(), V2) -> U43(isNatIList(activate(V2))), U41(tt(), V1, V2) -> U42(isNat(activate(V1)), activate(V2)), U43(tt()) -> tt(), isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)), isNatIList(n__zeros()) -> tt(), isNatIList(n__cons(V1, V2)) -> U41( and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2) ), U52(tt(), V2) -> U53(isNatList(activate(V2))), U51(tt(), V1, V2) -> U52(isNat(activate(V1)), activate(V2)), U53(tt()) -> tt(), U62(tt(), V2) -> U63(isNatIList(activate(V2))), U61(tt(), V1, V2) -> U62(isNat(activate(V1)), activate(V2)), U63(tt()) -> tt(), s(X) -> n__s(X), length(X) -> n__length(X), length(cons(N, L)) -> U71( and( and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N)) ), activate(L) ), length(nil()) -> 0(), U71(tt(), L) -> s(length(activate(L))), nil() -> n__nil(), U81(tt()) -> nil(), U91(tt(), IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))), and(X1, X2) -> n__and(X1, X2), and(tt(), X) -> activate(X), isNatIListKind(X) -> n__isNatIListKind(X), isNatIListKind(n__zeros()) -> tt(), isNatIListKind(n__take(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), isNatIListKind(n__cons(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), isNatIListKind(n__nil()) -> tt(), isNatKind(X) -> n__isNatKind(X), isNatKind(n__0()) -> tt(), isNatKind(n__length(V1)) -> isNatIListKind(activate(V1)), isNatKind(n__s(V1)) -> isNatKind(activate(V1)), take(X1, X2) -> n__take(X1, X2), take(0(), IL) -> U81(and(isNatIList(IL), n__isNatIListKind(IL))), take(s(M), cons(N, IL)) -> U91( and( and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and( and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)) ) ), activate(IL), M, N ) } DUP: We consider a duplicating system. Trs: { cons(X1, X2) -> n__cons(X1, X2), 0() -> n__0(), zeros() -> cons(0(), n__zeros()), zeros() -> n__zeros(), U12(tt()) -> tt(), isNatList(n__take(V1, V2)) -> U61( and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2) ), isNatList(n__cons(V1, V2)) -> U51( and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2) ), isNatList(n__nil()) -> tt(), activate(X) -> X, activate(n__zeros()) -> zeros(), activate(n__take(X1, X2)) -> take(X1, X2), activate(n__0()) -> 0(), activate(n__length(X)) -> length(X), activate(n__s(X)) -> s(X), activate(n__isNatIListKind(X)) -> isNatIListKind(X), activate(n__cons(X1, X2)) -> cons(X1, X2), activate(n__nil()) -> nil(), activate(n__and(X1, X2)) -> and(X1, X2), activate(n__isNatKind(X)) -> isNatKind(X), U11(tt(), V1) -> U12(isNatList(activate(V1))), U22(tt()) -> tt(), isNat(n__0()) -> tt(), isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)), isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)), U21(tt(), V1) -> U22(isNat(activate(V1))), U32(tt()) -> tt(), U31(tt(), V) -> U32(isNatList(activate(V))), U42(tt(), V2) -> U43(isNatIList(activate(V2))), U41(tt(), V1, V2) -> U42(isNat(activate(V1)), activate(V2)), U43(tt()) -> tt(), isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)), isNatIList(n__zeros()) -> tt(), isNatIList(n__cons(V1, V2)) -> U41( and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2) ), U52(tt(), V2) -> U53(isNatList(activate(V2))), U51(tt(), V1, V2) -> U52(isNat(activate(V1)), activate(V2)), U53(tt()) -> tt(), U62(tt(), V2) -> U63(isNatIList(activate(V2))), U61(tt(), V1, V2) -> U62(isNat(activate(V1)), activate(V2)), U63(tt()) -> tt(), s(X) -> n__s(X), length(X) -> n__length(X), length(cons(N, L)) -> U71( and( and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N)) ), activate(L) ), length(nil()) -> 0(), U71(tt(), L) -> s(length(activate(L))), nil() -> n__nil(), U81(tt()) -> nil(), U91(tt(), IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))), and(X1, X2) -> n__and(X1, X2), and(tt(), X) -> activate(X), isNatIListKind(X) -> n__isNatIListKind(X), isNatIListKind(n__zeros()) -> tt(), isNatIListKind(n__take(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), isNatIListKind(n__cons(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), isNatIListKind(n__nil()) -> tt(), isNatKind(X) -> n__isNatKind(X), isNatKind(n__0()) -> tt(), isNatKind(n__length(V1)) -> isNatIListKind(activate(V1)), isNatKind(n__s(V1)) -> isNatKind(activate(V1)), take(X1, X2) -> n__take(X1, X2), take(0(), IL) -> U81(and(isNatIList(IL), n__isNatIListKind(IL))), take(s(M), cons(N, IL)) -> U91( and( and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and( and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)) ) ), activate(IL), M, N ) } Fail