MAYBE MAYBE TRS: { U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)), U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)), isNatKind(ok(X)) -> ok(isNatKind(X)), active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3), active(U12(tt(), V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)), active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))), active(isNatKind(plus(V1, V2))) -> mark(U31(isNatKind(V1), V2)), active(isNatKind(0())) -> mark(tt()), active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3), active(U11(tt(), V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)), active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3), active(U13(tt(), V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)), active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3), active(U14(tt(), V1, V2)) -> mark(U15(isNat(V1), V2)), active(U15(X1, X2)) -> U15(active(X1), X2), active(U15(tt(), V2)) -> mark(U16(isNat(V2))), active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)), active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)), active(isNat(0())) -> mark(tt()), active(U16(X)) -> U16(active(X)), active(U16(tt())) -> mark(tt()), active(U22(X1, X2)) -> U22(active(X1), X2), active(U22(tt(), V1)) -> mark(U23(isNat(V1))), active(U21(X1, X2)) -> U21(active(X1), X2), active(U21(tt(), V1)) -> mark(U22(isNatKind(V1), V1)), active(U23(X)) -> U23(active(X)), active(U23(tt())) -> mark(tt()), active(U32(X)) -> U32(active(X)), active(U32(tt())) -> mark(tt()), active(U31(X1, X2)) -> U31(active(X1), X2), active(U31(tt(), V2)) -> mark(U32(isNatKind(V2))), active(U41(X)) -> U41(active(X)), active(U41(tt())) -> mark(tt()), active(U52(X1, X2)) -> U52(active(X1), X2), active(U52(tt(), N)) -> mark(N), active(U51(X1, X2)) -> U51(active(X1), X2), active(U51(tt(), N)) -> mark(U52(isNatKind(N), N)), active(U62(X1, X2, X3)) -> U62(active(X1), X2, X3), active(U62(tt(), M, N)) -> mark(U63(isNat(N), M, N)), active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3), active(U61(tt(), M, N)) -> mark(U62(isNatKind(M), M, N)), active(U63(X1, X2, X3)) -> U63(active(X1), X2, X3), active(U63(tt(), M, N)) -> mark(U64(isNatKind(N), M, N)), active(U64(X1, X2, X3)) -> U64(active(X1), X2, X3), active(U64(tt(), M, N)) -> mark(s(plus(N, M))), active(s(X)) -> s(active(X)), active(plus(N, s(M))) -> mark(U61(isNat(M), M, N)), active(plus(N, 0())) -> mark(U51(isNat(N), N)), active(plus(X1, X2)) -> plus(X1, active(X2)), active(plus(X1, X2)) -> plus(active(X1), X2), U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)), U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)), U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)), U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)), U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)), U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)), U15(mark(X1), X2) -> mark(U15(X1, X2)), U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)), isNat(ok(X)) -> ok(isNat(X)), U16(mark(X)) -> mark(U16(X)), U16(ok(X)) -> ok(U16(X)), U22(mark(X1), X2) -> mark(U22(X1, X2)), U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)), U21(mark(X1), X2) -> mark(U21(X1, X2)), U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)), U23(mark(X)) -> mark(U23(X)), U23(ok(X)) -> ok(U23(X)), 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)), U41(mark(X)) -> mark(U41(X)), U41(ok(X)) -> ok(U41(X)), U52(mark(X1), X2) -> mark(U52(X1, X2)), U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)), U51(mark(X1), X2) -> mark(U51(X1, X2)), U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)), U62(mark(X1), X2, X3) -> mark(U62(X1, X2, X3)), U62(ok(X1), ok(X2), ok(X3)) -> ok(U62(X1, X2, X3)), U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3)), U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3)), U63(mark(X1), X2, X3) -> mark(U63(X1, X2, X3)), U63(ok(X1), ok(X2), ok(X3)) -> ok(U63(X1, X2, X3)), U64(mark(X1), X2, X3) -> mark(U64(X1, X2, X3)), U64(ok(X1), ok(X2), ok(X3)) -> ok(U64(X1, X2, X3)), s(mark(X)) -> mark(s(X)), s(ok(X)) -> ok(s(X)), plus(X1, mark(X2)) -> mark(plus(X1, X2)), plus(mark(X1), X2) -> mark(plus(X1, X2)), plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)), proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)), proper(isNatKind(X)) -> isNatKind(proper(X)), proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)), proper(tt()) -> ok(tt()), proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)), proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)), proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)), proper(isNat(X)) -> isNat(proper(X)), proper(U16(X)) -> U16(proper(X)), proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)), proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)), proper(U23(X)) -> U23(proper(X)), proper(U32(X)) -> U32(proper(X)), proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)), proper(U41(X)) -> U41(proper(X)), proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)), proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)), proper(U62(X1, X2, X3)) -> U62(proper(X1), proper(X2), proper(X3)), proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3)), proper(U63(X1, X2, X3)) -> U63(proper(X1), proper(X2), proper(X3)), proper(U64(X1, X2, X3)) -> U64(proper(X1), proper(X2), proper(X3)), proper(s(X)) -> s(proper(X)), proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)), proper(0()) -> ok(0()), top(mark(X)) -> top(proper(X)), top(ok(X)) -> top(active(X)) } DUP: We consider a duplicating system. Trs: { U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)), U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)), isNatKind(ok(X)) -> ok(isNatKind(X)), active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3), active(U12(tt(), V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)), active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))), active(isNatKind(plus(V1, V2))) -> mark(U31(isNatKind(V1), V2)), active(isNatKind(0())) -> mark(tt()), active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3), active(U11(tt(), V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)), active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3), active(U13(tt(), V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)), active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3), active(U14(tt(), V1, V2)) -> mark(U15(isNat(V1), V2)), active(U15(X1, X2)) -> U15(active(X1), X2), active(U15(tt(), V2)) -> mark(U16(isNat(V2))), active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)), active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)), active(isNat(0())) -> mark(tt()), active(U16(X)) -> U16(active(X)), active(U16(tt())) -> mark(tt()), active(U22(X1, X2)) -> U22(active(X1), X2), active(U22(tt(), V1)) -> mark(U23(isNat(V1))), active(U21(X1, X2)) -> U21(active(X1), X2), active(U21(tt(), V1)) -> mark(U22(isNatKind(V1), V1)), active(U23(X)) -> U23(active(X)), active(U23(tt())) -> mark(tt()), active(U32(X)) -> U32(active(X)), active(U32(tt())) -> mark(tt()), active(U31(X1, X2)) -> U31(active(X1), X2), active(U31(tt(), V2)) -> mark(U32(isNatKind(V2))), active(U41(X)) -> U41(active(X)), active(U41(tt())) -> mark(tt()), active(U52(X1, X2)) -> U52(active(X1), X2), active(U52(tt(), N)) -> mark(N), active(U51(X1, X2)) -> U51(active(X1), X2), active(U51(tt(), N)) -> mark(U52(isNatKind(N), N)), active(U62(X1, X2, X3)) -> U62(active(X1), X2, X3), active(U62(tt(), M, N)) -> mark(U63(isNat(N), M, N)), active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3), active(U61(tt(), M, N)) -> mark(U62(isNatKind(M), M, N)), active(U63(X1, X2, X3)) -> U63(active(X1), X2, X3), active(U63(tt(), M, N)) -> mark(U64(isNatKind(N), M, N)), active(U64(X1, X2, X3)) -> U64(active(X1), X2, X3), active(U64(tt(), M, N)) -> mark(s(plus(N, M))), active(s(X)) -> s(active(X)), active(plus(N, s(M))) -> mark(U61(isNat(M), M, N)), active(plus(N, 0())) -> mark(U51(isNat(N), N)), active(plus(X1, X2)) -> plus(X1, active(X2)), active(plus(X1, X2)) -> plus(active(X1), X2), U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)), U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)), U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)), U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)), U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)), U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)), U15(mark(X1), X2) -> mark(U15(X1, X2)), U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)), isNat(ok(X)) -> ok(isNat(X)), U16(mark(X)) -> mark(U16(X)), U16(ok(X)) -> ok(U16(X)), U22(mark(X1), X2) -> mark(U22(X1, X2)), U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)), U21(mark(X1), X2) -> mark(U21(X1, X2)), U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)), U23(mark(X)) -> mark(U23(X)), U23(ok(X)) -> ok(U23(X)), 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)), U41(mark(X)) -> mark(U41(X)), U41(ok(X)) -> ok(U41(X)), U52(mark(X1), X2) -> mark(U52(X1, X2)), U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)), U51(mark(X1), X2) -> mark(U51(X1, X2)), U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)), U62(mark(X1), X2, X3) -> mark(U62(X1, X2, X3)), U62(ok(X1), ok(X2), ok(X3)) -> ok(U62(X1, X2, X3)), U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3)), U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3)), U63(mark(X1), X2, X3) -> mark(U63(X1, X2, X3)), U63(ok(X1), ok(X2), ok(X3)) -> ok(U63(X1, X2, X3)), U64(mark(X1), X2, X3) -> mark(U64(X1, X2, X3)), U64(ok(X1), ok(X2), ok(X3)) -> ok(U64(X1, X2, X3)), s(mark(X)) -> mark(s(X)), s(ok(X)) -> ok(s(X)), plus(X1, mark(X2)) -> mark(plus(X1, X2)), plus(mark(X1), X2) -> mark(plus(X1, X2)), plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)), proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)), proper(isNatKind(X)) -> isNatKind(proper(X)), proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)), proper(tt()) -> ok(tt()), proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)), proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)), proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)), proper(isNat(X)) -> isNat(proper(X)), proper(U16(X)) -> U16(proper(X)), proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)), proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)), proper(U23(X)) -> U23(proper(X)), proper(U32(X)) -> U32(proper(X)), proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)), proper(U41(X)) -> U41(proper(X)), proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)), proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)), proper(U62(X1, X2, X3)) -> U62(proper(X1), proper(X2), proper(X3)), proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3)), proper(U63(X1, X2, X3)) -> U63(proper(X1), proper(X2), proper(X3)), proper(U64(X1, X2, X3)) -> U64(proper(X1), proper(X2), proper(X3)), proper(s(X)) -> s(proper(X)), proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)), proper(0()) -> ok(0()), top(mark(X)) -> top(proper(X)), top(ok(X)) -> top(active(X)) } Fail