MAYBE MAYBE TRS: { U12(mark(X1), X2) -> mark(U12(X1, X2)), U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)), isNat(ok(X)) -> ok(isNat(X)), active(U12(X1, X2)) -> U12(active(X1), X2), active(U12(tt(), V2)) -> mark(U13(isNat(V2))), active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)), active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)), active(isNat(0())) -> mark(tt()), active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)), active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3), active(U11(tt(), V1, V2)) -> mark(U12(isNat(V1), V2)), active(U13(X)) -> U13(active(X)), active(U13(tt())) -> mark(tt()), active(U22(X)) -> U22(active(X)), active(U22(tt())) -> mark(tt()), active(U21(X1, X2)) -> U21(active(X1), X2), active(U21(tt(), V1)) -> mark(U22(isNat(V1))), active(U32(X1, X2)) -> U32(active(X1), X2), active(U32(tt(), V2)) -> mark(U33(isNat(V2))), active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3), active(U31(tt(), V1, V2)) -> mark(U32(isNat(V1), V2)), active(U33(X)) -> U33(active(X)), active(U33(tt())) -> mark(tt()), active(U41(X1, X2)) -> U41(active(X1), X2), active(U41(tt(), N)) -> mark(N), active(s(X)) -> s(active(X)), active(plus(N, s(M))) -> mark( U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N ) ), active(plus(N, 0())) -> mark(U41(and(isNat(N), isNatKind(N)), N)), active(plus(X1, X2)) -> plus(X1, active(X2)), active(plus(X1, X2)) -> plus(active(X1), X2), active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3), active(U51(tt(), M, N)) -> mark(s(plus(N, M))), active(U61(X)) -> U61(active(X)), active(U61(tt())) -> mark(0()), active(x(N, s(M))) -> mark( U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N ) ), active(x(N, 0())) -> mark(U61(and(isNat(N), isNatKind(N)))), active(x(X1, X2)) -> x(X1, active(X2)), active(x(X1, X2)) -> x(active(X1), X2), active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3), active(U71(tt(), M, N)) -> mark(plus(x(N, M), N)), active(and(X1, X2)) -> and(active(X1), X2), active(and(tt(), X)) -> mark(X), active(isNatKind(s(V1))) -> mark(isNatKind(V1)), active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))), active(isNatKind(0())) -> mark(tt()), active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))), U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)), U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)), U13(mark(X)) -> mark(U13(X)), U13(ok(X)) -> ok(U13(X)), U22(mark(X)) -> mark(U22(X)), U22(ok(X)) -> ok(U22(X)), U21(mark(X1), X2) -> mark(U21(X1, X2)), U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)), U32(mark(X1), X2) -> mark(U32(X1, X2)), U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)), U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)), U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)), U33(mark(X)) -> mark(U33(X)), U33(ok(X)) -> ok(U33(X)), U41(mark(X1), X2) -> mark(U41(X1, X2)), U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)), 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)), U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)), U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)), U61(mark(X)) -> mark(U61(X)), U61(ok(X)) -> ok(U61(X)), x(X1, mark(X2)) -> mark(x(X1, X2)), x(mark(X1), X2) -> mark(x(X1, X2)), x(ok(X1), ok(X2)) -> ok(x(X1, X2)), U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3)), U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3)), and(mark(X1), X2) -> mark(and(X1, X2)), and(ok(X1), ok(X2)) -> ok(and(X1, X2)), isNatKind(ok(X)) -> ok(isNatKind(X)), proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)), proper(isNat(X)) -> isNat(proper(X)), proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)), proper(tt()) -> ok(tt()), proper(U13(X)) -> U13(proper(X)), proper(U22(X)) -> U22(proper(X)), proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)), proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)), proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)), proper(U33(X)) -> U33(proper(X)), proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)), proper(s(X)) -> s(proper(X)), proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)), proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)), proper(0()) -> ok(0()), proper(U61(X)) -> U61(proper(X)), proper(x(X1, X2)) -> x(proper(X1), proper(X2)), proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3)), proper(and(X1, X2)) -> and(proper(X1), proper(X2)), 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: { U12(mark(X1), X2) -> mark(U12(X1, X2)), U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)), isNat(ok(X)) -> ok(isNat(X)), active(U12(X1, X2)) -> U12(active(X1), X2), active(U12(tt(), V2)) -> mark(U13(isNat(V2))), active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)), active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)), active(isNat(0())) -> mark(tt()), active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)), active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3), active(U11(tt(), V1, V2)) -> mark(U12(isNat(V1), V2)), active(U13(X)) -> U13(active(X)), active(U13(tt())) -> mark(tt()), active(U22(X)) -> U22(active(X)), active(U22(tt())) -> mark(tt()), active(U21(X1, X2)) -> U21(active(X1), X2), active(U21(tt(), V1)) -> mark(U22(isNat(V1))), active(U32(X1, X2)) -> U32(active(X1), X2), active(U32(tt(), V2)) -> mark(U33(isNat(V2))), active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3), active(U31(tt(), V1, V2)) -> mark(U32(isNat(V1), V2)), active(U33(X)) -> U33(active(X)), active(U33(tt())) -> mark(tt()), active(U41(X1, X2)) -> U41(active(X1), X2), active(U41(tt(), N)) -> mark(N), active(s(X)) -> s(active(X)), active(plus(N, s(M))) -> mark( U51( and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N ) ), active(plus(N, 0())) -> mark(U41(and(isNat(N), isNatKind(N)), N)), active(plus(X1, X2)) -> plus(X1, active(X2)), active(plus(X1, X2)) -> plus(active(X1), X2), active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3), active(U51(tt(), M, N)) -> mark(s(plus(N, M))), active(U61(X)) -> U61(active(X)), active(U61(tt())) -> mark(0()), active(x(N, s(M))) -> mark( U71( and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N ) ), active(x(N, 0())) -> mark(U61(and(isNat(N), isNatKind(N)))), active(x(X1, X2)) -> x(X1, active(X2)), active(x(X1, X2)) -> x(active(X1), X2), active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3), active(U71(tt(), M, N)) -> mark(plus(x(N, M), N)), active(and(X1, X2)) -> and(active(X1), X2), active(and(tt(), X)) -> mark(X), active(isNatKind(s(V1))) -> mark(isNatKind(V1)), active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))), active(isNatKind(0())) -> mark(tt()), active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))), U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)), U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)), U13(mark(X)) -> mark(U13(X)), U13(ok(X)) -> ok(U13(X)), U22(mark(X)) -> mark(U22(X)), U22(ok(X)) -> ok(U22(X)), U21(mark(X1), X2) -> mark(U21(X1, X2)), U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)), U32(mark(X1), X2) -> mark(U32(X1, X2)), U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)), U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)), U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)), U33(mark(X)) -> mark(U33(X)), U33(ok(X)) -> ok(U33(X)), U41(mark(X1), X2) -> mark(U41(X1, X2)), U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)), 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)), U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)), U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)), U61(mark(X)) -> mark(U61(X)), U61(ok(X)) -> ok(U61(X)), x(X1, mark(X2)) -> mark(x(X1, X2)), x(mark(X1), X2) -> mark(x(X1, X2)), x(ok(X1), ok(X2)) -> ok(x(X1, X2)), U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3)), U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3)), and(mark(X1), X2) -> mark(and(X1, X2)), and(ok(X1), ok(X2)) -> ok(and(X1, X2)), isNatKind(ok(X)) -> ok(isNatKind(X)), proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)), proper(isNat(X)) -> isNat(proper(X)), proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)), proper(tt()) -> ok(tt()), proper(U13(X)) -> U13(proper(X)), proper(U22(X)) -> U22(proper(X)), proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)), proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)), proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)), proper(U33(X)) -> U33(proper(X)), proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)), proper(s(X)) -> s(proper(X)), proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)), proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)), proper(0()) -> ok(0()), proper(U61(X)) -> U61(proper(X)), proper(x(X1, X2)) -> x(proper(X1), proper(X2)), proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3)), proper(and(X1, X2)) -> and(proper(X1), proper(X2)), proper(isNatKind(X)) -> isNatKind(proper(X)), top(mark(X)) -> top(proper(X)), top(ok(X)) -> top(active(X)) } Fail