(VAR M N V1 V2 X X1 X2 X3 )
(RULES 
        active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2))
        active(U12(tt, V2)) -> mark(U13(isNat(V2)))
        active(U13(tt)) -> mark(tt)
        active(U21(tt, V1)) -> mark(U22(isNat(V1)))
        active(U22(tt)) -> mark(tt)
        active(U31(tt, N)) -> mark(N)
        active(U41(tt, M, N)) -> mark(s(plus(N, M)))
        active(and(tt, X)) -> mark(X)
        active(isNat(0)) -> mark(tt)
        active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
        active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
        active(isNatKind(0)) -> mark(tt)
        active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
        active(isNatKind(s(V1))) -> mark(isNatKind(V1))
        active(plus(N, 0)) -> mark(U31(and(isNat(N), isNatKind(N)), N))
        active(plus(N, s(M))) -> mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
        active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
        active(U12(X1, X2)) -> U12(active(X1), X2)
        active(U13(X)) -> U13(active(X))
        active(U21(X1, X2)) -> U21(active(X1), X2)
        active(U22(X)) -> U22(active(X))
        active(U31(X1, X2)) -> U31(active(X1), X2)
        active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
        active(s(X)) -> s(active(X))
        active(plus(X1, X2)) -> plus(active(X1), X2)
        active(plus(X1, X2)) -> plus(X1, active(X2))
        active(and(X1, X2)) -> and(active(X1), X2)
        U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
        U12(mark(X1), X2) -> mark(U12(X1, X2))
        U13(mark(X)) -> mark(U13(X))
        U21(mark(X1), X2) -> mark(U21(X1, X2))
        U22(mark(X)) -> mark(U22(X))
        U31(mark(X1), X2) -> mark(U31(X1, X2))
        U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
        s(mark(X)) -> mark(s(X))
        plus(mark(X1), X2) -> mark(plus(X1, X2))
        plus(X1, mark(X2)) -> mark(plus(X1, X2))
        and(mark(X1), X2) -> mark(and(X1, X2))
        proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
        proper(tt) -> ok(tt)
        proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
        proper(isNat(X)) -> isNat(proper(X))
        proper(U13(X)) -> U13(proper(X))
        proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
        proper(U22(X)) -> U22(proper(X))
        proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
        proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
        proper(s(X)) -> s(proper(X))
        proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
        proper(and(X1, X2)) -> and(proper(X1), proper(X2))
        proper(0) -> ok(0)
        proper(isNatKind(X)) -> isNatKind(proper(X))
        U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
        U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
        isNat(ok(X)) -> ok(isNat(X))
        U13(ok(X)) -> ok(U13(X))
        U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
        U22(ok(X)) -> ok(U22(X))
        U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
        U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
        s(ok(X)) -> ok(s(X))
        plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
        and(ok(X1), ok(X2)) -> ok(and(X1, X2))
        isNatKind(ok(X)) -> ok(isNatKind(X))
        top(mark(X)) -> top(proper(X))
        top(ok(X)) -> top(active(X))
        
)