(VAR V1 V2 N M 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,V1,V2)) -> mark(U32(isNat(V1),V2))
active(U32(tt,V2)) -> mark(U33(isNat(V2)))
active(U33(tt)) -> mark(tt)
active(U41(tt,N)) -> mark(N)
active(U51(tt,M,N)) -> mark(s(plus(N,M)))
active(U61(tt)) -> mark(0)
active(U71(tt,M,N)) -> mark(plus(x(N,M),N))
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(isNat(x(V1,V2))) -> mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2))
active(isNatKind(0)) -> mark(tt)
active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
active(isNatKind(s(V1))) -> mark(isNatKind(V1))
active(isNatKind(x(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
active(plus(N,0)) -> mark(U41(and(isNat(N),isNatKind(N)),N))
active(plus(N,s(M))) -> mark(U51(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(N,s(M))) -> mark(U71(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,X3)) -> U31(active(X1),X2,X3)
active(U32(X1,X2)) -> U32(active(X1),X2)
active(U33(X)) -> U33(active(X))
active(U41(X1,X2)) -> U41(active(X1),X2)
active(U51(X1,X2,X3)) -> U51(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(U61(X)) -> U61(active(X))
active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
active(x(X1,X2)) -> x(active(X1),X2)
active(x(X1,X2)) -> x(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,X3) -> mark(U31(X1,X2,X3))
U32(mark(X1),X2) -> mark(U32(X1,X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1),X2) -> mark(U41(X1,X2))
U51(mark(X1),X2,X3) -> mark(U51(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))
U61(mark(X)) -> mark(U61(X))
U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
x(mark(X1),X2) -> mark(x(X1,X2))
x(X1,mark(X2)) -> mark(x(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,X3)) -> U31(proper(X1),proper(X2),proper(X3))
proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
proper(s(X)) -> s(proper(X))
proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
proper(U61(X)) -> U61(proper(X))
proper(0) -> ok(0)
proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
proper(x(X1,X2)) -> x(proper(X1),proper(X2))
proper(and(X1,X2)) -> and(proper(X1),proper(X2))
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(X3)) -> ok(U31(X1,X2,X3))
U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
s(ok(X)) -> ok(s(X))
plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
U61(ok(X)) -> ok(U61(X))
U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
x(ok(X1),ok(X2)) -> ok(x(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))
)