(VAR M N V1 V2 X X1 X2 )
(RULES 
        U101(tt, M, N) -> U102(isNatKind(activate(M)), activate(M), activate(N))
        U102(tt, M, N) -> U103(isNat(activate(N)), activate(M), activate(N))
        U103(tt, M, N) -> U104(isNatKind(activate(N)), activate(M), activate(N))
        U104(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N))
        U11(tt, V1, V2) -> U12(isNatKind(activate(V1)), activate(V1), activate(V2))
        U12(tt, V1, V2) -> U13(isNatKind(activate(V2)), activate(V1), activate(V2))
        U13(tt, V1, V2) -> U14(isNatKind(activate(V2)), activate(V1), activate(V2))
        U14(tt, V1, V2) -> U15(isNat(activate(V1)), activate(V2))
        U15(tt, V2) -> U16(isNat(activate(V2)))
        U16(tt) -> tt
        U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
        U22(tt, V1) -> U23(isNat(activate(V1)))
        U23(tt) -> tt
        U31(tt, V1, V2) -> U32(isNatKind(activate(V1)), activate(V1), activate(V2))
        U32(tt, V1, V2) -> U33(isNatKind(activate(V2)), activate(V1), activate(V2))
        U33(tt, V1, V2) -> U34(isNatKind(activate(V2)), activate(V1), activate(V2))
        U34(tt, V1, V2) -> U35(isNat(activate(V1)), activate(V2))
        U35(tt, V2) -> U36(isNat(activate(V2)))
        U36(tt) -> tt
        U41(tt, V2) -> U42(isNatKind(activate(V2)))
        U42(tt) -> tt
        U51(tt) -> tt
        U61(tt, V2) -> U62(isNatKind(activate(V2)))
        U62(tt) -> tt
        U71(tt, N) -> U72(isNatKind(activate(N)), activate(N))
        U72(tt, N) -> activate(N)
        U81(tt, M, N) -> U82(isNatKind(activate(M)), activate(M), activate(N))
        U82(tt, M, N) -> U83(isNat(activate(N)), activate(M), activate(N))
        U83(tt, M, N) -> U84(isNatKind(activate(N)), activate(M), activate(N))
        U84(tt, M, N) -> s(plus(activate(N), activate(M)))
        U91(tt, N) -> U92(isNatKind(activate(N)))
        U92(tt) -> 0
        isNat(n__0) -> tt
        isNat(n__plus(V1, V2)) -> U11(isNatKind(activate(V1)), activate(V1), activate(V2))
        isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
        isNat(n__x(V1, V2)) -> U31(isNatKind(activate(V1)), activate(V1), activate(V2))
        isNatKind(n__0) -> tt
        isNatKind(n__plus(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V2))
        isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1)))
        isNatKind(n__x(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
        plus(N, 0) -> U71(isNat(N), N)
        plus(N, s(M)) -> U81(isNat(M), M, N)
        x(N, 0) -> U91(isNat(N), N)
        x(N, s(M)) -> U101(isNat(M), M, N)
        0 -> n__0
        plus(X1, X2) -> n__plus(X1, X2)
        s(X) -> n__s(X)
        x(X1, X2) -> n__x(X1, X2)
        activate(n__0) -> 0
        activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
        activate(n__s(X)) -> s(activate(X))
        activate(n__x(X1, X2)) -> x(activate(X1), activate(X2))
        activate(X) -> X
        
)