(VAR M N V1 V2 X X1 X2 X3 )
(RULES 
        a__U11(tt, V1, V2) -> a__U12(a__isNat(V1), V2)
        a__U12(tt, V2) -> a__U13(a__isNat(V2))
        a__U13(tt) -> tt
        a__U21(tt, V1) -> a__U22(a__isNat(V1))
        a__U22(tt) -> tt
        a__U31(tt, V1, V2) -> a__U32(a__isNat(V1), V2)
        a__U32(tt, V2) -> a__U33(a__isNat(V2))
        a__U33(tt) -> tt
        a__U41(tt, N) -> mark(N)
        a__U51(tt, M, N) -> s(a__plus(mark(N), mark(M)))
        a__U61(tt) -> 0
        a__U71(tt, M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N))
        a__and(tt, X) -> mark(X)
        a__isNat(0) -> tt
        a__isNat(plus(V1, V2)) -> a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
        a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1)
        a__isNat(x(V1, V2)) -> a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
        a__isNatKind(0) -> tt
        a__isNatKind(plus(V1, V2)) -> a__and(a__isNatKind(V1), isNatKind(V2))
        a__isNatKind(s(V1)) -> a__isNatKind(V1)
        a__isNatKind(x(V1, V2)) -> a__and(a__isNatKind(V1), isNatKind(V2))
        a__plus(N, 0) -> a__U41(a__and(a__isNat(N), isNatKind(N)), N)
        a__plus(N, s(M)) -> a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
        a__x(N, 0) -> a__U61(a__and(a__isNat(N), isNatKind(N)))
        a__x(N, s(M)) -> a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
        mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
        mark(U12(X1, X2)) -> a__U12(mark(X1), X2)
        mark(isNat(X)) -> a__isNat(X)
        mark(U13(X)) -> a__U13(mark(X))
        mark(U21(X1, X2)) -> a__U21(mark(X1), X2)
        mark(U22(X)) -> a__U22(mark(X))
        mark(U31(X1, X2, X3)) -> a__U31(mark(X1), X2, X3)
        mark(U32(X1, X2)) -> a__U32(mark(X1), X2)
        mark(U33(X)) -> a__U33(mark(X))
        mark(U41(X1, X2)) -> a__U41(mark(X1), X2)
        mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3)
        mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2))
        mark(U61(X)) -> a__U61(mark(X))
        mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3)
        mark(x(X1, X2)) -> a__x(mark(X1), mark(X2))
        mark(and(X1, X2)) -> a__and(mark(X1), X2)
        mark(isNatKind(X)) -> a__isNatKind(X)
        mark(tt) -> tt
        mark(s(X)) -> s(mark(X))
        mark(0) -> 0
        a__U11(X1, X2, X3) -> U11(X1, X2, X3)
        a__U12(X1, X2) -> U12(X1, X2)
        a__isNat(X) -> isNat(X)
        a__U13(X) -> U13(X)
        a__U21(X1, X2) -> U21(X1, X2)
        a__U22(X) -> U22(X)
        a__U31(X1, X2, X3) -> U31(X1, X2, X3)
        a__U32(X1, X2) -> U32(X1, X2)
        a__U33(X) -> U33(X)
        a__U41(X1, X2) -> U41(X1, X2)
        a__U51(X1, X2, X3) -> U51(X1, X2, X3)
        a__plus(X1, X2) -> plus(X1, X2)
        a__U61(X) -> U61(X)
        a__U71(X1, X2, X3) -> U71(X1, X2, X3)
        a__x(X1, X2) -> x(X1, X2)
        a__and(X1, X2) -> and(X1, X2)
        a__isNatKind(X) -> isNatKind(X)
        
)