(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following 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(X1, X2)
activate(n__s(X)) → s(X)
activate(n__x(X1, X2)) → x(X1, X2)
activate(X) → X
Q is empty.